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Pathways in Mathematics
Peter Junghanns
Giuseppe Mastroianni
Incoronata Notarangelo
Weighted
Polynomial
Approximation
and Numerical
Methods for
Integral Equations
Series Editors
Takayuki Hibi, Department of Pure and Applied Mathematics, Osaka University,
Suita, Osaka, Japan
Wolfgang König, Weierstraß-Institut, Berlin, Germany
Johannes Zimmer, Fakultät für Mathematik, Technische Universität München,
Garching, Germany
Each “Pathways in Mathematics” book offers a roadmap to a currently well devel-
oping mathematical research field and is a first-hand information and inspiration
for further study, aimed both at students and researchers. It is written in an
educational style, i.e., in a way that is accessible for advanced undergraduate and
graduate students. It also serves as an introduction to and survey of the field for
researchers who want to be quickly informed about the state of the art. The point of
departure is typically a bachelor/masters level background, from which the reader
is expeditiously guided to the frontiers. This is achieved by focusing on ideas and
concepts underlying the development of the subject while keeping technicalities to
a minimum. Each volume contains an extensive annotated bibliography as well as a
discussion of open problems and future research directions as recommendations for
starting new projects. Titles from this series are indexed by Scopus.
More information about this series at http://www.springer.com/series/15133

Peter Junghanns • Giuseppe Mastroianni •
Weighted Polynomial
Approximation
and Numerical Methods
for Integral Equations
Peter Junghanns Giuseppe Mastroianni
Fakultät für Mathematik Department of Mathematics, Computer
Technische Universität Chemnitz Sciences and Economics
Chemnitz, Germany University of Basilicata
Potenza, Italy
Department of Mathematics
”Giuseppe Peano”
University of Turin
Turin, Italy
ISSN 2367-3451 ISSN 2367-346X (electronic)

ISBN 978-3-030-77496-7 ISBN 978-3-030-77497-4 (eBook)
https://doi.org/10.1007/978-3-030-77497-4
© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland
AG 2021
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This book is published under the imprint Birkhäuser, www.birkhauser-science.com, by the registered
company Springer Nature Switzerland AG.
The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland
Preface
The content of this book is a combination of two topics, one comes from the
theory of approximation of functions and integrals by interpolation and quadrature,
respectively, and the other from the numerical analysis of operator equations,
in particular, of integral and related equations. It is not necessary to point out
that integral equations play an important role in different mathematical areas. We
stress the connection between ordinary differential equations and Volterra-Fredholm
integral equations and the connection between boundary value problems for partial
differential equations and boundary integral equations, which are obtained by so-
called boundary integral methods. These methods lead to different classes of integral
equations like, for example, Fredholm integral equations of first and second kind
involving integral operators with both smooth and weakly singular kernel functions,
or Cauchy singular integral equations, as well as hypersingular integral equations,
or integro-differential equations.
Regarding interpolation and quadrature processes, we restrict ourselves to the
non periodic case, that means to the approximation and integration of functions
defined on bounded or unbounded intervals, where we attach particular value to
functions having singularities at the end points of the interval. This is due to our
further aim in this book, namely, to propose and to investigate numerical methods
for different classes of integral equations given on such intervals, where these
methods are based on the mentioned interpolation and quadrature processes. The
book contains classical results, but also very recent results. We thought it might be
worthwhile to publish a book in which these two topics are summarized.
In Chap. 1, the introduction, we give some hints for the use of the book and
introduce some general notations and agreements for the whole text. Chapter 2
collects the basic principles from linear functional analysis needed in the remaining
part of the book, gives definitions for different kinds of function spaces such as
weighted Lp spaces and weighted spaces of continuous functions as well as scales
of subspaces of them, which are important for our investigations. This chapter also
presents some concepts concerned with the stability and convergence of operator
sequences or, in other words, with numerical or approximation methods for operator
v
vi Preface
equations. Moreover, we recall some basic facts from fixed point theory and about
Newton’s method.
Chapters 3 and 4 are devoted to the study of interpolation processes and the
respective quadrature rules based on the zeros of orthogonal polynomials with
respect to certain weight functions on the interval (−1, 1), the half axis (0, ∞),
and the whole real axis. These chapters can be considered as a continuation of [121,
Chapter 2 and Chapter 4, Section 5.1], where the authors mainly consider classical
weights (also with additional inner singularities). In the present text we concentrate
on recent results and developments concerned with non classical weights like
exponential weights on (−1, 1) and on (0, ∞) and generalized Freud weights on
the real axis.
In Chap. 5, we provide mapping properties of various classes of integral operators
in certain Banach spaces of functions and with respect to appropriate scales of sub-
spaces of these Banach spaces, which are of interest for our further investigations.
Moreover, we discuss solvability properties of certain classes of nonlinear Cauchy
singular integral equations.
Chapters 6 and 7 deal with numerical methods for several classes of integral
equations based on some interpolation and quadrature processes considered in
Chaps. 3 and 4. While Chap. 6 concentrates on respective Nyström and collocation-
quadrature methods for Fredholm integral equations with continuous and weakly
singular kernel functions, in Chap. 7 collocation and collocation-quadrature meth-
ods are applied to strongly singular integral equations like linear and nonlinear
Cauchy singular integral equations, integral equations with strongly fixed singu-
larities, and hypersingular integral equations.
In Chap. 8, we investigate some concrete applications of the theory presented
in the previous chapter to examples from two-dimensional elasticity theory, airfoil
theory, and free boundary seepage flow problems. In the two final chapters, Chaps. 9
and 10, we give complete answers or detailed hints to the exercises and list a
series of inequalities, equivalences, and equalities used at many places in the book,
respectively.
The book is mainly addressed to graduate students familiar with the basics of real
and complex analysis, linear algebra, and functional analysis. But, the study of this
book is also worthwhile for researchers beginning to deal with the approximation
of functions and the numerical solution of operator equations, in particular integral
equations. Moreover, we hope that the present book is also suitable to give ideas for
handling further or new problems of interest, which can be solved with the help of
integral equations. The book should also reach engineers who are interested in the
solution of certain problems of similar kind as presented here.
We are deeply grateful to Birkhäuser for including this book in the very
successful series Pathways in Mathematics and, in particular, for the invaluable
assistance in preparing the final version of the text.
Chemnitz, Germany Peter Junghanns

Potenza, Italy Giuseppe Mastroianni
Turin, Italy Incoronata Notarangelo
Contents
1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1
2 Basics from Linear and Nonlinear Functional Analysis. . . . . . . . . . . . . . . . 5
2.1 Linear Operators, Banach and Hilbert Spaces . .. . . . . . . . . . . . . . . . . . . . 5
2.2 Fundamental Principles . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9
2.3 Compact Sets and Compact Operators . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 13
2.4 Function Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 16
2.4.1 Lp -Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 17
2.4.2 Spaces of Continuous Functions . . . . . . . .. . . . . . . . . . . . . . . . . . . . 23
2.4.3 Approximation Spaces and Unbounded
Linear Operators . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 34
2.5 Fredholm Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 42
2.6 Stability of Operator Sequences . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 43
2.7 Fixed Point Theorems and Newton’s Method.. .. . . . . . . . . . . . . . . . . . . . 50
3 Weighted Polynomial Approximation and Quadrature Rules
on (−1, 1) .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 57
3.1 Moduli of Smoothness, K-Functionals, and Best
Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 57
3.1.1 Moduli of Smoothness and K-Functionals.. . . . . . . . . . . . . . . . . 58
3.1.2 Moduli of Smoothness and Best Weighted
Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 65
3.1.3 Besov-Type Spaces . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 68
3.2 Polynomial Approximation with Doubling Weights on
the Interval (−1, 1) .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 78
3.2.1 Definitions .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 78
3.2.2 Polynomial Inequalities with Doubling Weights . . . . . . . . . . . 84
3.2.3 Christoffel Functions with Respect to Doubling
Weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 86
3.2.4 Convergence of Fourier Sums in Weighted Lp -Spaces . . . . 91
3.2.5 Lagrange Interpolation in Weighted Lp -Spaces .. . . . . . . . . . . 98
3.2.6 Hermite Interpolation . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 108
vii
viii Contents
3.2.7 Hermite-Fejér Interpolation . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 109

3.2.8 Lagrange-Hermite Interpolation.. . . . . . . .. . . . . . . . . . . . . . . . . . . . 110
3.3 Polynomial Approximation with Exponential
Weights on the Interval (−1, 1) . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 115
3.3.1 Polynomial Inequalities .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 116
3.3.2 K-Functionals and Moduli of Smoothness.. . . . . . . . . . . . . . . . . 122
3.3.3 Estimates for the Error of Best Weighted
Polynomial Approximation .. . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 124
3.3.4 Fourier Sums in Weighted Lp -Spaces . .. . . . . . . . . . . . . . . . . . . . 127
3.3.6 Gaussian Quadrature Rules . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 140
4 Weighted Polynomial Approximation and Quadrature Rules
on Unbounded Intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 145
4.1 Polynomial Approximation with Generalized Freud
Weights on the Real Line . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 145
4.1.1 The Case of Freud Weights . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 147
4.1.2 The Case of Generalized Freud Weights . . . . . . . . . . . . . . . . . . . . 150
4.2 Polynomial Approximation with Generalized Laguerre
Weights on the Half Line . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 195
4.2.2 Weighted Spaces of Functions . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 198
Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 199
4.3 Polynomial Approximation with Pollaczek–Laguerre
Weights on the Half Line . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 211
4.3.2 Weighted Spaces of Functions . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 214
Polynomial Approximation .. . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 217
4.3.5 Lagrange Interpolation in L2√w . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 224
4.3.6 Remarks on Numerical Realizations . . . .. . . . . . . . . . . . . . . . . . . . 227
5 Mapping Properties of Some Classes of Integral Operators . . . . . . . . . . 239
5.1 Some Properties of the Jacobi Polynomials .. . . .. . . . . . . . . . . . . . . . . . . . 239
5.2 Cauchy Singular Integral Operators .. . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 245
5.2.1 Weighted L2 -Spaces . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 245
5.2.2 Weighted Spaces of Continuous Functions . . . . . . . . . . . . . . . . . 252
Contents ix
5.2.3 On the Case of Variable Coefficients. . . .. . . . . . . . . . . . . . . . . . . . 262

5.2.4 Regularity Properties . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 268
5.3 Compact Integral Operators . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 276
5.4 Weakly Singular Integral Operators with Logarithmic Kernels .. . . 288
5.5 Singular Integro-Differential or Hypersingular Operators . . . . . . . . . 309
5.6 Operators with Fixed Singularities of Mellin Type .. . . . . . . . . . . . . . . . 313
5.7 A Note on the Invertibility of Singular Integral Operators
with Cauchy and Mellin Kernels . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 326
5.8 Solvability of Nonlinear Cauchy Singular Integral Equations . . . . . 329
5.8.1 Equations of the First Type .. . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 330
5.8.2 Equations of the Second Type . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 340
5.8.3 Equations of the Third Type . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 351
6 Numerical Methods for Fredholm Integral Equations . . . . . . . . . . . . . . . . . 355
6.1 Collectively Compact Sequences of Integral Operators . . . . . . . . . . . . 356
6.2 The Classical Nyström Method .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 360
6.2.1 The Case of Jacobi Weights . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 366
6.2.2 The Case of an Exponential Weight on (0, ∞) .. . . . . . . . . . . . 368
6.2.3 The Application of Truncated Quadrature Rules . . . . . . . . . . . 370
6.3 The Nyström Method Based on Product Integration Formulas . . . . 372
6.3.1 The Case of Jacobi Weights . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 374
6.3.2 The Case of an Exponential Weight on (0, ∞) .. . . . . . . . . . . . 380
6.3.3 Application to Weakly Singular Integral Equations .. . . . . . . 385
6.4 Integral Equations with Logarithmic Kernels .. .. . . . . . . . . . . . . . . . . . . . 392
6.4.1 The Well-posed Case . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 392
6.4.2 The Ill-posed Case . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 393
6.4.3 A Collocation-Quadrature Method .. . . . .. . . . . . . . . . . . . . . . . . . . 403
6.4.4 A Fast Algorithm.. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 409
7 Collocation and Collocation-Quadrature Methods
for Strongly Singular Integral Equations . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 419
7.1 Cauchy Singular Integral Equations on an Interval .. . . . . . . . . . . . . . . . 419
7.1.1 Collocation and Collocation-Quadrature Methods . . . . . . . . . 420
7.1.2 Weighted Uniform Convergence . . . . . . . .. . . . . . . . . . . . . . . . . . . . 424
7.1.3 Fast Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 443
7.2 Hypersingular Integral Equations . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 464
7.2.1 Collocation and Collocation-Quadrature Methods . . . . . . . . . 467
7.2.2 A Fast Algorithm.. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 474
7.3 Integral Equations with Mellin Type Kernels . . .. . . . . . . . . . . . . . . . . . . . 484
7.4 Nonlinear Cauchy Singular Integral Equations .. . . . . . . . . . . . . . . . . . . . 493
7.4.1 Asymptotic of the Solution .. . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 495
7.4.3 Convergence Analysis . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 500
7.4.4 A Further Class of Nonlinear Cauchy Singular
Integral Equations .. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 517
x Contents
8 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 537
8.1 A Cruciform Crack Problem .. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 537
8.1.1 The Integral Equations Under Consideration .. . . . . . . . . . . . . . 537
8.1.2 Solvability Properties of the Operator Equations .. . . . . . . . . . 542
8.1.3 A Quadrature Method.. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 547
8.2 The Drag Minimization Problem for a Wing . . .. . . . . . . . . . . . . . . . . . . . 549
8.2.1 Formulation of the Problem . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 549
8.2.2 Derivation of the Operator Equation . . . .. . . . . . . . . . . . . . . . . . . . 553
8.2.4 Numerical Examples .. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 565
8.3 Two-Dimensional Free Boundary Value Problems . . . . . . . . . . . . . . . . . 570
8.3.1 Seepage Flow from a Dam . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 570
8.3.2 Seepage Flow from a Channel.. . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 582
9 Hints and Answers to the Exercises . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 601
10 Equalities and Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 637
10.1 Equalities and Equivalences . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 637
10.2 General Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 638
10.3 Marcinkiewicz Inequalities . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 640
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 643
Index . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 653
Chapter 1
Introduction
In this book we are mainly interested in the approximation of (in general, complex
valued) functions, defined on bounded or unbounded intervals of the real line, with
possible singularities (like unboundedness or nonsmoothness) at the endpoints of
these intervals (cf. Chaps. 3 and 4). In particular, we focus on situations, where such
functions are (unknown) solutions of different kinds of integral equations, namely
Fredholm integral equations (Chap. 6), linear and nonlinear Cauchy singular integral
equations, hypersingular integral equations, and integral equations with Mellin
kernels (Chap. 7). We propose and investigate numerical methods for solving these
operator equations. Thereby, we concentrate on so-called global ansatz functions
for the unknown solutions by searching an approximate solution as a finite linear
combination of (in general, weighted) polynomials (cf. Chaps. 6 and 7).
Consequently, our main goals are, firstly, studying (weighted) polynomial
approximation of functions from weighted spaces of continuous functions and
weighted Lp -spaces (cf. Chaps. 3 and 4) and, secondly, the stability and convergence
of certain approximation methods for integral equations basing on interpolation and
quadrature processes as investigated in Chaps. 3 and 4. Thus, if readers are mainly
interested in the approximation of functions and integrals, then, after reading this
introduction and Chap. 2, they can concentrate on studying Chaps. 3 and 4 on
weighted polynomial approximation. Readers, who are more interested in numerics
for integral equations, can skip Chaps. 3 and 4 and directly proceed with Chap. 5
on mapping properties of some classes of integral operators, followed by Chap. 6
or/and Chap. 7 (consulting certain results from Chaps. 3 and 4 if necessary). They
should end up reading Chap. 8 on some Applications devoted to fracture mechanics,
wing theory, and free boundary value problems (cf. the picture at the end of this
introduction). Moreover, many sections of the book are written in such a way that,
for a reader who would not go too much into the details, it is also possible to study
a single section with a certain success and without taking into account too much
references to other places in the book. That means, that definitions and notations
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 1

P. Junghanns et al., Weighted Polynomial Approximation and Numerical
Methods for Integral Equations, Pathways in Mathematics,
https://doi.org/10.1007/978-3-030-77497-4_1
2 1 Introduction
are repeated at appropriate places. Anyway we recommend the readers to apply

themselves to the exercises.
Finally, let us make some agreements concerning notations and formulations,
which are in force throughout the whole book. By Z, N0 , N, R, and C we denote
the sets of integer numbers, nonnegative integers, positive integers, real and complex
numbers, respectively. By capital and boldface letters X, Y, . . . from the end of the
Latin alphabet, we will denote spaces (of functions), in particular metric spaces,
linear spaces, vector spaces, normed spaces, and Banach spaces. We have to take
into account that a metric space is an ordered pair (X, d) of a set X and a metric
d : X × X −→ [0, ∞) or that a normed space is an ordered pair (X, .) of a
linear space X over a field K (here usually K = R or K = C) and a norm function
. : X −→ [0, ∞) satisfying the respective axioms.
A function f or map from a set A into a set B is often written as f : A −→ B,
a → f (a) or, shortly, as f : A −→ B. We use calligraphic letters A, B, . . . to
denote operators A : X −→ Y, B : Z −→ W, . . .
In all what follows, we denote by c, c1 , c2 , . . . positive real constants which
can have different values at different places. Moreover, by c = c(n, f, x, . . .) we
indicate that c is independent of the variables n, f, x, . . . If A(n, f, x, . . .) and
B(n, f, x, . . .) are two positive functions depending on certain variables n, f, x, . . .,
then we will write A ∼n,f,x,... B, if there exists a positive constant c =
c(n, f, x, . . .) such that c−1 B(n, f, x, . . .) ≤ A(n, f, x, . . .) ≤ c B(n, f, x, . . .) is
satisfied.
∞ ∞
Sometimes, for sequences (ξn )n=0 and (ηn )n=0 of positive numbers, we will use
the notion ξn = O(ηn ) in order to say that there is a positive constant C such that
ξn ≤ C ηn for all n ∈ N0 . If in such a situation in a finite number of elements
of these sequences some numbers are not well defined, then these numbers have
∞
to be set equal to 1. Hence, for example, we write shortly (ln n) = (ln n)n=0 for
∞
(1, 1, ln 2, ln 3 . . .) or (ξn ln n) = (ξn ln n)n=0 for (ξ0 , ξ1 , ξ2 ln 2, ξ3 ln 3, . . .).
Moreover, sometimes function values at special points (for example, endpoints
of an interval) are defined as limits (such that the respective function is continuous
in that points). Thus, the formulation “Let ρ(x) = (1 − x)α (1 + x)β be a Jacobi
weight and let f : (−1, 1) −→ C be a function such that ρf : [−1, 1] −→ C
is continuous.” means that f : (−1, 1) −→ C is continuous and that the finite
limits (ρf )(±1) := limx→±1,x∈(−1,1) ρ(x)f (x) exist. By Greek letters σ , μ, . . . we
will denote weight functions σ (x), μ(x), . . ., while the letters α, β, . . . from the
beginning of the Greek alphabet are used for special parameters in these weights.
Additionally, in Chap. 9 we give complete answers or detailed hints to the
exercises and in Chap. 10, as a quick reference, we collect a series of inequalities
important for a lot of our considerations.
1 Introduction 3
Chapter 1 : Introduction
Chapter 2 : Basics from linear functional analysis
Chapters 3 and 4 : Chapter 5 :
Weighted polynomial approximation Mapping properties of integral operators
Chapter 6 : Chapter 7 :
Fredholm integral equations Strongly singular integral equations
Chapter 8 : Applications
Two proposals for reading parts of the book

Chapter 2
Basics from Linear and Nonlinear
Functional Analysis
In this chapter we collect the basic principles from linear functional analysis needed
in the remaining part of the book. We give definitions for different kinds of function
spaces such as weighted Lp -spaces and weighted spaces of continuous functions as
well as scales of subspaces of them, which are important for our investigations. We
also present some concepts concerned with the stability and convergence of operator
sequences or, in other words, with numerical or approximation methods for operator
equations. Moreover, we recall some basic facts from fixed point theory, namely
Banach’s and Schauder’s fixed point theorems, and discuss a few aspects on the
convergence of Newton’s iteration method.
2.1 Linear Operators, Banach and Hilbert Spaces
By boldfaced capital letters X, Y, . . . we will denote (linear) spaces (in most

cases, of functions), while by calligraphic letters A, B, . . . we will refer to (linear)
operators. The notion A : X −→ Y means that the operator A is defined on X
and the images Af (for all f ∈ X) belong to Y. For this, we also write shortly
A : X −→ Y, f → Af . The set of linear operators A : X −→ Y from a linear
space X into a linear space Y will be denoted by L(X, Y).
Let us recall some basic concepts and facts from linear functional analysis. A
normed linear space (X, .) is a linear space X over an field K (here K = R, the
field of real numbers, or K = C, the field of complex numbers) equipped with a so
called norm . : X −→ [0, ∞), x → x satisfying
(N1) f = 0 if and only if f = (the zero element in X),
(N2) αf = |α|f ∀ α ∈ K, ∀ x ∈ X,
(N3) f + g ≤ f + g ∀ f, g ∈ X (triangle inequality).
We call (X, .) a real or complex normed space if K = R or K = C, respectively.
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 5

P. Junghanns et al., Weighted Polynomial Approximation and Numerical
Methods for Integral Equations, Pathways in Mathematics,
https://doi.org/10.1007/978-3-030-77497-4_2
6 2 Basics from Linear and Nonlinear Functional Analysis
A normed linear space (X, .) is automatically considered as a metric space

(X, ρ), where the metric ρ : X × X −→ [0, ∞) is defined by ρ(f, g) = f − g.
With the help of this metric the notions of bounded, open, closed, and compact
∞
sets as well as of convergent or Cauchy sequences (fn )n=1 are given. A normed
linear space (X, .) is called Banach space if the respective metric space (X, ρ)
∞
is complete, i.e., every Cauchy sequence (fn )n=1 ⊂ X is convergent in X. We say
that two norms .1 and .2 on a linear space X are equivalent if there is a real
constant γ ≥ 1 such that
γ −1 f 2 ≤ f 1 ≤ γ f 2 ∀f ∈ X.
In this case, a sequence is convergent in (X, .1 ) if and only if it is convergent in

(X, .2 ). In other words, these two spaces have the same topological properties. If
X and Y are two normed spaces over the same field K, then the product space X × Y
can be equipped with several equivalent norms, for example with
p p1
(f, g)p = f X + gY p
for some p ∈ [1, ∞). We agree that, if in the sequel there occur products of normed
spaces, we will consider them equipped with one of these norms. Moreover, K is
also considered as the Banach space (K, |.|) with the modulus as norm. Clearly,
if both spaces X and Y are Banach spaces, then so is their product. With these
definitions, one can easily see that the maps X × X −→ X, (f, g) → f + g and
K × X −→ X, (α, f ) → αf are continuous maps.
Let X and Y be normed linear spaces over the same field K. A linear operator
A ∈ L(X, Y) is called bounded if it maps bounded sets into bounded sets, which is
equivalent to

AX→Y := sup Af Y : f ∈ X, f X ≤ 1 < ∞ .
The boundedness of A ∈ L(X, Y) is equivalent to the continuity of the operator

A : X −→ Y (as a map between the respective metric spaces), and the number
M = AX→Y is the smallest one for which
Af Y ≤ M f X ∀f ∈ X.
The set of all bounded linear operators from X into Y is denoted by L(X, Y), in case
Y = X by L(X). The linear combination of two linear operators A, B ∈ L(X, Y)
is defined by (αA + βB)f := (αAf ) + (βBf ) ∀f ∈ X, where α, β ∈ K. Then
L(X, Y) is a linear space, and L(X, Y), .X→Y = L(X, Y), .L(X,Y) is a
normed linear space, which is a Banach space if and only if Y is a Banach space
(see the following three exercises).
Let (X, .X ) and (Y, .Y ) be two normed spaces. We say that Y is continu-
ously embedded into X, if Y ⊂ X and the operator E : Y −→ X, y → y is
2.1 Linear Operators, Banach and Hilbert Spaces 7
bounded, i.e., if yX ≤ cyY holds true for all y ∈ Y with a finite constant c ∈ R
not depending on y.
In all what follows, we abstain from writing the index in a norm (for example, in
.X or .L(X,Y)), if there is no possibility of misunderstandings.
Exercise 2.1.1 Let X0 be a dense linear subspace of X and let Y be a Banach space.
Assume A0 ∈ L(X0 , Y). Show that there exists a unique extension A ∈ L(X, Y) of
A0 , i.e., Af = A0 f for all f ∈ X0 .
∞
Exercise 2.1.2 For a sequence (xn )n=1 of elements xn ∈ X of a normed space
∞
n ∞

(X, .), we say that the series xn is convergent if the sequence xk
n=1 k=1 n=1
∞

converges. It is called absolutely convergent, if the number series xn
n=1
converges. Show that X is a Banach space if and only if every absolutely convergent
series of elements of X is convergent in X.
Exercise 2.1.3 Prove that L(X, Y) is a Banach space if and only if Y is a Banach
space. (Hint: For the ⇒-direction, use Corollary 2.2.10 from Sect. 2.2 below.)
An operator P ∈ L(X, X) is called projection if P 2 = P. Since in this case, for
g = Pf , we have PgX = gX , we see that PX→X ≥ 1 for every continuous
projection P ∈ L(X, X).
Let H be a linear space over the field K = R or K = C, equipped with a so-called
inner product ., . : H × H −→ K having the following properties:
(I1) f, f ≥ 0 ∀ f ∈ H, and f, f = 0 if and only if f = ,
(I2) f, g = g, f ∀ f, g ∈ H (α denotes the complex conjugate number of
α ∈ C),
(I3) αf + βg, h = α f, h + β g, h ∀ f, g, h ∈ H, ∀ α, β ∈ K.
A linear space (H, ., .)
√with inner product is considered as a normed linear space
(H, .) with f := f, f . In such a space the Cauchy-Schwarz inequality
holds true,
| f, g | ≤ f g ∀ f, g ∈ H .
An immediate consequence of this inequality is, that the map H × H −→ K,

(f, g) → f, g is continuous. A infinite dimensional linear space (H, ., .) with
inner product is called Hilbert space if the respective normed space (H, .) is a
Banach space.
∞
In what follows, let (H, ., .) be a Hilbert space. A sequence (en )n=0 ⊂ H is
called orthonormal system if
1 : m = n,
em , en = δmn := (2.1.1)
0 : m = n .
We remark that in this case the system {e0 , e1 , . . . , em } is linearly independent for
every m ∈ N0 . We denote the linear hull of this system by Hm and define the
operators Pm : H −→ H by

m
Pm f = f, ej ej .
j =0
Then the image R(Pm ) := Pm (H) is equal to Hm . Moreover, Pm : H −→ H is an

orthoprojection, which means that
Pm f, g = f, Pm g and Pm

2
f = Pm f ∀ f, g ∈ H .
This implies Pm ∈ L(H, H) and Pm H→H = 1. We list further important

∞
properties of an orthonormal system (en )n=0 : For every f ∈ H,
(P1) f − Pm f = inf {f − p : p ∈ Hm },
m
(P2) |f, ek |2 ≤ f 2 ∀ m ∈ N0 .
k=0
Property (P2) is known as Bessel’s inequality and can be written equivalently as
∞

(P3) |f, ek |2 ≤ f 2 .
k=0
∞
An orthonormal system (en )n=0 is named complete if it meets Parseval’s equality
∞

|f, ek |2 = f 2 ∀f ∈ H. (2.1.2)
k=0
If this is the case, then
lim f − Pm f = 0 ∀f ∈ H, (2.1.3)
m→∞
∞

which can also be written as f = f, ek ek (in the sense of convergence in H)
k=0
and which is a consequence of the relation

m
f − Pm f 2 = f − Pm f, f − Pm f = f 2 − Pm f 2 = f 2 − | f, ek |2 .
k=0
(2.1.4)
2.2 Fundamental Principles 9
∞
Exercise 2.1.4 Show that the orthonormal system (en )n=0 is complete if and only
∞
if the set H := Hm is dense in H, which means that every f ∈ H is the limit of
m=0
a sequence of elements of H.
2.2 Fundamental Principles
There exist three fundamental principles in linear functional analysis, the principle
of uniform boundedness, the closed graph theorem, and the theorem on the sufficient
number of bounded linear functionals, which are closely connected with the Banach-
Steinhaus theorem, Banach’s theorem, and the Hahn-Banach theorem, respectively.
In the sequel, we will describe these principles in a short manner.
Principle of Uniform Boundedness Let X be a Banach space and Y be a normed
space. If a family F ⊂ L(X, Y) of bounded linear operators is pointwise bounded,
i.e.,

sup Af Y : A ∈ F < ∞ ∀f ∈ X,
then F is uniformly bounded, i.e., sup {AX→Y : A ∈ F} < ∞.

The following Banach-Steinhaus theorem is an application of this principle to
strongly convergent operator sequences. A sequence of operators An : X −→ Y is
called strongly convergent to the operator A : X −→ Y if
lim An f − Af Y = 0 ∀f ∈ X.
n→∞
Theorem 2.2.1 Let X and Y be Banach spaces, X0 ⊂ X be a dense subset, and

∞ ∞
An ∈ L(X, Y), n ∈ N. Then (An )n=1 is strongly convergent if and only if (An f )n=1
∞
is a Cauchy sequence in Y for all f ∈ X0 and the number sequence (An X→Y )n=1
is bounded.
Exercise 2.2.2 Give a proof of Theorem 2.2.1 using the principle of uniform
boundedness.
The Closed Graph Theorem A linear operator A : D(A) −→ Y, where the
domain D(A) of A is a linear subspace of the normed space X, is called closed , if
xn ∈ D(A), xn −→ x in X, and Axn −→ y in Y imply x ∈ D(A) and Ax = y.
The following closed graph theorem deals with the case D(A) = X.
Theorem 2.2.3 If X, Y are Banach spaces and if the linear operator A : X −→ Y
is closed, then A : X −→ Y is continuous, i.e., A ∈ L(X, Y).
An immediate consequence of the closed graph theorem is Banach’s theorem.

Theorem 2.2.4 If X and Y are Banach spaces and if A ∈ L(X, Y) realizes a
bijective mapping A : X −→ Y, then A is continuously invertible, i.e., A−1 belongs
to L(Y, X).
The subset of L(X, Y) of all invertible operators we will denote by GL(X, Y), in
case X = Y by GL(X).
Exercise 2.2.5 Prove Theorem 2.2.4 with the help of Theorem 2.2.3.
Sufficient Number of Bounded Linear Functionals For the Hahn-Banach
theorem, there are a real version and a complex version.
Theorem 2.2.6 (Real Version) Let X be a real normed space and p : X −→ R be
a sublinear functional, i.e., p(x + y) ≤ p(x) + p(y) and p(αx) = αp(x) for all
x, y ∈ X and all α ∈ [0, ∞). If X0 ⊂ X is a linear subspace and if f : X0 −→ R
is a linear functional satisfying f (x) ≤ p(x) for all x ∈ X0 , then there is a linear
functional F : X −→ R having the properties F (x) = f (x) for all x ∈ X0 and
F (x) ≤ p(x) for all x ∈ X.
Theorem 2.2.7 (Complex Version) Let X be a normed space over K = R or
K = C and X0 ⊂ X a linear subspace. If f : X0 −→ K is a linear functional
with |f (x)| ≤ x for all x ∈ X0 , then there exists a linear functional F : X −→ K
satisfying F (x) = f (x) for all x ∈ X0 and |F (x)| ≤ x for all x ∈ X.
The set of all linear and bounded functionals on a normed space X, i.e.,
(L(X, K), .X→K ), is also denoted by X∗ and called the dual space of X.
If a normed space Y is continuously embedded into a normed space X, then it is
easily to see that X∗ is continuously embedded into Y∗ . The following first corollary
of the Hahn-Banach theorems shows that, for a linear subspace X0 ⊂ X equipped
with the induced norm of X, the dual space X∗0 is “not greater” than X∗ .
Corollary 2.2.8 Assume that X is a normed space and X0 ⊂ X a linear subspace
equipped with the norm from X.Then, for every f0 ∈ X∗0 , there is an f ∈ X∗ with
the properties f (x) = f0 (x) for all x ∈ X0 and f X∗ = f0 X∗0 .
Proof The case f0 X∗ = 0 is trivial. Let f0 X∗ > 0. By x∗ := f0 X∗ xX
0 0 0
we define a norm on X, where
|f0 (x)| ≤ f0 X∗ xX = x∗ ∀ x ∈ X0

0
holds true. In view of Theorem 2.2.7, there exists a functional f ∈ X∗ such that
f (x) = f0 (x) for all x ∈ X0 and |f (x)| ≤ x∗ = f0 X∗ xX for all x ∈ X.
0
Thus, on the one hand f X∗ ≤ f0 X∗ . On the other hand, we have
0
f X∗ = sup {|f (x)| : x ∈ X, xX ≤ 1} ≥ sup {|f0 (x)| : x ∈ X0 , xX ≤ 1} = f0 X∗ ,
0
and the corollary is proved.

2.2 Fundamental Principles 11
Corollary 2.2.9 Let X be a normed space and X0 a linear subspace of X. If x ∗ ∈

X with d = dist(x ∗ , X0 ) := inf {x0 − x ∗ : x0 ∈ X0 } > 0 then there exists a
functional f0 ∈ X∗ such that f (x0 ) = 0 for all x0 ∈ X0 , f0 X∗ = 1, and f0 (x ∗ ) =
d.
Proof Set X1 = {x = x0 + αx ∗ : x0 ∈ X0 , α ∈ K} and f1 (x0 + αx ∗ ) = αd for
x0 ∈ X0 and α ∈ K. This definition is correct, since the relation x0 + α x ∗ =
x0 + αx ∗ for some x0 , x0 ∈ X0 , α, α ∈ K, implies x0 − x0 = (α − α )x ∗ ∈ X0 and
hence α = α. In case α = 0 we get

|f1 (x0 + αx ∗ ) = |α|d = |α| inf −α −1 y0 − x ∗ : y0 ∈ X0

= inf y0 + αx ∗ : y0 ∈ X0 ≤ x0 + αx ∗ ∀ x 0 ∈ X0 .
Consequently, f1 X∗ ≤ 1. Furthermore, there are xn ∈ X0 such that

1
xn − x ∗ −→ d if n tends to infinity. This implies

d = |f1 (xn − x ∗ )| ≤ f1 X∗ xn − x ∗ −→ f1 X∗ d ,
1 1
i.e., 1 ≤ f1 X∗ . Finally, we have to use Corollary 2.2.8.

1
The following conclusion of the Hahn-Banach theorem is also known as

proposition on the sufficient number of bounded linear functionals.
Corollary 2.2.10 Let X be a normed space and x0 ∈ X \ {}. Then there is a
functional f0 ∈ X∗ with the properties f0 X∗ = 1 and f0 (x0 ) = x0 .
The phrase “sufficient number of bounded linear functionals” describes the
separation property of the dual space X∗ formulated in the following exercise.
Exercise 2.2.11 Use Theorem 2.2.7 to give a proof of Corollary 2.2.10 and show
that, as a consequence of this corollary, the functionals from X∗ separate the points
of X, which means that, for arbitrary different points x1 and x2 of X, there is a
functional f ∈ X∗ such that f (x1 ) = f (x2 ).
For 1 < p < ∞, let us consider the sequence space
∞

∞
p
= ξ= (ξn )n=0 : ξn ∈ C , |ξn | < ∞
p
, (2.2.1)
n=0
which is, equipped with the norm

∞ 1
p
ξ p = |ξn | p
, (2.2.2)
n=0
a Banach space. As a consequence of the Hölder inequality

∞ ∞ 1 ∞ 1
p q

ξn ηn ≤ |ξn | p
|ηn | q
, ξ∈ p
, η∈ q
(2.2.3)

n=0 n=0 n=0
1
+ q1 = 1 , its dual space ( p )∗ can be identified with q via the isometrical
p
isomorphism

p ∗ ∞
J : −→ q
, f → J f = η = (ηn )n=0 (2.2.4)

∞
p )∗
with f (ξ ) = ξn ηn for all ξ ∈ p (i.e., J : ( −→ q is linear and bijective
n=0
with J f = f ( for all f ∈ ( p )∗ ).
q p )∗
Exercise 2.2.12 Use Hölder’s inequality (2.2.3) to show that the map J :
( p )∗ −→ q defined in (2.2.4) is an isometrical isomorphism.
∞ p
For a sequence ω = (ωn )n=0 of positive numbers, by ω we denote the weighted
p ∞
√ ∞
space ω = x = (ξn )n=0 : p ωn ξn n=0 ∈ p equipped with the norm
∞
1
p
ξ p
ω
= ωn |ξn | p
. (2.2.5)
n=0
Of course,

p ∗ ∞
Jω : ω −→ q
ω, f → Jω f = η = (ηn )n=0 (2.2.6)

∞
p
with f (ξ ) = ωn ξn ηn for all ξ ∈ ω is an isometrical isomorphism (cf.
n=0
Exercise 2.2.12).
Corollary 2.2.13 Let 1 < p < ∞ and p1 + q1 = 1, and let αk ∈ C, γk > 0, k ∈ N0 ,
be given numbers. If there is a constant A ∈ R such that, for all βk ∈ C and all
n ∈ N,
n n 1
p

γk αk βk ≤ A γk |βk |p , (2.2.7)

k=0 k=0
then
∞ 1
q
γk |αk |q ≤ A. (2.2.8)
k=0
2.3 Compact Sets and Compact Operators 13
∞
Proof For γ = (γn )n=0 , define the linear functional

n
∞
fn : p
γ −→ C, ξ = (ξn )n=0 → γk αk ξk .
k=0
p ∗
By (2.2.7) we have fn ∈ γ and, since Jω in (2.2.6) is an isometrical
1

n q
isomorphism, fn p ∗
γ
= γk |αk |q ≤ A, which yields (2.2.8).

k=0
2.3 Compact Sets and Compact Operators
Let E be a metric space with the distance function d : E × E −→ [0, ∞). Recall
that a subset A ⊂ E is called compact if every covering of A by open sets contains a
finite covering of A and that A is called relatively compact if its closure is compact.
For ε > 0, by an ε-net for a nonempty subset A ⊂ E we mean a set Aε ⊂ E such
that for every x ∈ A there exists an xε ∈ Aε with d(x, xε ) < ε. This condition can
also be written in the formula
Aε ∩ Uε (x) = ∅ ∀x ∈ A,
where Uε (x) = UE
ε (x) denotes the (open) ε-neighbourhood of x ∈ E,
Uε (x) = {y ∈ E : d(y, x) < ε} .
Exercise 2.3.1 Let A ⊂ E be a nonempty subset of a metric space (E, d). Show
that, if for every ε > 0 there exists a finite ε-net Aε ⊂ E for A, then, for every
ε > 0, there exists a finite ε-net Bε ⊂ A for A.
Exercise 2.3.2 Prove that A ⊂ E is relatively compact if and only if every sequence
∞
(xn )n=1 of points xn ∈ A possesses a convergent subsequence. Moreover, show that
a set A ⊂ E is relatively compact if, for every ε > 0, there exists a finite ε-net for
A, and that the reverse conclusion is true, if E is a complete metric space.
Note, that a subset of a finite dimensional normed space is relatively compact if
and only if it is bounded.
Lemma 2.3.3 Let X be a Banach space and Xn ⊂ X, n = 1, 2, . . . be a sequence
of finite dimensional subspaces of X. If A is a bounded subset of X and if
lim sup {En (g) : g ∈ A} = 0 , (2.3.1)

n→∞
where En (g) = inf {g − fn : fn ∈ Xn }, then A is relatively compact.

Proof There exists an R ∈ R with A ⊂ UR () := {f ∈ X : f < R}. If ε > 0

then there is an N ∈ N such that EN (g) < ε2 for all g ∈ A. Thus, for every
g ∈ A, there exists an fN,g ∈ XN such that g − fN,g < 2ε . The set AN :=

fN,g : g ∈ A ⊂ XN is bounded, namely AN ⊂ UR+ ε2 (). Consequently, there
is a finite ε2 -net AεN ⊂ XN for AN , which is an ε-net for A. It remains to refer to
Exercise 2.3.2.

Lemma 2.3.4 Assume that X and Xn satisfy the conditions of Lemma 2.3.3 with
∞
Xn ⊂ Xn+1 , n ∈ N, and that Xn is dense in X. If A ⊂ X is relatively compact,
n=1
then (2.3.1) is satisfied.
Proof Assume that there are an ε > 0 and a sequence n1 < n2 < . . . of positive
integers such that

sup Enk (g) : g ∈ A ≥ 2ε > 0 ∀ k ∈ N .
Then there exist gk ∈ A with Enk (gk ) ≥ ε, where we can assume that (due to the
relative compactness of A, cf. Exercise 2.3.2) gk −→ g ∗ for k −→ ∞ and for some
g ∗ ∈ X. Let k be sufficiently large, such that gk − g ∗ < ε2 . Then, for fnk ∈ Xnk ,
ε
ε ≤ fnk − gnk ≤ fnk − g ∗ + g ∗ − gnk < fnk − g ∗ +
2

implying that 2ε ≤ fnk − g ∗ for all fnk ∈ Xnk and all sufficiently large k. Hence,
∞
taking into account Xn ⊂ Xn+1 , we have ε
2 ≤ f − g ∗ for all f ∈ X0 := Xn ,
n=1
which contradicts the assumed density of X0 in X.

Let E be a compact metric space and denote by C(E) the Banach space of all
continuous functions f : E −→ C, where the norm is given by
f ∞ = f ∞,E := max {|f (x)| : x ∈ E} .
A family F of functions from C(E) is called uniformly bounded if it is a bounded

subset of C(E), i.e., if there exists a positive number M ∈ R such that |f (x)| ≤ M
for all x ∈ E and for all f ∈ F . A family F of continuous functions from C(E) is
called equicontinuous if, for every ε > 0, there is a δ > 0 such that
|f (x1 ) − f (x2 )| < ε ∀ f ∈ F, ∀ x1 , x2 ∈ E : d(x1 , x2 ) < δ .
Theorem 2.3.5 (Arzela-Ascoli) Let E be a compact metric space. A set F ⊂ C(E)

is relatively compact if and only if it is a uniformly bounded and equicontinuous
family.
2.3 Compact Sets and Compact Operators 15
Exercise 2.3.6 Let (E, d) be a metric space. A family F ⊂ C(E) is called (locally)
equicontinuous in x0 ∈ E, if for every ε > 0 there exists a δ > 0 such that
|f (x) − f (x0 )| < ε ∀ f ∈ F , ∀ x ∈ Uδ (x0 ) .
Show that, for a compact metric space (E, d), a set F ⊂ C(E) is equicontinuous if
and only if it is equicontinuous in every x0 ∈ E.
Exercise 2.3.7 For I = [−1, 1], I = [0, ∞], or I = [−∞, ∞], by (I, db ) we
denote the respective metric spaces defined by db (x, y) = | arctan(x) − arctan(y)|
π
with arctan(±∞) = ± . Prove that (I, db ) is a compact metric space.
2
Exercise 2.3.8 Show that, if E is one of the (compact) metric spaces from
Exercise 2.3.7, then the Banach space (C(E), .∞ ) is given by
⎧
⎪
⎪ C[−1, 1] : I = [−1, 1] ,
⎪
⎨
C(I) = {f ∈ C[0, ∞) : ∃ f (∞) := limx→∞ f (x)} : I = [0, ∞] ,
⎪
⎪
⎪
⎩
{f ∈ C(−∞, ∞) : ∃ f (±∞) := limx→±∞ f (x)} : I = [−∞, ∞] ,
and f ∞ = f C(I) = sup {|f (x)| : x ∈ I}.

∞
Exercise 2.3.9 Let (fn )n=1 be a sequence of functions fn : E −→ C on the
compact metric space E, which are equicontinuous, and assume that there is a
continuous function f : E −→ C such that
lim fn (x) = f (x) for all x ∈ E .

n→∞
Show that lim fn − f ∞,E = 0.

n→∞
An operator T ∈ L(X, Y) is called compact, if it maps bounded subsets of X into
relatively compact subsets of Y, which is equivalent tothe condition, that the image

of the unit ball of X is relatively compact, i.e., that T f : f ∈ X, f X ≤ 1 is
relatively compact in Y. The subset of L(X, Y) of all compact operators we denote
by K(X, Y) or, in case of X = Y, by K(X). Examples of compact operators will be
considered in Sect. 5.3.
We say that a normed space (X0 , .X0 ) is compactly embedded into a normed
space (X, .X ), if the embedding operator E : X0 −→ X, f → f is compact.
Exercise 2.3.10 Let Y be a Banach space. Show that (K(X, Y), .X→Y ) is a
closed linear subspace of (L(X, Y), .X→Y ).
The following lemma and the following corollary are consequences of the
Banach-Steinhaus theorem and combine strong convergence with compactness of
operators.
Lemma 2.3.11 Assume that X and Y are Banach spaces, An , A ∈ L(X, Y),
An −→ A strongly, and M ⊂ X is relatively compact. Then
lim sup {(An − A)xY : x ∈ M} = 0 ,

n→∞
i.e., strong convergence is uniform on relatively compact subsets.

Proof Let M ⊂ X be relatively compact and assume that, for n −→ ∞,
sup {(An − A)xY : x ∈ M} does not tend to zero. Then there are an ε > 0,
nk > nk−1 ∈ N0 , n0 := 0, and xk ∈ M such that (Ank − A)xk Y ≥ 2ε, k =
1, 2, . . . We can assume, due to the relative compactness of M, that xk −→ x ∗ in X
for k −→ ∞. Since, in virtue of Theorem 2.2.1, γ := sup An L(X,Y) : n ∈ N is
finite, we conclude

2ε ≤ (Ank − A)xk Y ≤ (Ank − A)(xk − x ∗ )Y + (Ank − A)x ∗ Y

≤ γ + AL(X,Y) xk − x ∗ Y + Ank − A x ∗ Y .

Hence, the inequality ε ≤ Ank − A x ∗ Y is valid for all k ≥ k0 in contradiction
to the strong convergence Ank −→ A.

Corollary 2.3.12 If X, Y, and Z are Banach spaces, T ∈ K(X, Y), An , A ∈
L(Y, Z), and if An −→ A strongly, then lim (An − A)T L(X,Z) = 0.
n→∞
Proof Since by assumption the set M0 := {T x : x ∈ X, xX ≤ 1} is relatively
compact, in virtue of Lemma 2.3.11 we obtain

(An − A)T L(X,Z) = sup (An − A)yZ : y ∈ M0 −→ 0 if n −→ ∞ ,

Exercise 2.3.13 For Banach spaces X and Y, show that T ∈ K(X, Y) implies T ∈ ∗
L(Y∗ , X∗ ). Using this and taking into account Corollary 2.3.12, prove the following:
If X, Y, Z, and W are Banach spaces, T ∈ K(X, Y), An , A ∈ L(Y, Z), and if
An −→ A strongly, as well as Bn , B ∈ L(W, X) and Bn∗ −→ B ∗ strongly, then
lim An T Bn − AT BL(W,Z) = 0. (For the definition of the adjoint operator B ∗
n→∞
of B, see Sect. 2.5.)
2.4 Function Spaces
In this section we give an overview on the types of function spaces playing an

essential role in this book. Thereby, we concentrate on typical examples of such
spaces, while further particular spaces will be introduced at that places of the book,
where they are needed.
2.4 Function Spaces 17
2.4.1 Lp -Spaces
Here and in what follows, by a weight function u : I −→ R on an interval

I ⊂ R we mean a nonnegative, measurable,
and integrable1 function u(x), for which
u(x) dx > 0 and all moments x k u(x) dx, k ∈ N0 , are finite. If 1 ≤ p < ∞,
I I
then, as usual, by Lp (I ) we denote the Banach space of all (classes of) measurable
functions f : I −→ C, for which the Lp -norm
1
p
f Lp (I ) = |f (x)|p dx
I
is finite. In case p = ∞, by L∞ (I )
the space of all (classes of) measurable and
essentially bounded functions equipped with the norm
f L∞ (I ) = ess sup {|f (x)| : x ∈ I } := inf {sup {|f (x)| : x ∈ A} : A ⊂ I, m(A) = 0}
is meant, where by m(A) the Lebesgue measureof theset A is denoted. In case

I = (a, b), we also write Lp (a, b) instead of Lp (a, b) , and if (a, b) = (−1, 1),
p
Lp instead of Lp (−1, 1). If u is a weight function on I , then by Lu (I ) we refer to
the Banach space of all (classes of) functions f : I −→ C for which f u belongs to
Lp (I ) with the norm f Lpu (I ) = f uLp (I ) .
If u(x) = v α,β (x) = (1 − x)α (1 + x)β with α, β > −1 is a Jacobi weight,
p p
then we use the notation Lα,β := Lv α/p,β/p together with f α,β,(p) := f Lp . In
α,β
particular, the inner product and the norm in the Hilbert space L2α,β are defined by

1 1
f, gα,β := f (x)g(x)v α,β
(x) dx and f α,β := f α,β,(2) = |f (x)|2 v α,β (x) dx ,
−1 −1
respectively. Moreover, for a real number s ≥ 0, we define the Sobolev-type

subspace L2,s 2
α,β of Lα,β as

∞
2

L2,s
α,β := f ∈ L2α,β : (n + 1)2s f, pnα,β α,β
<∞ , (2.4.1)
n=0
α,β
where pn (x) denotes the normalized Jacobi polynomial with respect to v α,β (x) of
degree n (cf. Sect. 5.1). Equipped with the inner product
∞
α,β
f, gα,β,s := (n + 1)2s f, pnα,β α,β
g, pn α,β
(2.4.2)
n=0
1 Here, measurability and integrability is meant in the Lebesgue sense.


and the respective norm f α,β,s := f, f α,β,s , the space L2,s
α,β is again a Hilbert
2,0
space. Note that Lα,β = L2α,β .

Exercise 2.4.1 Show that L2,s α,β , ., . α,β,s is a Hilbert space.
Remark 2.4.2 It is well-known that, for 1 < p < ∞ and p1 + q1 = 1, the map J :
1
∗
Lq −→ Lp with (J g)(f ) = f (x)g(x) dx is an isometric isomorphism. For
p ∗ −1
this, we write L = Lq . Consequently, in the same way of identification we have
∗ 1
p q
Lv α,β = Lv −α,−β . Moreover, if we use (Jα,β g)(f ) = g(x)f (x)v α,β (x) dx,
∗ −1
q p
then Jα,β : Lα(q−1),β(q−1) −→ Lα,β is also an isometric isomorphism.
Example 2.4.3 Let −1 < α, β < 1, p > 1, and p1 + q1 = 1 and consider the integral
operator K defined by
1
(Kf )(x) = K(x, y)u(y)v α,β (y) dy , −1 < x < 1 ,
−1
where the measurable function K : (−1, 1)2 \ {(x, x) : x ∈ (−1, 1)} −→ C is

assumed to satisfy
|K(x, y)| ≤ c|x − y|−η , (x, y) ∈ (−1, 1)2 \ {(x, x) : x ∈ (−1, 1)} ,

p
for some η ∈ 0, q1 . By Hölder’s inequality we get, for f ∈ Lα,β and −1 < x < 1,
1 q1 1
−ηq α,β − −
f α,β,(p) ≤ c v −α ,−β (x)
q
|(Kf )(x)| ≤ c |x − y| v (y) dy f α,β,(p) ,
−1
where α± = max {0, ±α}, β± = max {0, ±β}, and where we took into account
Lemma 5.2.10. Consequently,
1 q1 1 q1
q −α,−β −α + ,−β +
Kf −α,−β,(q) = |(Kf )(x)| v (x) dx ≤c v (x) dx f α,β,(p) ,
−1 −1
p + +
1
≤ c v −α ,−β Lq 1 (−1,1).
q
i.e., K ∈ L Lα,β , L−α,−β with KLp q
α,β →L−α,−β
Lemma 2.4.4 The norm in L2,s

α,β is equivalent to the norm
∞
1
! "2 2
f α,β,s,∼ := (m + 1) 2s−1 α,β
Em (f )2 ,
m=0
α,β
where Em (f )p := inf f − P α,β,(p) : P ∈ Pm with Pm being the set of
α,β
polynomials of degree less than m and E0 (f )2 := f α,β . Since, in case γ ≤ α
and δ ≤ β, L2γ ,δ is continuously embedded into L2α,β , we have also the continuous
embedding L2,s 2,s
γ ,δ into Lα,β in this case.
Proof It is well known that, for f ∈ L2α,β ,

∞
1
2 2

α,β
Em (f )2 = f, pnα,β α,β
.
n=m
Consequently,
∞
∞
2 ∞ 2
n

f 2α,β,s,∼ = (m + 1)2s−1 f, pnα,β α,β = f, pnα,β α,β (m + 1)2s−1 ,
m=0 n=m n=0 m=0
and the relation

n n
1
1 m + 1 2s−1 1
lim (m + 1) 2s−1
= lim = x 2s−1 dx
n→∞ (n + 1)2s n→∞ n+1 n+1 0
m=0 m=0
proves the equivalence of the two norms.

Remark 2.4.5 The spaces L2,s
α,β , s ≥ 0 can be considered as a Hilbert scale generated
by the operator
∞

Ef = (1 + n) f, pnα,β α,β
pnα,β
n=0
2,1
with domain D(E) = Lα,β (cf. [18, Chapter III, §6.9]). Hence, these spaces have
the interpolation property, i.e., if the linear operator A is bounded from L2,s 1
α1 ,β1
2,s (τ )
into L2,s 2,t1 2,t2
α2 ,β2 and from Lα1 ,β1 into Lα2 ,β2 , then A is also bounded from Lα1 ,β1 into
2 1
2,s (τ )
2
Lα2 ,β 2
, where sj (τ ) = (1 − τ )sj + τ tj and 0 < τ < 1 [18, Chapter III, §6.9,
Theorem 6.10].
Lemma 2.4.6 (cf. [20, Lemma 4.2]) If h(x, .) ∈ L2α,β for all (or almost all) x ∈
[−1, 1] and h(., y) ∈ L2,s
γ ,δ uniformly with respect to y ∈ [−1, 1], then the linear
operator H : L2α,β −→ L2,s
γ ,δ defined by
1
(Hf )(x) = h(x, y)f (y)v α,β (y) dy , −1 < x < 1 ,
−1
is bounded.
Proof For f ∈ L2α,β and m ∈ N0 , using Fubini’s theorem and the Cauchy-Schwarz
inequality we can estimate
2 1 1
2

= (y) dy pm (x)v (x) dx
γ ,δ γ ,δ
Hf, pm γ ,δ
h(x, y)f (y)v α,β γ ,δ
−1 −1
2
1
=
γ ,δ α,β
h(., y), pm γ ,δ
f (y)v (y) dy
−1
1 2
γ ,δ α,β
≤ h(., y), pm γ ,δ
v (y) dyf 2α,β .
−1
Hence,
∞
2
γ ,δ
Hf 2γ ,δ,s = (1 + m)2s Hf, pm γ ,δ
m=0
∞
1 2
γ ,δ α,β
≤ (1 + m)2s h(., y), pm γ ,δ
v (y) dyf 2α,β
−1 m=0
1
= h(., y)2γ ,δ,s v α,β (y) dyf 2α,β ≤ c f 2α,β ,
−1
and the lemma is proved.

Lemma 2.4.7 ([20, Conclusion 2.3, pp. 196, 197]) For 0 ≤ s < t, the space L2,t
α,β
is compactly embedded into L2,s N, then f ∈ L2,r
α,β . If r ∈ √ α,β if and only if f
(k) ϕ k ∈
L2α,β for all k = 0, . . . , r, where ϕ(x) = 1 − x 2 . Moreover, the norm f α,β,r is

r
(k) k
equivalent to f ϕ .
α,β
k=0
A proof of the second part of the previous lemma will be given in Sect. 3.1.3 (see
Corollary 3.1.30).
For u = v γ ,δ with γ , δ > − p1 and in the general case 1 ≤ p ≤ ∞, the Sobolev-
type spaces are defined by
# $
p,r p (r) r
Wu = f ∈ Lu : f (r−1)
∈ AC(−1, 1), f ϕ u < ∞ , (2.4.3)
p
√
r ∈ N, where again ϕ(x) := 1 − x 2 and where AC(−1, 1) = ACloc (−1, 1)
denotes the set of all functions f : (−1, 1), which are absolutely continuous on
every closed subinterval of (−1, 1).
Remark 2.4.8 Concerning the properties of absolutely continuous functions we

refer the reader to the famous book of Natanson [173, Chapter IX, §1 and §2]. In par-
ticular, we mention that, for every absolutely continuous function f : [a, b] −→ C,
the derivative f (x) exists for almost all x ∈ [a, b], where f ∈ L1 [a, b].
The spaces defined in (2.4.3) become Banach spaces if we equip them with the
norm

f Wp,r
u
= f up + f (r)ϕ r up (2.4.4)

(cf. Exercise 2.4.10). The set f ∈ Cr−1 (−1, 1) : f (r−1) ∈ AC(−1, 1) is also
p
denoted by ACr−1 loc . By Lloc we refer to the set of all measurable functions
f : (−1, 1) −→ C satisfying f ∈ Lp (a, b) for all closed intervals [a, b] ⊂ (−1, 1).
Note that the following lemma is also true for f ∈ C(−1, 1) and C[a, b] instead
of Lp (a, b).
p
Lemma 2.4.9 Assume that g, fn ∈ ACr−1
loc and f ∈ Lloc satisfy

lim fn − f Lp (a,b) = 0 and lim fn(r) − g (r) Lp (a,b) = 0
n→∞ n→∞
for every interval [a, b] ⊂ (−1, 1). Then f ∈ ACr−1

loc and f
(r) = g (r) .
Proof We define Pn ∈ Pr , n ∈ N, by Pn(r) ≡ 0 and

x
Pn(k) (x) = Pn(k+1) (y) dy + fn(k) (0) − g (k) (0) , k = r − 1, . . . , 0 .
0
This implies
x ! (k+1) "
fn(k) (x) − Pn(k) (x) − g (k)
(x) = fn (y) − Pn(k+1) (y) − g (k+1) (y) dy
0
and, for every interval [a, b] ⊂ (−1, 1),

(k) p−1
f −P (k) −g (k) ≤ (b−a) p fn(k+1) −Pn(k+1) −g (k+1) Lp (a,b) , k = r−1, . . . , 0 .
n n [a,b],∞
Hence, by induction, lim fn − Pn − g∞,[a,b] = 0 and, since fn −→ f in

n→∞
Lp (a, b), additionally we have Pn −→ f −g in Lp (a, b). This yields f (x)−g(x) =
P (x) for almost all x ∈ (−1, 1) with P ∈ Pr .

α β
Note that, in case u = v 2 , 2 , we have W2,r 2,r
u = Lα,β due to Lemma 2.4.7 and the
following exercise, which can be solved with the help of Lemma 2.4.9.
p,r
Exercise 2.4.10 Let 1 ≤ p < ∞ and u = v γ ,δ , γ , δ > − p1 . Prove that Wu
equipped with the norm defined in (2.4.4) as well as with the norm
r

(k) k
f ϕ u (2.4.5)
p
k=0
(cf. Lemma 2.4.7) becomes a Banach space and that these two norms are equivalent
p,r
on Wu .
As a conclusion of Exercise 2.4.10 we have the following property of a
multiplication operator. For this, we define the space Crϕ of all r times continuously
differentiable functions f : (−1, 1) −→ C satisfying the conditions f (k) ϕ k ∈
C[−1, 1] for k = 0, 1, . . . , r. This space, equipped with the norm f Crϕ =
r
(k) k
f ϕ , is a Banach space.
∞
k=0
Corollary 2.4.11 Let r ≥ 0 be an integer and a ∈ Crϕ . Then the multiplication

p,r p,r p,r
operator aI : Wu −→ Wu , u → au belongs to L Wu , where
aIWp,r p,r ≤ ca r

u →Wu Cϕ
with a constant c = c(a).

p,r
Proof For f ∈ Wu we obtain, in view of (2.4.4),
r

af Wp,r
u
≤ (af )(k) ϕ k .
p
k=0
Using
k
k (j ) j (k−j ) k−j
(af ) ϕ =
(k) k
a ϕ f ϕ ,
j
j =0
and taking into account Exercise 2.4.10, we get
k
k (j ) j
r

af Wp,r ≤ a ϕ f (k−j ) ϕ k−j
u j ∞ p
k=0 j =0

≤ max a (j )ϕ j max f (k−j ) ϕ k−j 2r+1
0≤j ≤r ∞ 0≤j ≤r p
≤ caCrϕ f Wp,r
u
,

γ δ
Corollary 2.4.12 Since we have W2,ru = L2,r
γ ,δ for r ∈ N0 and u = v
2 , 2 , we
can apply Corollary 2.4.11 together with Remark 2.4.5. Hence, for a ∈ Crϕ , the
multiplication operator aI : L2,s 2,s
γ ,δ −→ Lγ ,δ is continuous for 0 ≤ s ≤ r.
p,r
The Besov-type space Bq,u basing on the kth weighted modulus of continuity
p
of a function f ∈ Lu given by
# $

kϕ (f, t)u,p = sup khϕ f u :0<h≤t , (2.4.6)
Lp (Ihk )
where, for k ∈ N,

k
k k
khϕ f (x) = j
(−1) f x+ − j hϕ(x) (2.4.7)
j 2
j =0
! "
and Ihk = −1 + 2h2 k 2 , 1 − 2h2 k 2 , can be defined as follows. If we set, for k ≥ r,
⎧
⎪
⎪ 1% k &q q1
⎪
⎪
ϕ (f, t)u,p
⎪
⎨ dt : 1 ≤ q < ∞,
0 r+ 1
|f |u,p,q,r := t q
⎪
⎪
⎪
⎪ kϕ (f, t)u,p
⎪
⎩ sup : q=∞
t >0 tr
and f Bp,r
q,u
:= f up + |f |u,p,q,r , then, for 1 ≤ p ≤ ∞, 1 ≤ q ≤ ∞, and r > 0,

p,r p
Bq,u := f ∈ Lu : f Bp,r
q,u
< ∞ . (2.4.8)
Note, that

p,r
(a) Bq,u , .Bp,r
q,u
is a Banach space,
p,r p,r
(b) in case r ∈ N, we have the equality B∞,u = Wu with equivalent norms,
p,[r]+1
(c) for 1 ≤ q ≤ ∞ and r > 0, we have the continuous embeddings Wu ⊂
p,r p,[r]
Bq,u ⊂ Wu (see [139, Proposition 3.3]).
We refer also to Sect. 3.1.3.
2.4.2 Spaces of Continuous Functions
For a compact interval [a, b] ⊂ R, by C[a, b] = (C[a, b], .∞ ) we refer to the
linear space of all continuous functions f : [a, b] −→ C, which is, equipped
with the infinity or Chebyshev-norm f ∞,[a,b] := max {|f (x)| : a ≤ x ≤ b}, a
Banach space, even a Banach algebra. (See also Exercise 2.4.19 below as well as
Sect. 2.3 and Exercise 2.3.8.) The linear subspace Cr,λ [a, b] of C[a, b] is defined
for r ∈ N0 and 0 < λ ≤ 1 and consists of those functions f : [a, b] −→ C, which
are r times continuously differentiable and the rth derivative of which is Hölder
continuous on [a, b], i.e.,
(r)
f (x1 ) − f (r) (x2 )
cr,λ,[a,b](f ) := sup : a ≤ x1 < x2 ≤ b < ∞ .
|x1 − x2 |λ
In the space Cr,λ [a, b] a norm is given by

r
(k)
f Cr,λ [a,b] = f + cr,λ,[a,b] (f ) (2.4.9)
∞,[a,b]
k=0
and makes it to a Banach space (see Exercise 2.4.17). In case of [a, b] = [−1, 1],
we will write C, Cr,λ , f ∞ , and cr,λ (f ) instead of C[−1, 1], Cr,λ [−1, 1],
f ∞,[−1,1] , and cr,λ,[−1,1](f ), respectively.
Exercise 2.4.13 Show that, for 0 < λ < 1 and 0 < x < 1,
1 − x λ < (1 − x)λ and 1 + x λ < 21−λ (1 + x)λ (2.4.10)
and that the function ϕλ (x) = |x|λ , −1 ≤ x ≤ 1, belongs to C0,λ [−1, 1].

If A ⊂ C0,λ [a, b] is a bounded set in C0,λ [a, b], .C0,λ [a,b] , i.e.,
f C0,λ [a,b] ≤ cA < ∞ for all f ∈ A, then f ∞,[a,b] ≤ cA and |f (x1 ) − f (x2 )| ≤
cA |x1 − x2 |λ for all f ∈ A, which implies that the set A is a uniformly bounded
set of equicontinuous functions. Thus, by the Arzela-Ascoli Theorem 2.3.5, the set
A is relatively compact in C[a, b], i.e., we get the following corollary (see also
Exercise 2.4.18).

Corollary 2.4.14 C0,λ [a, b], .C0,λ [a,b] is compactly embedded into C[a, b],

.∞,[a,b] .
Exercise 2.4.15 Prove that Cr,μ [a, b] ⊂ Cr,λ [a, b] for 0 < λ ≤ μ ≤ 1.
Exercise 2.4.16 For a < b < c, prove that

f ∈ Cr,λ [a, b] ∩ Cr,λ [b, c] : f (j) (b − 0) = f (j) (b + 0), j = 0, . . . , r = Cr,λ [a, c] .
Exercise
r,λ 2.4.17 Show that (2.4.9) defines a norm in Cr,λ [a, b] and that
C [a, b], .Cr,λ [a,b] is a Banach space.
Exercise 2.4.18 Let 0 < λ < μ. By using the compact embedding C0,μ [a, b] ⊂
C[a, b] (see Corollary 2.4.14), prove that C0,μ [a, b] is compactly embedded into
C0,λ [a, b].
Exercise 2.4.19 Prove that C0,λ [a, b] is a Banach algebra, i.e., that additionally
to the properties of a Banach space we have, for f, g ∈ C0,λ [a, b],
fg ∈ C0,λ [a, b] and fgCr,λ [a,b] ≤ f C0,λ [a,b] gC0,λ [a,b] .
Exercise 2.4.20 Let 0 < λ < 1 and, for x, y ∈ [−1, 1],
(x − y)|x − y|−λ : x = y ,
h(x, y) =
0 : x =y.
Show that h(., y) ∈ C0,1−λ uniformly with respect to y ∈ [−1, 1], i.e.,
|h(x1 , y) − h(x2 , y)| ≤ c |x1 − x2 |1−λ , (xj , y) ∈ [−1, 1]2 , j = 1, 2 ,
with c = c(x1, x2 , y).

Exercise 2.4.21 For f (x) = x x = ex ln x , prove that f ∈ C0,ε [0, a] for all ε ∈
(0, 1) and all a ∈ (0, ∞).
Exercise 2.4.22 Let 0 < a < ∞ and 0 < μ ≤ 1. Show that the function
f : [0, a] −→ R, x → x μ ln x (f (0) := 0)
belongs to C0,λ [0, a] for every λ ∈ (0, μ). Moreover, prove that, for every λ ∈
(0, 1), there is a constant cλ such that
|h(x1 , y) − h(x2 , y)| ≤ cλ |xx − x2 |λ , (xj , y) ∈ [−1, 1]2 ,
where h(x, y) = (x − y) ln |x − y|.

For a Jacobi weight v γ ,δ (x) = (1 − x)γ (1 + x)δ with γ , δ ≥ 0, by
Cbγ ,δ= Cbγ ,δ (−1, 1) we denote the Banach space of all continuous functions
f : (−1, 1) −→ C, for which v γ ,δ f : (−1, 1) −→ C is bounded and for which the
respective norm is defined by

f γ ,δ,∞ = v γ ,δ f ∞ = sup v γ ,δ (x)f (x) : x ∈ Iγ ,δ .
By Cγ ,δ = Cγ ,δ (−1, 1) we refer to the closed subspace of Cbγ ,δ (hence, also a

Banach space) of all continuous functions f : (−1, 1) −→ C, for which the finite
limits lim v γ ,δ (x)f (x) exist. The closed subspace Cγ ,δ = Cγ ,δ (−1, 1) of Cγ ,δ
x→±1∓0
consists of all functions f ∈ Cγ ,δ satisfying
lim v γ ,δ (x)f (x) = 0 if γ > 0 and lim v γ ,δ (x)f (x) = 0 if δ > 0 .

x→1−0 x→−1+0
Note that the set P of all algebraic polynomials (with complex coefficients) is
dense in Cγ ,δ (cf. the following Exercise 2.4.23 for the case 0 ≤ γ , δ ≤ 1 and
Exercise 6.2.5 for the general case γ , δ ≥ 0). For f ∈ Cbγ ,δ , we define the best
γ ,δ γ ,δ
weighted uniform approximation Em (f ) = Em (f )∞ = Em (f )v γ ,δ ,∞ of f by
polynomials, belonging to the subset Pm of P of all polynomials of degree less than
m ∈ N, as
γ ,δ
Em (f ) = inf f − pγ ,δ,∞ : p ∈ Pm .
γ ,δ
In case m = 0, we set E0 (f ) := f γ ,δ,∞ .
Exercise 2.4.23 Show that, for 0 ≤ γ , δ ≤ 1 and f ∈ Cbγ ,δ , we have
γ ,δ
lim Em (f ) = 0 if and only if f ∈ Cγ ,δ . Conclude that Cγ ,δ is equal to the
m→∞
smallest closed subspace of Cbγ ,δ containing the set P of all polynomials.
Exercise 2.4.24 Prove that, for all f ∈ Cbγ ,δ and all n ∈ N, there exists a

f f γ ,δ
polynomial Pn ∈ Pn such that f − Pn = En (f ).
γ ,δ,∞
Exercise 2.4.25 For −1 < xN < . . . < x1 < 1 and αk ∈ C \ {0}, consider the

N
linear functional ϕ : Cbγ ,δ −→ C, f → αk f (xk ) and show that
k=1

N
ϕ(Cb ∗ = v −γ ,−δ (xk )|αk |.
γ ,δ )
k=1
Lemma 2.4.26 For α, β > −1 and s > 12 , the space L2,s is continuously embed-
α,β
ded into the space Cγ ,δ , where γ = 12 max 0, α + 12 and δ = 12 max 0, β + 12 .
α,β α,β
Proof Let f ∈ L2,s
α,β , s > 2 , and f0 = v
1 γ ,δ f as well as p
n =v
γ ,δ p
n . Moreover,
define
# $ # $
1 1
γ1 := α−2γ = α−max 0, α + and δ1 := β−2δ = β−max 0, β + .
2 2
α,β α,β α,β α,β α,β α,β

Then pn , pm γ ,δ = pn , pm α,β
= δmn and f0 , pn γ1 ,δ1
= f, pn α,β
.
1 1
Since (see [16, Theorem 1.1])
1 1
α,β √ 1 α+ 2 √ 1 β+ 2
p (x) 1 − x + 1 + x + ≤ c = c(n, x) , (2.4.11)
n
n n
α,β
−1 < x < 1, we have pn ∞ ≤ c = c(n) and, for k ∈ N0 ,
∞
∞ 1
∞
2
−2s
f0 , pnα,β pα,β (x)
γ1 ,δ1 n
≤c f0 , pnα,β γ1 ,δ1 ≤ c f α,β,s (n + 1) .
n=k n=k n=k
Thus,
the Fourier
series of the function f0 with respect to the orthonormal system
α,β
pn : n ∈ N0 in L2γ1 ,δ1 converges uniformly on [−1, 1], which implies that f0 :
[−1, 1] −→ C is continuous. Moreover, the last estimate shows that f γ ,δ,∞ ≤
c f α,β,s .

∞
For a sequence ξ = (ξn )n=0 of positive real numbers tending to zero for n −→
ξ
∞, by Cγ ,δ we refer to the space of all functions f ∈ Cγ ,δ , for which the number

f γ ,δ,ξ := sup ξn−1 En (f ) : n ∈ N0
γ ,δ
(2.4.12)
γ ,δ γ ,δ γ ,δ γ ,δ γ ,δ
is finite. Since En (f +g) ≤ En (f )+En (g) and En (λf ) = |λ|En (f ) is true
ξ
for all f, g ∈ Cγ ,δ and all λ ∈ C, by (2.4.12) a norm is defined on Cγ ,δ . Therefore,
ξ
by Cγ ,δ we will refer to the space equipped with this norm. In case ξ0 = ξ1 = 1 and
ξn = n−ρ lnτ n, n ≥ 2, ρ > 0, τ ∈ R, we write Cγ ,δ instead of Cγ ,δ and .γ ,δ,ρ,τ
ρ,τ ξ
instead of .γ ,δ,ξ , respectively.

As already mentioned in the introductory Chap. 1, we denote by c a positive
constant which can have different values at different places. Moreover, by c =
c(n, f, x, . . .) we indicate that c is independent of n, f, x, . . . If A(n, f, x, . . .) and
B(n, f, x, . . .) are two positive functions depending on certain variables n, f, x, . . .,
then we will write A ∼n,f,x,... B, if there exists a positive constant c =
c(n, f, x, . . .) such that c−1 B(n, f, x, . . .) ≤ A(n, f, x, . . .) ≤ c B(n, f, x, . . .)
holds true.
Lemma 2.4.27 Let γ , δ ≥ 0, α > 2γ − 1, β > 2δ − 1, ρ ≥ 0, and ε > 0. Then the
ρ+ε,τ 2,ρ
space Cγ ,δ is continuously embedded into the space Lα,β .
Proof Since
1 12
f α,β ≤ (1 − x) α−2γ
(1 + x) β−2δ
dx f γ ,δ,∞ = c f γ ,δ,∞ ,
−1
α,β γ ,δ ρ+ε,τ
we have also En (f )2 ≤ c En (f ) with c = c(n). Consequently, for f ∈ Cγ ,δ ,
due to Lemma 2.4.4 we can estimate
∞
! "2
f 2α,β,ρ ≤ c (n + 1)2ρ−1 Enα,β (f )2
n=0
# $
∞
2
≤ c sup (n + 1)2(ρ+ε) ln−2τ (n + 2) En (f ) (n + 1)−2ε−1 ln2τ (n + 2)
γ,δ
: n ∈ N0
n=0
2
≤ c sup (n + 1)ρ+ε ln−τ (n + 2)En (f ) : n ∈ N0
γ,δ
≤ c f 2γ,δ,ρ+ε,τ ,
which proves the claimed embedding.

The following proposition presents a sequence of properties of the just defined

subspaces of Cγ ,δ . By c+ ∞
0 we denote the set of all zero sequences ξ = (ξn )n=0 of
+
positive real numbers ξn , i.e., ξ ∈ c0 if and only if ξn > 0 for all n ∈ N0 and
∞
lim ξn = 0. Moreover, by ∞ we refer to the set all bounded sequences (ξn )n=0 of
n→∞
complex numbers.
∞ ∞
Proposition 2.4.28 Let ξ = (ξn )n=0 and η = (ηn )n=0 be sequences of positive real
numbers and let γ , δ ≥ 0. Then
ξ
(a) Cγ ,δ is a Banach space,
(b) Cγ ,δ is compactly embedded into Cγ ,δ if ξ ∈ c+
ξ
0,
ξ η ξn
(c) Cγ ,δ is compactly embedded into Cγ ,δ if lim = 0,
n→∞ ηn
∞
(d) if ξn
∈ ∞, ξ η
the embedding Cγ ,δ ⊂ Cγ ,δ is continuous,
ηn n=0
ξ ξ
(e) if γ ≤ ρ and δ ≤ τ , the embedding Cγ ,δ ⊂ Cρ,τ is continuous.
Proof
∞ ξ
(a) Let (fm )m=1 be a Cauchy sequence in Cγ ,δ . Since Cγ ,δ is a Banach space,
there exists a function f ∈ Cγ ,δ with lim fm − f γ ,δ,∞ = 0. Let M :=
m→∞
sup fm γ ,δ,ξ: m ∈ N and choose, for every n ∈ N0 , a number mn ∈ N such
that fmn − f γ ,δ,∞ ≤ ξn . Then
γ ,δ γ ,δ γ ,δ
En (f ) ≤ En f − fmn + En fmn ≤ (1 + M)ξn .
ξ
Hence, the function f belongs to Cγ ,δ . Moreover, for every ε > 0, there exists
an m0 ∈ N such that fk − fm γ ,δ,ξ ≤ ε for all k,m ≥ m0 . For n ∈ N0 ,

choose a natural number kn ≥ m0 satisfying f − fkn γ ,δ,∞ ≤ ε ξn . Then, for
all m ≥ m0 ,
γ ,δ γ ,δ γ ,δ
En (f − fm ) ≤ En f − fkn + En fkn − fm ≤ 2 ε ξn , n ∈ N,
ξ
which shows that fm converges to f in the norm of Cγ ,δ .
ξ
(b) Since ξn −→ 0, we have Cγ ,δ ⊂ Cγ ,δ (cf. Exercise 2.4.23). Let M be a bounded
ξ f
subset of Cγ ,δ , say f γ ,δ,ξ ≤ c0 < ∞ for all f ∈ M. Moreover, let Pn ∈
f
Pn be a polynomial with f − Pn γ ,δ,∞ = En (f ) (cf. Exercise 2.4.24). It
γ ,δ
follows f γ ,δ,∞ ≤ c1 < ∞ and, for all n ∈ N and f ∈ M,

f f
Pn ≤ Pn − f γ ,δ,∞ + f γ ,δ,∞ ≤ c0 ξn + c1 ≤ c2 < ∞ ,
γ ,δ,∞
f
which implies Pn ∞ ≤ c3 n2 max{γ ,δ} , since, for each polynomial p ∈ Pn , we
have the Remez inequality (cf. Proposition 3.2.21)
# $
1
p∞ ≤ c sup |p(x)| : |x| ≤ 1 − 2 , c = c(n, p) . (2.4.13)
2n
In view of the Arzela-Ascoli

theorem (cf.
Theorem 2.3.5), it remains to show
that the function family f v γ ,δ : f ∈ M is equicontinuous. For this, define
min {γ , 1} : γ > 0 , min {δ, 1} : δ > 0 ,

γ0 = and δ0 = (2.4.14)
1 : γ = 0, 1 : δ = 0.
Then v γ ,δ ∈ C0,λ for λ = min {γ0 , δ0 }. For ε > 0, we can choose n0 ∈ N, such
that c0 ξn0 < 4ε , and set
1
λ
ε
η := 2(1+max{γ ,δ}) .
6 c3 n0 v γ ,δ 0,λ
C
For f ∈ M, x1 , x2 ∈ [−1, 1], and |x1 − x2 | < η, we get

|f (x1 )vγ ,δ (x1 ) − f (x2 )vγ ,δ (x2 )

f f f f
≤ f (x1 ) − pn0 (x1 ) vγ ,δ (x1 ) + pn0 (x1 )vγ ,δ (x1 ) − pn0 (x2 )vγ ,δ (x2 ) + pn0 (x2 ) − f (x2 ) vγ ,δ (x2 )
f
≤ 2En0 (f ) + pn0 C0,λ vγ ,δ C0,λ |x1 − x2 |λ
γ ,δ
f
≤ 2 c0 ξn0 + 3 n20 pn0 ∞ vγ ,δ C0,λ ηλ ≤ ε ,
where we took into account that, for a polynomial pm ∈ Pm , due to Markov’s

inequality

p ≤ (m − 1)2 pm ∞ ≤ m2 pm ∞ , (2.4.15)
m ∞
we have
1−λ
pm C0,λ ≤ pm ∞ + pm 2
∞
≤ (1 + 21−λ m2 )pm ∞ ≤ 3 m2 pm ∞ .
ξ
(c) Let M be a bounded subset of Cγ ,δ as in the proof of (b), from which we
also conclude that every sequence of elements from M has a subsequence
∞
(fm )m=1 convergent in Cγ ,δ , i.e., there exists a function f ∈ Cγ ,δ with
lim fm − f γ ,δ,∞ = 0. For each n ∈ N, choose mn ∈ N such that
m→∞
mn > mn−1 (m0 := 0) and fmn − f γ ,δ,∞ ≤ ξn and let pn∗ be a polynomial

from Pn with pn∗ − fmn γ ,δ,∞ = En (fmn ). Then
γ ,δ
∗
p − f ≤ pn∗ − fmn γ ,δ,∞ + fmn − f γ ,δ,∞ ≤ (c0 + 1)ξn ,
n γ ,δ,∞
ξ
which means that f belongs to Cγ ,δ . For ε > 0, choose an n0 ∈ N, such that

ξn
< ε for all n > n0 , and an n1 ∈ N, such that fm − f <
ηn 2 f γ ,δ,ξ +c0 n γ ,δ,∞

ε min 1,η0 ,...,ηn0
2 for all n > n1 . Then, for all n > n1 ,
εηn
En (f − fmn ) ≤ f − fmn γ ,δ,∞ <
γ ,δ
, n = 0, 1, 2, . . . , n0 ,
2
and
γ ,δ εηn
En (f − fmn ) ≤ f γ ,δ,ξ + c0 ξn < , n = n0 + 1, n0 + 2, . . .
2

Consequently, f − fmn γ ,δ,η < ε for all n > n1 .
(d) Since ξn ≤ c0 ηn , n ∈ N0 with some real constant c0 , the claimed embedding
ξ η
follows immediately from the definition of the norms in Cγ ,δ and Cγ ,δ .
(e) The assertion is a conclusion of the continuous embedding Cγ ,δ ⊂ Cρ,τ .

For a√function f : (−1, 1) −→ C, for r ∈ N, h, t > 0, γ , δ ≥ 0, and for
ϕ(x) = 1 − x 2 , define the symmetric difference of order r (cf. (2.4.7))

r r
k r
hϕ f (x) :=
r
(−1) f x+ − k h ϕ(x) , (2.4.16)
k 2
k=0
−1 + 2r 2 h2 ≤ x ≤ 1 − 2r 2 h2 , and the main part of the weighted modulus of

continuity rϕ (f, t)γ ,δ,∞ by (cf. (2.4.6))
⎧ √ −1
⎪
⎪
⎨ sup v γ ,δ (x) rhϕ f (x) : h < r 2 ,
sup −1+2r 2 h2 ≤x≤1−2r 2 h2 (2.4.17)
√ −1
0<h≤t ⎪
⎪
⎩ 0 :h≥ r 2 .
The following proposition enables us to obtain criterions for a continuous function

ξ
f : (−1, 1) −→ C to belong to Cγ ,δ .
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covered hill in the midst of forests. Two hundred coolies worked
forty-five days in clearing away vegetation and excavating the buried
terraces. Measurements and drawings were made, and twelve plates
from them accompany Sir Stamford Raffles’s work. After the Dutch
recovered possession of Java, their artists and archæologists gave
careful study to this monument of earlier civilization and arts. Further
excavations showed that the great platform or broad terrace around
the temple mass was of later construction than the body of the
pyramid, that a flooring nine feet deep had been put entirely around
the lower walls, presumably to brace them, and thus covering many
inscriptions the meanings of which have not yet been given, not to
English readers at least. Dutch scientists devoted many seasons to
the study of these ruins, and Herr Brumund’s scholarly text,
completed and edited by Dr. Leemans of Leyden, accompanies and
explains the great folio volumes of four hundred plates, after
Wilsen’s drawings, published by the Dutch government in 1874.
Since their uncovering the ruins have been kept free from
vegetation, but no other care has been taken. In this comparatively
short time legends have grown up, local customs have become
fixed, and Boro Boedor holds something of the importance it should
in its immediate human relations.
For more than six centuries the hill-temple was lost to sight,
covered with trees and rank vegetation; and when the Englishmen
brought the great sculptured monument to light, the gentle, easily
superstitious Javanese of the neighborhood regarded these recha—
statues and relics of the ancient, unknown cult—with the greatest
reverence. They adopted them as tutelary divinities, as it were,
indigenous to their own soil. While Wilsen lived there the people
brought daily offerings of flowers. The statue on the first circular
terrace at the right of the east staircase, and the secluded image at
the very summit, were always surrounded with heaps of stemless
flowers laid on moss and plantain-leaves. Incense was burned to
these recha, and the people daubed them with the yellow powder
with which princes formerly painted, and even humble bridegrooms
now paint, themselves on festal days, just as Burmese Buddhists
daub gold-leaf on their shrines, and, like the Cingalese Buddhists,
heap champak and tulse, jasmine, rose, and frangipani flowers,
before their altars. When questioned, the people owned that the
offerings at Boro Boedor were in fulfilment of a vow or in
thanksgiving for some event in their lives—a birth, death, marriage,
unexpected good fortune, or recovery from illness. Other worshipers
made the rounds of the circular terraces, reaching to touch each
image in its latticed bell, and many kept all-night vigils among the
dagobas of the Nirvana circles. Less appealing was the custom, that
grew up among the Chinese residents of Djokjakarta and its
neighborhood, of making the temple the goal of general pilgrimage
on the Chinese New Year’s day. They made food and incense
offerings to the images, and celebrated with fireworks, feasts, and a
general May-fair and popular outdoor fête.
THE LATTICED DAGOBAS ON THE CIRCULAR TERRACES.
After the temple was uncovered the natives considered it a free

quarry, and carried off carved stones for door-steps, gate-posts,
foundations, and fences. Every visitor, tourist or antiquarian, scientist
or relic-hunter, helped himself; and every residency, native prince’s
garden, and plantation lawn, far and near, is still ornamented with
Boro Boedor’s sculptures. In the garden of the Magelang Residency,
Miss Marianne North found a Chinese artist employed in “restoring”
Boro Boedor images, touching up the Hindu countenances with a
chisel until their eyes wore the proper Chinese slant. The museum at
Batavia has a full collection of recha, and all about the foundation
platform of the temple itself, and along the path to the passagrahan,
the way is lined with displaced images and fragments, statues, lions,
elephants, horses; the hansa, or emblematic geese of Buddhism; the
Garouda, or sacred birds of Vishnu; and giant genii that probably
guarded some outer gates of approach. A captain of Dutch hussars
told Herr Brumund that, when camping at Boro Boedor during the
Javanese war, his men amused themselves by striking off the heads
of statues with single lance- or saber-strokes. Conspicuous heads
made fine targets for rifle and pistol practice. Native boys, playing on
the terraces while watching cattle, broke off tiny heads and
detachable bits of carving, and threw them at one another; and a few
such playful shepherds could effect as much ruin as any of the
imaginary bands of fanatic Moslems or Brahmans. One can better
accept the plain, rural story of the boy herders’ destructiveness than
those elaborately built up tales of the religious wars, when priests
and people, driven to Boro Boedor as their last refuge, retreated,
fighting, from terrace to terrace, hurling stones and statues down
upon their pursuers, the last heroic believers dying martyrs before
the summit dagoba. Fanatic Mohammedans in other countries
doubtless would destroy the shrines of a rival, heretic creed; but
there is most evidence in the history and character of the Javanese
people that they simply left their old shrines, let them alone, and
allowed the jungle to claim at its will what no longer had any interest
or sacredness for them. To this day the Javanese takes his religion
easily, and it is known that at one time Buddhism and Brahmanism
flourished in peace side by side, and that conversion from one faith
to the other, and back again, and then to Mohammedanism, was
peaceful and gradual, and the result of suasion and fashion, and not
of force. The old cults faded, lost prestige, and vanished without
stress of arms or an inquisition.
XVI
BORO BOEDOR AND MENDOET
With five hundred Buddhas in near neighborhood, one might expect

a little of the atmosphere of Nirvana, and the looking at so many
repetitions of one object might well produce the hypnotic stage akin
to it. The cool, shady passagrahan at Boro Boedor affords as much
of earthly quiet and absolute calm, as entire a retreat from the outer,
modern world, as one could ever expect to find now in any land of
the lotus. This government rest-house is maintained by the resident
of Kedu, and every accommodation is provided for the pilgrim, at a
fixed charge of six florins the day. The keeper of the passagrahan
was a slow-spoken, lethargic, meditative old Hollander, with whom it
was always afternoon. One half expected him to change from battek
pajamas to yellow draperies, climb up on some vacant lotus
pedestal, and, posing his fingers, drop away into eternal meditation,
like his stony neighbors. Tropic life and isolation had reduced him to
that mental stagnation, torpor, or depression so common with single
Europeans in far Asia, isolated from all social friction, active, human
interests, and natural sympathies, and so far out of touch with the
living, moving world of the nineteenth century. Life goes on in
placidity, endless quiet, and routine at Boro Boedor. Visitors come
rarely; they most often stop only for riz tavel, and drive on; and not a
half-dozen American names appear in the visitors’ book, the first
entry in which is dated 1869.
I remember the first still, long lotus afternoon in the passagrahan’s
portico, when my companions napped, and not a sound broke the
stillness save the slow, occasional rustle of palm-branches and the
whistle of birds. In that damp, heated silence, where even the mental
effort of recalling the attitude of Buddha elsewhere threw one into a
bath of perspiration, there was exertion enough in tracing the
courses and projections of the terraced temple with the eye. Even
this easy rocking-chair study of the blackened ruins, empty niches,
broken statues, and shattered and crumbling terraces, worked a
spell. The dread genii by the doorway and the grotesque animals
along the path seemed living monsters, the meditating statues even
seemed to breathe, until some “chuck-chucking” lizard ran over them
and dispelled the half-dream.
In those hazy, hypnotic hours of the long afternoon one could best
believe the tradition that the temple rose in a night at miraculous
bidding, and was not built by human hands; that it was built by the
son of the Prince of Boro Boedor, as a condition to his receiving the
daughter of the Prince of Mendoet for a wife. The suitor was to build
it within a given time, and every detail was rigidly prescribed. The
princess came with her father to inspect the great work of art, with its
miles of bas-reliefs and hundreds of statues fresh from the sculptor’s
chisel. “Without doubt these images are beautiful,” she said coldly,
“but they are dead. I can no more love you than they can love you”;
and she turned and left her lover to brood in eternal sorrow and
meditation upon that puzzle of all the centuries—the Eternal
Feminine.
At last the shadows began to stretch; a cooler breath came;
cocoanut-leaves began to rustle and lash with force, and the musical
rhythm of distant, soft Malay voices broke the stillness that had been
that of the Sleeping Beauty’s enchanted castle. A boy crept out of a
basket house in the palm-grove behind the passagrahan, and
walked up a palm-tree with that deliberate ease and nonchalance
that is not altogether human or two-footed, and makes one rub his
eyes doubtingly at the unprepared sight. He carried a bunch of
bamboo tubes at his belt, and when he reached the top of the
smooth stem began letting down bamboo cups, fastening one at the
base of each leaf-stalk to collect the sap.
Everywhere in Java we saw them collecting the sap of the true
sugar-palm and the toddy-palm, that bear such gorgeous spathes of
blossoms; but it is only in this region of Middle Java that sugar is
made from the cocoa-palm. Each tree yields daily about two quarts
of sap that reduce to three or four ounces of sugar. The common
palm-sugar of the passers looks and tastes like other brown sugar,
but this from cocoa-palms has a delicious, nutty fragrance and flavor,
as unique as maple-sugar. We were not long in the land before we
learned to melt cocoa-palm sugar and pour it on grated ripe
cocoanut, thus achieving a sweet supreme.
The level valley about Boro Boedor is tilled in such fine lines that it
seems in perspective to have been etched or hatched with finer tools
than plow and hoe. There is a little Malay temple surrounded by
graves in a frangipani-grove near the great pyramid, where the
ground is white with the fallen “blossoms of the dead,” and the tree-
trunks are decked with trails of white and palest pink orchids. The
little kampong of Boro Boedor hides in a deep green grove—such a
pretty, picturesque little lot of basket houses, such a carefully painted
village in a painted grove,—the village of the Midway Plaisance, only
more so,—such a set scene and ideal picture of Java, as ought to
have wings and footlights, and be looked at to slow music. And
there, in the early summer mornings, is a busy passer in a grove that
presents more and more attractive pictures of Javanese life, as the
people come from miles around to buy and to sell the necessaries
and luxuries of their picturesque, primitive life, so near to nature’s
warmest heart.
All the neighborhood is full of beauty and interest, and there are
smaller shrines at each side of Boro Boedor, where pilgrims in
ancient times were supposed to make first and farewell prayers. One
is called Chandi Pawon, or more commonly Dapor, the kitchen,
because of its empty, smoke-blackened interior resulting from the
incense of the centuries of living faith, and of the later centuries
when superstitious habit, and not any surviving Buddhism, led the
humble people to make offerings to the recha, the unknown,
mysterious gods of the past.
THE RIGHT-HAND IMAGE AT MENDOET.
Chandi Mendoet, two miles the other side of Boro Boedor, is an

exquisite pyramidal temple in a green quadrangle of the forest, with
a walled foss and bridges. Long lost and hidden in the jungle, it was
accidentally discovered by the Dutch resident Hartman in 1835, and
a space cleared about it. The natives had never known of or
suspected its existence, but the investigators determined that this
gem of Hindu art was erected between 750 and 800 A. D. The
workmanship proves a continued progress in the arts employed at
Boro Boedor, and the sculptures show that the popular faith was
then passing through Jainism back to Brahmanism. The body of the
temple is forty-five feet square as it stands on its walled platform,
and rises to a height of seventy feet. A terrace, or raised
processional path, around the temple walls is faced with bas-reliefs
and ornamental stones, and great bas-reliefs decorate the upper
walls. The square interior chapel is entered through a stepped arch
or door, and the finest of the Mendoet bas-reliefs, commonly spoken
of as the “Tree of Knowledge,” is in this entrance-way. There Buddha
sits beneath the bo-tree, the trunk of which supports a pajong, or
state umbrella, teaching those who approach him and kneel with
offerings and incense. These figures, as well as the angels
overhead, the birds in the trees, and the lambs on their rocky shelf,
listening to the great teacher, are worked out with a grace and skill
beyond compare. Three colossal images are seated in the chapel, all
with Buddha’s attributes, and Brahmanic cords as well, and the long
Nepal ears of the Dhyani ones. They are variously explained as the
Hindu trinity, as the Buddhist trinity, as Buddha and his disciples, and
local legends try to explain them even more romantically. One
literary pilgrim describes the central Adi Buddha as the statue of a
beautiful young woman “counting her fingers,” the mild, benign, and
sweetly smiling faces of all three easily suggesting femininity.
One legend tells that this marvel of a temple was built by a rajah
who, when once summoned to aid or save the goddess Durga, was
followed by two of his wives. To rid himself of them, he tied one wife
and nailed the other to a rock. Years afterward he built this temple in
expiation, and put their images in it. An avenging rival, who had
loved one of the women, at last found the rajah, killed him, turned
him to stone, and condemned him to sit forever between his abused
partners.
A legend related to Herr Brumund told that “once upon a time” the
two-year-old daughter of the great Prince Dewa Kosoumi was stolen
by a revengeful courtier. The broken-hearted father wandered all
over the country seeking his daughter, but at the end of twelve years
met and, forgetting his grief, demanded and married the most
beautiful young girl he had ever seen. Soon after a child had been
born to them, the revengeful courtier of years before told the prince
that his beautiful wife was his own daughter. The priests assured
Prince Dewa that no forgiveness was possible to one who had so
offended the gods, and that his only course of expiation lay in
shutting himself, with the mother and child, in a walled cell, and there
ending their days in penitence and prayer. As a last divine favor, he
was told that the crime would be forgiven if within ten days he could
construct a Boro Boedor. All the artists and workmen of the kingdom
were summoned, and working with zeal and frenzy to save their
ruler, completed the temple, with its hundreds of statues and its
miles of carvings, within the fixed time. But it was then found that the
pile was incomplete, lacking just one statue of the full number
required. Prayers and appeals were useless, and the gods turned
the prince, the mother, and the child to stone, and they sit in the cell
at Mendoet as proof of the tale for all time.
With such interests we quite forgot the disagreeable episode in the
steaming, provincial town beyond the mountains, and cared not for
toelatings-kaart or assistant resident. Nothing from the outer world
disturbed the peace of our Nirvana. No solitary horseman bringing
reprieve was ever descried from the summit dagoba. No file of
soldiers grounded arms and demanded us for Dutch dungeons. Life
held every tropic charm, and Boro Boedor constituted an ideal world
entirely our own. The sculptured galleries drew us to them at the
beginning and end of every stroll, and demanded always another
and another look. A thousand Mona Lisas smiled upon us with
impassive, mysterious, inscrutable smiles, as they have smiled
during all these twelve centuries, and often the realization, the
atmosphere of antiquity was overpowering in sensation and weird
effect.
Boro Boedor is most mysterious and impressive in the gray of
dawn, in the unearthly light and stillness of that eerie hour. Sunrise
touches the old walls and statues to something of life; and sunset,
when all the palms are silhouetted against skies of tenderest rose,
and the warm light flushes the hoary gray pile, is the time when the
green valley of Eden about the temple adds all of charm and poetic
suggestion. Pitch-darkness so quickly follows the tropic sunset that
when we left the upper platform of the temple in the last rose-light,
we found the lamps lighted, and huge moths and beetles flying in
and about the passagrahan’s portico. Then lizards “chuck-chucked,”
and ran over the walls; and the invisible gecko, gasping, called, it
seemed to me, “Becky! Becky! Becky! Becky! Becky! Becky!” and
Rebecca answered never to those breathless, exhausted, appealing
cries, always six times repeated, slowly over and over again, by the
fatigued soul doomed to a lizard’s form in its last incarnation. There
was infinite mystery and witchery in the darkness and sounds of the
tropic night—sudden calls of birds, and always the stiff rustling,
rustling of the cocoa-palms, and the softer sounds of other trees, the
shadows of which made inky blackness about the passagrahan;
while out over the temple the open sky, full of huge, yellow, steadily
glowing stars, shed radiance sufficient for one to distinguish the
mass and lines of the great pyramid. Villagers came silently from out
the darkness, stood motionless beside the grim stone images, and
advanced slowly into the circle of light before the portico. They knelt
with many homages, and laid out the cakes of palm-sugar, the
baskets and sarongs, we had bought at their toy village. Others
brought frangipani blossoms that they heaped in mounds at our feet.
They sat on their heels, and with muttered whispers watched us as
we dined and went about our affairs on the raised platform of the
portico, presenting to them a living drama of foreign life on that
regularly built stage without footlights. One of the audience pierced a
fresh cocoanut, drank the milk, and then rolling kanari and benzoin
gum in corn-fiber, lighted the fragrant cigarette, and puffed the
smoke into the cocoa-shell. “It is good for the stomach, and will keep
off fever,” they answered, when we asked about this incantation-like
proceeding; and all took a turn at puffing into the shell and reinhaling
the incense-clouds. The gentle little Javanese who provided better
dinners for passagrahan guests than any island hotel had offered us,
came into the circle of light, with her mite of a brown baby sleeping in
the slandang knotted across her shoulder. The old landlord could be
heard as he came back far enough from his Nirvana to call for the
boy to light a fresh pipe; and one felt a little of the gaze and
presence of all the Dhyani Buddhas on the sculptured terraces in the
strange atmosphere of such far-away tropic nights by the Boedor of
Boro.
When we came “gree-ing” back by those beautiful roads to Djokja,
and drew up with a whirl at the portico of the Hotel Toegoe, the
landlord of beaming countenance ran to meet us, greet us with
effusion, and give us a handful of mail—long, official envelops with
seals, and square envelops of social usage.
“Your passports are here. They came the next day. They are so
chagrined that it was all a stupid mistake. The assistant resident at
Buitenzorg telegraphed to the resident here to tell the three
American ladies who were to arrive in Djokja that he had posted their
passports, and to have every attention paid you. He wished to
commend you and put you en rapport with the Djokja officials, that
you might enjoy their courtesies. Then the telegraph operator
changed the message so as not to have to send so many words on
the wire, and he made them all think you were some very dangerous
people whom they must arrest and send back. The assistant resident
knew there was some mistake as soon as he saw you.” (Did he?)
“He is so chagrined. And it was all the telegraph operator’s fault, and
you must not blame our Djokja Residency.”
Instead of mollifying, this rather irritated us the more, and the
assistant resident’s long, formal note was fuel to the flame.
“Ladies: This morning I telegraphed to the secretary-

general what in heaven’s name could be the reason you were
not to go to Djokja. I got no answer from him, but received a
letter from the chief of the telegraph, who had received a
telegram from the telegraph office of Buitenzorg, to tell me
there had been a mistake in the telegram. Instead of ‘The
permission is not given,’ there should have been written, ‘The
papers of permission I have myself this moment posted. Do
all you can in the matter,’ etc. Perhaps you will have received
them the moment you get this my letter.
“So I am so happy I did not insist upon your returning to
Buitenzorg, and so sorry you had so long stay at Boro
Boedor; and I hope you will forget the fatal mistake, and feel
yourself at ease now,” etc.
Evidently the little episode was confined to the bureau of

telegraphs entirely, the messages to the American consul, secretary-
general, and Buitenzorg resident all suppressed before reaching
them. Certainly this was no argument for the government ownership
and control of telegraphs in the United States. There were regrets
and social consolations offered, but no distinct apology; and we were
quite in the mood for having the American consul demand apology,
reparation, and indemnity, on pain of bombardment, as is the foreign
custom in all Asia. Pacification by small courtesies did not pacify.
Proffered presentation to native princes, visits to their bizarre
palaces, and attendance at a great performance by the sultan’s
actors, dancers, musicians, and swordsmen, would hardly offset
being arrested, brought up in an informal police-court, cross-
questioned, bullied, and regularly ordered to Boro Boedor under
parole. We would not remain tacitly to accept the olive-branch—not
then. The profuse landlord was nonplussed that we did not humbly
and gratefully accept these amenities.
“You will not go back to Buitenzorg now, with only such unhappy
experience of Djokja! Every one is so chagrined, so anxious that you
should forget the little contretemps. Surely you will stay now for the
great topeng [lyric drama], and the wedding of Pakoe Alam’s
daughter!”
“No; we have our toelatings-kaarten, and we leave on the noon
train.”
And then the landlord knew that we should have been locked up
for other reasons, since sane folk are never in a hurry under the
equator. They consider the thermometer, treat the zenith sun with
respect, and do not trifle with the tropics.
XVII
BRAMBANAM
“In the whole course of my life I have never met with such stupendous
and finished specimens of human labor and of the science and taste of
ages long since forgot, crowded together in so small a compass, as in this
little spot [Brambanam], which, to use a military phrase, I deem to have
been the headquarters of Hinduism in Java.” (Report to Sir Stamford
Raffles by Captain George Baker of the Bengal establishment.)
There are ruins of more than one hundred and fifty temples in the
historic region lying between Djokjakarta and Soerakarta, or Djokja
and Solo, as common usage abbreviates those syllables of
unnecessary exertion in this steaming, endless mid-summer land of
Middle Java. As the train races on the twenty miles from Djokja to
Brambanam, there is a tantalizing glimpse of the ruined temples at
Kalasan; and one small temple there, the Chandi Kali Bening, ranks
as the gem of Hindu art in Java. It is entirely covered, inside as well
as outside, with bas-reliefs and ornamental carvings which surpass
in elaboration and artistic merit everything else in this region, where
refined ornament and lavish decoration reached their limit at the
hands of the early Hindu sculptors. The Sepoy soldiers who came
with the British engineers were lost in wonder at Kalasan, where the
remains of Hindu art so far surpassed anything they knew in India
itself; while the extent and magnificence of Brambanam’s Brahmanic
and Buddhist temple ruins amaze every visitor—even after Boro
Boedor.
TEMPLE OF LORO JONGGRAN AT BRAMBANAM.
We had intended to drive from Boro Boedor across country to

Brambanam, but, affairs of state obliging us to return from our
Nirvana directly to Djokja, we fell back upon the railroad’s promised
convenience. In this guide-bookless land, where every white resident
knows every crook and turn in Amsterdam’s streets, and next to
nothing about the island of Java, a kind dispenser of misinformation
had told us that the railway-station of Brambanam was close beside
the temple ruins; and we had believed him. The railway had been
completed and formally opened but a few days before our visit, and
our Malay servant was also quite sure that the road ran past the
temples, and that the station was at their very gates.
When the train had shrieked away from the lone little station
building, we learned that the ruins were a mile distant, with no sort of
a vehicle nor an animal nor a palanquin to be had; and
archæological zeal suffered a chill even in that tropic noonday. The
station-master was all courtesy and sympathy; but the choice for us
lay between walking or waiting at the station four hours for the next
train on to Solo.
We strolled very slowly along the broad, open country road under
the deadly, direct rays of the midday sun,—at the time when, as the
Hindus say, “only Englishmen and dogs are abroad,”—reaching at
last a pretty village and the grateful shade of tall kanari-trees, where
the people were lounging at ease at the close of the morning’s busy
passer. Every house, shed, and stall had made use of carved temple
stones for its foundations, and the road was lined with more such
recha—artistic remains from the inexhaustible storehouse and
quarry of the neighboring ruins. Piles of tempting fruit remained for
sale, and brown babies sprawled content on the warm lap of earth,
the tiniest ones eating the green edge of watermelon-rind with
avidity, and tender mothers cramming cold sweet potato into the
mouths of infants two and four months old. There was such an easy,
enviable tropical calm of abundant living and leisure in that Lilliput
village under Brobdingnag trees that I longed to fling away my
“Fergusson,” let slip life’s one golden, glowing, scorching opportunity
to be informed on ninth-century Brahmanic temples, and, putting off
all starched and unnecessary garments of white civilization, join that
lifelong, happy-go-lucky, care-free picnic party under the kanari-trees
of Brambanam; but—
A turn in the road, a break in the jungle at one side of the highway,
disclosed three pyramidal temples in a vast square court, with the
ruins of three corresponding temples, all fallen to rubbish-heaps,
ranged in line facing them. These ruined piles alone remain of the
group of twenty temples dedicated to Loro Jonggran, “the pure,
exalted virgin” of the Javanese, worshiped in India as Deva, Durga,
Kali, or Parvati. Even the three temples that are best preserved have
crumbled at their summits and lost their angles; but enough remains
for the eye to reconstruct the symmetrical piles and carry out the
once perfect lines. The structures rise in terraces and broad courses,
tapering like the Dravidian gopuras of southern India, and covered,
like them, with images, bas-reliefs, and ornamental carvings. Grand
staircases ascend from each of the four sides to square chapels or
alcoves half-way up in the solid body of the pyramid, and each
chapel once contained an image. The main or central temple now
remaining still enshrines in its west or farther chamber an image of
Ganesha, the hideous elephant-headed son of Siva and Parvati.
Broken images of Siva and Parvati were found in the south and north
chambers, and Brahma is supposed to have been enshrined in the
great east chapel. An adjoining temple holds an exquisite statue of
Loro Jonggran, “the maiden with the beautiful hips,” who stands,
graceful and serene, in a roofless chamber, smiling down like a true
goddess upon those who toil up the long carved staircase of
approach. Her particular temple is adorned with bas-reliefs, where
the gopis, or houris, who accompany Krishna, the dancing youth, are
grouped in graceful poses. One of these bas-reliefs, commonly
known as the “Three Graces” has great fame, and one and two
thousand gulden have been vainly offered by British travelers
anxious to transport it to London. Another temple contains an image
of Nandi, the sacred bull; but the other shrines have fallen in
shapeless ruins, and nothing of their altar-images is to be gathered
from the rubbish-heaps that cover the vast temple court.
CLEARING AWAY RUBBISH AND VEGETATION AT BRAMBANAM TEMPLES.
The pity of all this ruined splendor moves one strongly, and one
deplores the impossibility of reconstructing, even on paper, the
whole magnificent place of worship. The wealth of ornament makes
all other temple buildings seem plain and featureless, and one set of
bas-reliefs just rescued and set up in line, depicting scenes from the
Ramayan, would be treasure enough for an art museum. On this
long series of carved stones disconsolate Rama is shown searching
everywhere for Sita, his stolen wife, until the king of the monkeys,
espousing his cause, leads him to success. The story is wonderfully
told in stone, the chisel as eloquent as the pen, and everywhere one
reads as plainly the sacred tales and ancient records. The graceful
figures and their draperies tell of Greek influences acting upon those
northern Hindus who brought the religion to the island; and the
beautifully conventionalized trees and fruits and flowers, the mythical
animals and gaping monsters along the staircases, the masks,
arabesques, bands, scrolls, ornamental keystones, and all the
elaborate symbols and attributes of deities lavished on this group of
temples, constitute a whole gallery of Hindu art, and a complete
grammar of its ornament.
KRISHNA AND THE THREE GRACES.
These temples, it is believed, were erected at the beginning of the

ninth century, and fixed dates in the eleventh century are also
claimed; but at least they were built soon after the completion of
Boro Boedor, when the people were turning back to Brahmanism,
and Hindu arts had reached their richest development at this great
capital of Mendang Kumulan, since called Brambanam. The fame of
the Javanese empire had then gone abroad, and greed for its riches
led Khublai Khan to despatch an armada to its shores; but his
Chinese commander, Mengki, returned without ships or men, his
face branded like a thief’s. Another expedition was defeated, with a
loss of three thousand men, and the Great Khan’s death put an end
to further schemes of conquest. Marco Polo, windbound for five
months on Sumatra, then Odoric, and the Arab Ibn Batuta, who
visited Java in the fourteenth century, continued to celebrate the
riches and splendor of this empire, and invite its conquest, until Arab
priests and traders began its overthrow. Its princes were conquered,
its splendid capitals destroyed, and with the conversion of the people
to Mohammedanism the shrines were deserted, soon overgrown,
and became hillocks of vegetation. The waringen-tree’s fibrous roots,
penetrating the crevices of stones that were only fitted together, and
not cemented, have done most damage, and the shrines of Loro
Jonggran went fast to utter ruin.
A Dutch engineer, seeking to build a fort in the disturbed country
between the two native capitals, first reported these Brambanam
temples in 1797; but it was left for Sir Stamford Raffles to have them
excavated, surveyed, sketched, and reported upon. Then for eighty
years—until the year of our visit—they had again been forgotten, and
the jungle claimed and covered the beautiful monuments. The
Archæological Society of Djokja had just begun the work of clearing
off and rescuing the wonderful carvings, and groups of coolies were
resting in the shade, while others pottered around, setting bas-reliefs
in regular lines around the rubbish-heaps they had been taken from.
This salvage corps chattered and watched us with well-contained
interest, as we, literally at the very boiling-point of enthusiasm, at
three o’clock of an equatorial afternoon, toiled up the magnificent
staircases, peered into each shrine, made the rounds of the
sculptured terraces, or processional paths, and explored the whole
splendid trio of temples, without pause.
Herr Perk, the director of the works, and curator of this
monumental museum, roused by the rumors of foreign invasion,
welcomed us to the grateful shade of his temporary quarters beside
the temple, and hospitably shared his afternoon tea and bananas
with us, there surrounded by a small museum of the finest and most
delicately carved fragments, that could not safely be left unprotected.
While we cooled, and rested from the long walk and the eager
scramble over the ruins, we enjoyed too the series of Cephas’s
photographs made for the Djokja Society, and in them had evidence
how the insidious roots of the graceful waringen-trees had split and
scattered the fitted stones as thoroughly as an earthquake; yet each
waringen-gripped ruin, the clustered roots streaming, as if once
liquid, over angles and carvings, was so picturesque that we half
regretted the entire uprooting of these lovely trees.
LORO JONGGRAN AND HER ATTENDANTS.
When the director was called away to his workmen, we bade our
guiding Mohammedan lead the way to Chandi Sewou, the
“Thousand Temples,” or great Buddhist shrine of the ancient capital.
“Oh,” he cried, “it is far, far from here—an hour to walk. You must go
to Chandi Sewou in a boat. The water is up to here,” touching his
waist, “and there are many, many snakes.” Distrusting, we made him
lead on in the direction of Chandi Sewou; perhaps we might get at
least a distant view. When we had walked the length of a city block
down a shady road, with carved fragments and overgrown stones
scattered along the way and through the young jungle at one side,
we turned a corner, walked another block, and stood between the
giant images that guard the entrance of Chandi Sewou’s great
quadrangle.
The “Thousand Temples” were really but two hundred and thirty-
six temples, built in five quadrilateral lines around a central cruciform
temple, the whole walled inclosure measuring five hundred feet
either way. Many of these lesser shrines—mere confessional boxes
in size—are now ruined or sunk entirely in the level turf that covers

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3.2.7 Hermite-Fejér Interpolation . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 109

5.2.3 On the Case of Variable Coefficients. . . .. . . . . . . . . . . . . . . . . . . . 262

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 1

are repeated at appropriate places. Anyway we recommend the readers to apply

Chapter 2 : Basics from linear functional analysis

Chapters 3 and 4 : Chapter 5 :

Weighted polynomial approximation Mapping properties of integral operators

Fredholm integral equations Strongly singular integral equations

Two proposals for reading parts of the book

2.1 Linear Operators, Banach and Hilbert Spaces

By boldfaced capital letters X, Y, . . . we will denote (linear) spaces (in most

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 5

A normed linear space (X, .) is automatically considered as a metric space

In this case, a sequence is convergent in (X, .1 ) if and only if it is convergent in

The boundedness of A ∈ L(X, Y) is equivalent to the continuity of the operator

An immediate consequence of this inequality is, that the map H × H −→ K,

Then the image R(Pm ) := Pm (H) is equal to Hm . Moreover, Pm : H −→ H is an

Pm f, g = f, Pm g and Pm

This implies Pm ∈ L(H, H) and Pm H→H = 1. We list further important

If this is the case, then

2.2 Fundamental Principles

then F is uniformly bounded, i.e., sup {AX→Y : A ∈ F} < ∞.

Theorem 2.2.1 Let X and Y be Banach spaces, X0 ⊂ X be a dense subset, and

An immediate consequence of the closed graph theorem is Banach’s theorem.

|f0 (x)| ≤ f0 X∗ xX = x∗ ∀ x ∈ X0

and the corollary is proved. 

Corollary 2.2.9 Let X be a normed space and X0 a linear subspace of X. If x ∗ ∈

Consequently, f1 X∗ ≤ 1. Furthermore, there are xn ∈ X0 such that

i.e., 1 ≤ f1 X∗ . Finally, we have to use Corollary 2.2.8. 

The following conclusion of the Hahn-Banach theorem is also known as

which is, equipped with the norm

a Banach space. As a consequence of the Hölder inequality

2.3 Compact Sets and Compact Operators

Uε (x) = {y ∈ E : d(y, x) < ε} .

lim sup {En (g) : g ∈ A} = 0 , (2.3.1)

where En (g) = inf {g − fn  : fn ∈ Xn }, then A is relatively compact.

Proof There exists an R ∈ R with A ⊂ UR () := {f ∈ X : f  < R}. If ε > 0

f ∞ = f ∞,E := max {|f (x)| : x ∈ E} .

A family F of functions from C(E) is called uniformly bounded if it is a bounded

|f (x1 ) − f (x2 )| < ε ∀ f ∈ F, ∀ x1 , x2 ∈ E : d(x1 , x2 ) < δ .

Theorem 2.3.5 (Arzela-Ascoli) Let E be a compact metric space. A set F ⊂ C(E)

|f (x) − f (x0 )| < ε ∀ f ∈ F , ∀ x ∈ Uδ (x0 ) .

and f ∞ = f C(I) = sup {|f (x)| : x ∈ I}.

lim fn (x) = f (x) for all x ∈ E .

Show that lim fn − f ∞,E = 0.

A normed linear space (X, .) is automatically considered as a metric space

In this case, a sequence is convergent in (X, .1 ) if and only if it is convergent in

Pm f, g = f, Pm g and Pm

This implies Pm ∈ L(H, H) and Pm H→H = 1. We list further important

then F is uniformly bounded, i.e., sup {AX→Y : A ∈ F} < ∞.

|f0 (x)| ≤ f0 X∗ xX = x∗ ∀ x ∈ X0

and the corollary is proved.

Consequently, f1 X∗ ≤ 1. Furthermore, there are xn ∈ X0 such that

i.e., 1 ≤ f1 X∗ . Finally, we have to use Corollary 2.2.8.

where En (g) = inf {g − fn : fn ∈ Xn }, then A is relatively compact.

Proof There exists an R ∈ R with A ⊂ UR () := {f ∈ X : f < R}. If ε > 0

f ∞ = f ∞,E := max {|f (x)| : x ∈ E} .

and f ∞ = f C(I) = sup {|f (x)| : x ∈ I}.

Show that lim fn − f ∞,E = 0.

lim sup {(An − A)xY : x ∈ M} = 0 ,

and the corollary is proved.

proves the equivalence of the two norms.

and the lemma is proved.

L2α,β for all k = 0, . . . , r, where ϕ(x) = 1 − x 2 . Moreover, the norm f α,β,r is

Hence, by induction, lim fn − Pn − g∞,[a,b] = 0 and, since fn −→ f in

aIWp,r p,r ≤ ca r

with a constant c = c(a).

and the corollary is proved.

fg ∈ C0,λ [a, b] and fgCr,λ [a,b] ≤ f C0,λ [a,b] gC0,λ [a,b] .

with c = c(x1, x2 , y).

instead of .γ ,δ,ξ , respectively.

which proves the claimed embedding.

follows f γ ,δ,∞ ≤ c1 < ∞ and, for all n ∈ N and f ∈ M,