ADVANCES IN APPLIED MATHEMATICS 12, 215-234 (1991)

The Rational Approximation of Circuit Design

DANIEL E. WULBERT
University of California, San Diego, La Jolla, California 92093

A characterization is proved for approximations from classes of functions that

arise in circuit theory. The approximating families include gain functions and
rational functions satisfying stability and passivity restraints. 0 1991 Academic
Press, Inc.

1. INTRODUCTION

Approximation by classes of rational functions arises in the design of

electrical filters. This paper is an analysis of such circuit design motivated
approximates. Although, the connections to electrical engineering are of
interest, the paper is intended to stand alone as a mathematical paper,
and we use mathematical conventions and notation.
We consider two kinds of approximation and combinations of four
hypotheses. Let f be a real continuous function on [a, b]. We seek an r
satisfying a combination of hypothesesbelow that also:

(A2) minimizes Ilf - lr1211,or

(8) minimizes Ilf - %rll.
Suppose r is in R;(C). The additional hypothesis are:

(P) (passivity) r(x) 2 0 for all real x,

(R) (reflexive) r(x) is real for all imaginary x, and
(S) (stability) r has no poles in the open upper half plane.
(G) (gain functions) either:
(i) 1 2 %r(x) 2 0 for all real X, (for (A2) approximation), or
(ii) 1 2 Ir(x)l 2 2 0 for all real x (for (8) approximation).

Before describing the results, we state two consequences. One shows

that in the most regular cases, the approximates behave as in the classical
215
0196~8858/91 $7.50
Copyright Q 1991 by Academic Press, Inc.
All rights of reproduction in any form reserved.
216 DANIEL E. WULBERT

Rr setting. The other illustrates behavior not anticipated from classical

results.
COROLLARY. Let p and q be polynomials such that:
(i) p and q have no common factors,
(ii) p and q have no real zeros,
(iii) deg p = deg q.
Then p(x *)/4(x *) E Riz is a best approximation to a continuous real f from
%PRSR$C) if and only ifp(x)/q(x) is a best approximation to f from RI.
EXAMPLE. There is a continuous function real f and a rational func-
tion r in %PRSR$C) such that r is a best approximation to f from
3PRSR$C), but -r is not a best approximation to f - 2r (although the
settings produce identical error functions).
The combinations of the above approximation problems are reduced to
a few variations of one problem. We characterize the best approximations
and employ the characterization to prove: when best approximations are
unique; when local best approximations are global; and at which functions
the best approximation operator is continuous.
Section three contains technical results used to represent polynomials
and rational functions that satisfy the restraints. Section four reduces the
approximation problems to approximation from a family, Qr (definitions
are in the notation section), and variations of it. The families involved with
‘3%are not closed and hence do not admit best approximations to all
continuous functions. Therefore all the approximations considered here
are from the closures of these subclasses of rational functions. The
closures are characterized in Section four. A simple characterization arises
only if n I m + 1, and that is the only case considered. (cf. [Wulbert,
19781for the kinds of problems that arise when n > m + 1.)
Section five contains the characterizations from Q: and its subfamilies.
The first is a perturbation characterization that says that r is a best
approximation to f if and only if the error function, f - r, has zero as a
best approximation from a situation specific convex set. Proving this for a
convex set with the “right” properties is the theoretical base for all the
results. However the set is cumbersome for authenticating a specific
approximation. Alternation theorems are derived from the perturbation
theorem. The alternations are more complicated then in the classical
setting. The number of alternations required is reduced by an alternation
behavior (off the domain of definition of the functions) at the zeros and
poles (including at infinity) of r. To show that the alternation is not as
unwieldy as its statement appears, an example of a specific approximate r
is defined on [O,11.The alternations on [O,11required to verify that r is a
 CIRCUIT APPROXIMATIONS 217

best approximation from 15 different families of approximates are tabu-

lated. Section six includes the corollaries to the characterizations, such as
the unicity of best approximations.

2. NOTATION AND DEFINITIONS

All functions are normed with the supremum norm. For a function f
crit(f) = {x: If(x)1 = Ilfll} and Z(f) = (x: f(x) = O}
are called the critical point set and the zero set of f, and
1 if f(x) > 0,
wf(x) = 0 iff(x) =O,
i -1 if0 >f(x).
The real and imaginary parts of a function f are abbreviated %if and D:f,
and
%K={%k:kEK) and 23K = {Bk: k E K}.
For a subset of functions K
dist( f, K) = inf( llf - kll: k E K} .
A member k in K is a best approximation to f if,
Ilf - kll = dist( f, K).
If there is a neighborhood U of k such that k is a best approximation to f
from U n K, then k is a local best approximation to f from K. The closure
of K is abbreviated cl K.
If E is a property of functions (such as G, P, R, or S from the
Introduction) then
EK = {k E K: k satisfiesE}.
The families of rational functions used in this paper are:
P,, = ( p a polynomial with real coefficients, and deg p I n} ,
P,,(C) = { p a polynomial with complex coefficients, and deg p I n} ,

Q;= ~:pEP,,q~P,,andq>Oforallrealnumbers
9
218 DANIEL E. WULBERT

We define R;(C) analogously. We assume that if p/q E Rr, then p and

q have no common factors.
Domain of finctzbm. All the functions are defined on subsets of the
real line. However, often properties of the approximates, which are
polynomials and quotients of polynomials, on other points of the complex
plane are used. We will assume that the functions are defined on a real
interval [a, b]. If we are approximating by functions with property R then
we assume that a > 0. It is not essential that the domain be an interval.
We could use any compact subset of reals (or the nonnegative reals for
property R).

3. REPRESENTATION OF APPROXIMATES

We use this section to rewrite the classes of polynomials and rational

functions satisfying the extra conditions P, R, and S. The first two lemmas
are elementary observations.
LEMMA 3.1.
RP,, = RPzn+r = {p(x2): p E P,}.

LEMMA 3.2.

(9 RP2,(C) = {p(x’) + &3(x2): p E P,, t E Pnpl},

(ii) RP,,+,(C) = {p(x2) + &(x2): p E P,, t E Pn},
(iii) p E RP,(C) = {w E Z(p) = -I? E Z(p)}.

LEMMA 3.3. RR,“(C) = {(p/q): p E RP,(C), q E RP,(C), q # 0 on

[O,111.
Proof. Suppose that

P(x) + it(x)
r(x) = E RR,“(C),
4(x) + fi(x>
where p, t, q, and s are polynomials that are real on the imaginary axis.
Since T(X) E RR,“(C), for any purely imaginary number, z, [p(z) +
it(z)l/[q(z) + is(z)1 must be real, and the argument of the numerator
must equal that of the denominator. Except where s or t vanish, p/t =
q/s, and in all cases ps = qt. Let fi and f2 be the common factors (if
they exist) of p and t and of q and s, respectively. Hence there are
polynomials, fi, f2, pl, si, ql, and t, all real on the imaginary axis and
 CIRCUIT APPROXIMATIONS 219

such that
p + it = fl[ p1 + it11 and 4 + fi =f2[q1+ &I.
By absorbing a constant into the factors fi and f2, we may also assume
that p1 and q1 are manic polynomials. So for all purely imaginary
numbers, and hence for all numbers,
PS = 4t * PlS, = q,t, ==+.
Pl = 41 and t, = sl.
We conclude that r = fi/f2, and this proves the representation. 0
LEMMA 3.4. Zf p E P, + I and q E P,,+ I are polynomials with no com-
mon factors, and if

M= max{n + degp, m + degq}

then
pP, + qP, = PM.

Comment. This lemma is well known and can be proved by computing

the dimensions of both spaces (cf. [3 or 11, Lemma 1.11).Given w E PM, a
constructive method to find polynomials u E P, and u E P,,, with pu +
qu = w is to equate coefficients of like powers of x on both sides of the
equation and solve for the coefficients of u and U. This produces M
equations in m + n + 2 unknowns. Although the equations are not linear
they can be solved iteratively beginning with the constant coefficients.
LEMMA 3.5.

(9 PP,, = PPzn+r = (42 + s2: q E P,, s E P,),

(ii) PRP,, = PRP,,,, = {q2(x2) + x2s2(x2): q E P,, s E P,-,},
(iii) PRP4n+2 = PRP4n+3 = {q2(x2) + x2s2(x2): q E P,, sin Pn}.

Proof. Part (i) is proven in [121. Parts (ii) and (iii) are similar. We will
prove part (ii). A quadratic polynomial in PRP, is of the form x2 + cx,
and the result is satisfied by taking q = 6 and s = 1. Irreducible
fourth-degree nonnegative polynomials in PRP, can be written:
(x + a + ip)(x + CT-i/3)(x - ff + ip)(x - a$). (3-l)
This is equal to

[x2 - (p” + (Y2j2 + x2[2p12, (3.2)

which also has the proposed form.

220 DANIEL E. WULBERT

Now consider

[ a’( x’) + nW( x”)] [ a*( X2) + x’p”( x”)] . (3.3)

This is equivalent to

[a( +X*b(X*)p(x*)]*+x*[a(x*)b(x’) - a(x’)p(x*)]‘.

(3.4)
Now suppose we are given a polynomial in PRP’,. Line (3.2) shows that its
irreducible factors are of the stated form, and line (3.4) shows that the
products of these factors retain the same form.
LEMMA 3.6.

(0 w3Rf(C) = m?;(C),
(ii) %SRR;(C) = !ltRR,m(C),
(iii) asPR;(c) = %PR;(C),
(iv) %SPRR;(C) = %PRR,m(C).

Proof. L&t
p + it
r* = - E R,“(C).
q+is

We are going to construct an r E SRz(C) with ‘8tr = ‘Br *. It will of

course suffice to prove the result for rational functions r* which have no
real factors. Hence we will assume that q + is has no real zeroes.
Suppose that

where
3Pj < O, 3uj > 0, 3Yj < O, (3.6)
and
{Pj} n Icj} n (Yjyij = 0*
Let
41 + is1 = n(Z -Pj>nCZ - 'j;.)* (3.8)
 CIRCUIT APPROXIMATIONS 221

Then on the real line,

1 Wl + ts,
!Jtr* = (3.9)
l7( Z - rj)( Z- Tj) 4: + sf *

Finally let qz and s2 be polynomials in P,, defined by

q2+iS2= n(Z-CLj)n(Z-~j)n(Z-Yj>. (3.10)

By (3.10) qz and sr have no common factors. (A zero common to both of

these real polynomials would have its conjugate be a zero to both. Then
both would be zeros for q2 + is,. However, by its definition q2 + is, has
no conjugate pairs of zeros.) By Lemma 3.4 there are u and u E P,,, such
that
q2u + s2u = pq1 + ts,. (3.11)
SO

u + iv q2u + s2u
!J? -=
q2 + is2 422 + s22

s;+s;
= l-I( z-Pj)(z-cLj)n(z- vj)(Z-~j)Il(Z-~j)(Z - yj) ’

= (45 + Sf)TI( Z - rj)( z - rj) ’

= !Br*. (3.12)
We let
u + iv
r=-
q2 + is2

to finish the proof of part (i).

Now suppose that r* E RR,“; we want to show that the constructed
function r is also in RR:. By Lemma 3.3 we assume that the numerator
and denominators p + it and q + is are in R P,(C) and RPJC), respec-
tively. Since a polynomial is in RP,(C) if and only if its zeros are
symmetric about the imaginary axis (i.e., z is a zero if and only if -Z is a
zero), we have that both
q1 + is, and q2 + is, E RP,(C).
We need to conclude that u + iv E RP,,,(C). From the representation
222 DANIEL E. WlJLBERT

(Lemma 3.2) of members of RP,(C) we can rewrite q2 + kFz in the form

q&a + ~s&2) and observe that pqr + ts, is a polynomial in x2. We
still have that q&x2) and xs,(x2) have no common factors. It follows that
q&x2) and x*s,(x*) a1so h ave no common factors. There are polynomials
ui and ui of appropriate degree such that
q3(x*)u1(x2) +x*s&2)u1(x2) =pql + ts,.

We let
U(X) = u1(x2) and u(x) = q(2),
and again r = [u + iu]/[q2 + kz] satisfies the conditions proving part (ii>.
Since both r* and r have the same real part, it is trivial that
r* E PR,” implies r E PR,“.
This proves part (iii), and hence part (iv). 0
The last result of this section is an observation about the form of the
subclassesof the rational functions Qr.
LEMMA 3.7.

(9 RQ,“= $:p~RP,,q~PRP,,qfOon[a,b] ,
( 1

(ii) PQ,” = ~:pEPP,,qEPP,,qfoon[a,b] ,

i I

(iii) PRQ,” = ;: p E PRP,, q E PRP,, q f 0 on [a,b] ,

I

(9
The subcases of GQ,” and GRQ,” with p = 1 are themselves of inter-
est.

4. REDUCTION OF APPROXIMATES TO Qr

We use the representations of the last section to show that approxima-

tions, by both the real parts of classes of rational functions and by the
 CIRCUIT APPROXIhJATIONS 223

squares of absolute values of such functions, are equivalent to approxima-

tions by classesof Qr.
PROPOSITION 4.1.

(9 lR;(C)12 = lSR,m(C)12 = PQ;,“,

(ii) IRR;;(C)12 = IRSR;;(C)12 = PRQ;,“.
Proof.
p+it 2 q2 + t2
vqC)12 = p+is = ~ (4.1)
I I q2 + s2 ’
where p and t E Pm, and q and s E P,,. By Lemma 3.5(i) this is equivalent
to PQ;,“. Using Lemmas 3.2, 3.3, and 3.5,

q(2) + ixt(x”) 2 q2( 2) + XV( x2)

IRR;;(C)l’ = = PRQ;,“.
q( x2) + ks( x2) = q2(x2) + x2s2(x2)
(4.4
Now suppose that
p2 + t2
E IR,“(C)l”.
q2 + s2
We wish to show that it is also in ISZ?F(C>I2. Suppose that

q(Z) +lS(Z)=,fJ(zmajeiPj)*

Put

q*(Z) + is*(Z) = ,fJ(Z -“j + i&l).

Although q* + Is* has no zeros in the upper half-plane,

14*(Z) + iS*(Z)12 = ,fi(Z - “j + i18,1),~I(i - aj - ilPjl)Y

= ,fi (Iz - ajl2 + PT),

=,~(z-~j+ia,)~~(i-~j-iaj),

= lq2( z) + s2(z)12. (4.3)

224 DANIEL E. WIJLBERT

This proves that

IR,“(C)l’ c ISqC)12.
For the final inclusion of the proof suppose that q + is above is real on
the imaginary axis. We want to conclude that q * + is* shares this prop-
erty. A polynomial is real on the imaginary axis if and only if its zeros are
symmetric about the imaginary axis, Lemma 3.2(iii). The construction of
q* + is* reflects about the real axis all zeros with positive imaginary part.
It leaves others undisturbed. Hence q * + b * is real on the imaginary axis,
and

lRR;f(C)12 c lRSR;;(C)12. 0
The real parts of the classes of rational functions do not form a closed
set. The next result identifies the closures with spaces of the form QT.
The results are only tidy when n I m + 1. In [12] a characterization is
given for cl %Rr for all m and n, and the statement of part (i) below
(with II I m + 1) is derived from it there. But by immediately restricting
ourselves to n I m + 1 we will build a less technical and more pleasant
proof.
PROPOSITION 4.2. For n I m + 1:

(9 cl ‘M;(C) = cl %SRn”(C) = Q;+“,

(ii) cl %PR;;(C) = cl %PSR;;(C) = PQ;,“+2n,
(iii) cl %RR;;(C) = cl !RRSR;,m(C) = RQ;;+2n,
(iv> cl %PRR;;(C) = cl !J?PRSR;;(C) = PRQ,2,“+2”.

Proof The proofs follow the same pattern for all cases. We will show
that cl %RRzz(C) = RQ,m,“12.Since

p( x2) + kt( x2) p( x2)4( x2) + x23( x2)t( x2)

3 (4.4)
q(2) + ius = q2( x’) + x2s2(x2) ’
%RR;;(C) c RQ2m,++“. (4.5)
Because R Qz++” is closed we have the inclusion of sets in one direction.
To show the converse conclusion we first assume that q(x) and MT(X)have
no common zeros and that max{deg q, 1 + deg s} = n. Then since n I
m + 1 by Lemma 3.4,

dim(pq+xst:pEP,,tEP,+,) =m+n+l. (4.6)

 CIRCUIT APPROXIMATIONS 225

so

{p(x2)q(x2) +x2s(x2)t(x2): p E Pm, t E P,,,-l} = RP2m+2,. (4.7)

Again using Lemma 3.4,

RP2,+2n
c cl %RR;;(C). (4.8)
q2( x”) + x22( 2)

From Lemma 3.5,

PRP,, = (q2(x2) + x2s2( x2): deg q I n, deg s 5 12- I}. (4.9)

The proof will be completed by showing that line (4.8) remains valid even
when q and s have common zeros or max{deg q, 1 + deg s) < n. Since
members of PR Pan are the uniform limit of functions with those two
properties, there exist {dj) c PRP,, which converge uniformly to q2(x2)
+ x2s2(x2). Since q2(x2) + x2s2(x2) has no zeros on [a, b], we have that

f E fiRR;;(C) for all g E RP2,+2,, (4.10)

J

g converges to
g (4.11)
Tq q2( x’) + xV( x”) *

so
g
E cl %RZ?;;(C), (4.12)
q2( x2) + x2s2( x’)

and

RQZ,+”c cl ‘%RR;;(C). 0 (4.13)

5. CHARACTERIZATION OF APPROXIMATIONS FROM RQ,”

To approximate from spaces of the type Qr we replace the space of

approximates by a related family that is easier to manipulate. For example
suppose that
p2 + t2
r=- E PQ;,“.
q2 + s2
226 DANIEL E. WULBERT

Then the mapping

maps P,, x P,, x P,, x P,, onto PQz,.

2n Taking the Frechet derivative, we
compute the tangent space at [p2 + t21/[q2 + s21 to be

2ph, + 2th, _ (P2 + f2>

,[2qh,+2sh,]:hjEP,,j=1...4
q2 + s2 (q2 + s2)

Using Lemma 3.4 several times we have that if: 111each of the three
pairs (i) p and t, (ii) q and s, and (iii) p2 + t2 and q2 + s2 are relatively
prime, and [2] both the numerator and the denominator of r have degree
2n, then the tangent space equals P4,/(q2 + s212.It then follows [2 or 111
that r is a best approximation from P&z,” to a continuous function f if
and only if there are 4n + 2 points, where the error function has maximal
amplitude but with alternating sign. (Equivalently, r is the unique best
approximation to f from R$;.) However, it is an involved argument to
prove a characterization without the above restrictions, and the statement
of the characterization is not as simple as the special case above. The
proofs are parallel to those in [12] and are not included.
We need some technical definitions to state the characterization theo-
rems. Although necessary for the statement, they become less intrusive
when applied to specific functions. For instance, the example given after
the characterization theorem does not involve such technicalities. Suppose
that r E RQ;,“. From Lemmas 3.7 and 3.1 we may assume that r(x) =
p(x2)/q(x2) for appropriate polynomials p and q. We may assume that p
and q have no common quadratic factors, but they may have some
common factors which if canceled would leave a rational function that is
not of the form stated for RQZ,“.
Let F, p,,, and q,, be polynomials so that p. and q,, are manic and have
no common factors and so that

P = FP, and q = ho. (5-l)

Since q E PP,, any real zero of q is a double zero. Since p and q have no
common quadratic factors, Z(q) and Z(q,) intersect the real line in
identical sets. It also follows that F has no zeros on [a, bl. These
observations are relevant to a comparison with the results in [12].
Put
M = max{deg q,, + m, deg q. + n} = dim{q,P,,, + POP,) - 1. (5.2)
 CIRCUIT APPROXIMATIONS 227

Define

Z(q,) n Reals, if n + deg p

-qp,q) = <rn + degq; (5.3)
{Z( q,,) n Reals] u {m} u { -m} otherwise,

and

z(q,) n KW7 if n + degp 5 m + degq;

WPdl) =
{Z(q,) n [O,m)} u (m} otherwise.
(5.4)

Now let

H( p, 4) = (h(x): h E PMand sgn h(x) = - sgn po( x)

for x E Z( p, q)} (5.5)

and
PH(p,q) = {h(x2): h E PM andsgnh(x) = -sgnp,(x)
for x E PZ( p, q)}. (5.6)

We also need

qp,q) = H(p,q) n H(q,d (5.7)

and
Pqp,q) = pH(p,q) n pH(qd). (5.8)

LEMMA 5.1. r(x) = p(x)/q(x) is a best approximation to a continuous

function f from Qr if and only if for each h in H(p, q>,

Ilf - rll I Ilf - r - hll.

LEMMA 5.2. r(x) = p(x2>/q(x2) is a best approximation to a continu-
ous function f from RQ;,” if and only if for each h in PH(p, q),

Ilf - rll < Ilf - r - Ml.

Approximation from PQ2, 2m has a technical problem. That is, suppose
that a continuous f has the value - 1 at 1. For example let f = - 1x1.
Then a lot of functions, for example, r,(x) = cx2, 0 I c I 1, can be best
approximations, and none of these exhibit conceivably interesting altema-
228 DANIEL E. WULBERT

tion characterizations. Such a characterization is a goal of this section. A

general idea in the proof is that if r does not have “enough” alternations,
one wishes to “perturbate” r with “an acceptable function” at the critical
points in the direction of the function being approximated. We make two
obvious adjustments. We make an hypothesis.
HYPOTHESIS FOR PQ,” AND PRQ;,” APPROXIMATION. when character-
izing an approximation, r, from PQ,” or PRQ$,” to a continuous function,
f, we will assumethat either :
[ll f 2 0 on 10,11,or
[21 r > 0 on [O,11.
LEMMA 5.3. Under the PQ,” hypothesis, r(x) = p(x)/q(x) is a best
approximation to a function f from PQ,” if and only if for each k in K(p, q),

Ilf - rll 5 Ilf - r - kll.

LEMMA 5.4. Under the PRQZZ hypothesis, r(x) =p(x2)/q(x2) is a

best approximation to a function f from PRQZ,” if and only if for each k in
PK( P, q),
Ilf - rll I Ilf - r - kll.
Similarly if we are approximating by the members of Q that are
bounded between zero and one on the entire real line (GQ,” and GRQr>
we make some restrictions.
HYPOTHESIS FOR GQ,” AND GRQZ,” APPROXIMATION. When charucter-
izing an approximation, r, from GQ,” or GRQZ,” to a continuous function,
f, we will assume that either:
Ill 1 21f 2 0 on LO,11, or
[2] 1 > r > 0 on [0, 11.
LEMMA 5.5. Under the GQ,” hypothesis, r(x) = p(x)/[p(x) + q(x)] is
a best approximation to a function f from GQ,” if and only if for each k in
KCP, q),
Ilf - rll I Ilf - r - kll.
LEMMA 5.6. Under the GRQ;,” hypothesis, r(x) =p(x2)/[p(x2) +
q(x2)] is a best approximation to a function f from CR&z,” if and only if for
each k in PK(p, q),

Ilf - rll I Ilf - r - kll.

 CIRCUIT APPROXIMATIONS 229

The perturbation lemmas above are used to prove more verifiable

alternation type characterization. We will need yet more technical defini-
tions. Put
D = Z(p,q) U crit(f- r),
PD =PZ(p,q) U crit(f- r),
(5.9)
E = Z(p,q) U Z(q,p) U crit(f- r),
PE = PZ(p,q) U PZ(q,p) U crit(f- r).

The “P” designation denotes the use of the nonnegative real axis in
the domain of the functions. It is not linked with the function spaces
having property P. In fact it will be associated with those spaces having
property R.
Define [, P[, 5, and Pt on D, PD, E, and PE, respectively, with values
in (-1,l) by

-wvdx) for x E Z(p,q),

5= for x E crit(f-
i sgn(f- r)(x) r).

-smp,(x) forx Epz(~,q),

Pl =
i sgn(f- r)(x) forx E crit(f- r).

-sgn PO(x) forx E Z(p,q), (5.10)

5= sgn 4dx) for x E Z(q, p),
1 sgn(f- r)(x) for x E crit(f- r),

-sgn PO(x) for x E PZ(p,q),

P5 = w 4dx) for x E PZ(q,p),
1 sgn(f- r)(x) for x E crit(f- r).

THEOREM 5.7. p(x)/q(x) is a best approximation to a continuous f

from Qr if and only if th ere are M + 2 points (xj},!!~’ c D such that

0 IX, <Xl < *** <xM+l 503 and l(xj) = -ll(xj+l)*

THEOREM 5.8. p(x2)/q(x2> is a best approximation to a continuous f

from RQZ,” if and only if there are M + 2 points (xj},eil c PD such that

OIx,<x,< *+a <xw+lIm and pl(xj) = -1p5(xj+l)*

THEOREM 5.9. Under the PQ,” hypothesis, p(x)/q(x) is a best approxi-

mation to a continuous f from PQ,” if and only if there are M + 2 points
230 DANIEL E. WULBERT

(Xj}~~~’ c E such that

0 < x0 < x1 < * * * < xM+l I OJ and (("j) = -15(xj+l)*

THEOREM 5.10. Under the PRQZZ hypothesis, p(x’>/q(x*) is a best

approximation to a continuous f from PRQZ,” if and only if there are A4 + 2
points {xi},?:’ c PE such that

0 I x0 < x1 < - * - < xM+l I CQ and pS(xj) = -lpS(xj+l)*

THEOREM 5.11. Under the GQ,” hypothesis, p(x)/[p(x) + q(x)] is a

best approximation to a continuous f from GQ,” if and only if there are
A4 + 2 points {xi},?:’ L E such that

0 IX, <x, < --* <xM+l I CQ and 5(xj) = -15Cxj+l)-

THEOREM 5.12. Under the GRQ,,2m hypothesis, p(x*>/[p(x) + q(x*>l

is a best approximation to a continuous f from GRQ;,” if and only if there
are M + 2 points {xj},?il c PE such that

0 I x0 < x1 < . . . < xM+l I * and p5(xj) = -lpS(xj+l)*

DEFINITION. The sequences of points, {xi},?:‘, in the above theorems

are called points of alternation.
PROPOSITION 5.13. Every continuous function, f, on [a, bl has a best
approximation from each of the spaces Qr, PQ,“, RQ,“, PRQ,“, GQ,“, and
GRQ,“.
Proof This is a standard argument. Suppose that

r.I = ~ Pj E Qr,
qi’ + si”

and that
llrj - f II * min{llm - f II: m E Qr}.

By absorbing a constant into the numerators we may assume that the

norms of the denominators equal one. Hence the norms of qj and sj are
bounded. Since llrjll is bounded, so is IIp,II. There are subsequences of pi,
qj, and sj-which we assume we already have-that converge to say p, q,
and s. Since llr,ll is bounded, every zero of q2 + s2 is a zero of p,
Cancelling the corresponding factors still leaves rational function r in QF.
That is, the denominator is positive and free of zeros on [a, bl. Further-
 CIRCUIT APPROXIMATIONS 231

more, on the entire complex plane, except the finite set Z<q,? + s,?), rj
converges pointwise to r. It follows that r is a best approximation to f,
that r is (i) nonnegative on the real line, (ii) bounded by 1 on the real line,
and (iii) real on the imaginary axis if each rj is. 0

6. CONSEQUENCES

We present corollaries and examples that come from the characteriza-

tion theorems. These applications will be free from the technical defini-
tions used in the characterizations. The first result is an example of the
characterization theorems applied to a specific function as an approximate
from several spaces of approximates. The characterizations reduce to
simply counting alternations on [O,11.
EXAMPLE 6.1. We compare the alternations theorems using a specific
function. From Lemma 3.7 and Lemma 3.5:

(x2 - 4)2
r(x) = 2 E PRQ,4 c RQ: c R:.
(x2 - 3)

Approximating a continuous function on [O,11by r, we expect, and obtain,

different characterizations depending on the class of approximates being
assumed. The different characterizations are tabulated below. By inspect-
ing r, we automatically get some points of alternation at f. fi and +2,
the zeros of the numerator and denominator of r. In some cases we
automatically get points of alternation at the points at infinity. The rest of
the alternations (those listed in Table I) are on the critical point set in
[O,11.
COROLLARY 6.2. Every continuous function on [a, b] has a unique best
approximation from Qr and from RQ,“. Every nonnegative continuous
function on [a, bl has a unique best approximation from PQ,” and from
PRQ,“. Every continuous function, f, on [a, b] such that 1 2 f 2 0, has a
unique best approximation from GQ,” and from GRQ,“.
Idea of Proof. The idea in this well-known style of proof is to show
that the difference of two best approximations must switch sign “an
impossible” number of times on the points of alternation (cf. [12]).
COROLLARY 6.3. Zf r E Qr, RQ,“, PQ,“, PRQ,“, GQ,“, or GRQ: is a
local best approximation to f then it is the global best approximation.
Proof If r is a local best approximation to f, then for sufficiently small
t > 0, r is a global best approximation to tf + (1 - t)r. Hence the
232 DANIEL E. WULBERT

TABLE I

Approximates Necessaty and sufficient critical point alternations

PRQ:
RQ: 4 alternation points, the first having a minus sign

PQ:
Q:
5 alternation points, the first having a plus sign
7 alternation points, the first having a minus sign

R"4
PRQ:
9 alternation points, the first having a plus sign
10 alternation points

PRQ:
RQ:
5 alternation points, the first having a plus sign
6 alternation points, the tirst having a minus sign

RQ:
PQ:
6 alternation points, the first having a minus sign
7 alternation points, the first having a plus sign

PQ,"
4"
9 alternation points, the first having a minus sign
11 alternation points, the first having a minus sign

net 13 alternation points, the first having a plus sign

13 alternation points, the first having a plus sign
14 alternation points
14 alternation points

alternation characterization applies to r and tf + (1 - t)r - r = t<f - r).

Of course f - r exhibits the same alternation pattern, and this identifies r
as the best approximation to f. This argument works for PQ,” approxima-
tion as long as r is not equal to 0 at a critical point of f - r, where f < 0.
But, if r is zero at such a point, no function in PQ,” can be a better
approximation, and r is a global best approximation.
COROLLARY 6.4. Suppose p and q are in P,, that deg p I deg q and
that q # 0 on [O,oo]. Then p/q is a best approximation to f from Rt if and
only if p(x2>/q(x2) is a best approximation to f from RQ;,“.
COROLLARY 6.5. Suppose that p and q are in P,,, that deg p = deg q,
and that p and q are not = 0 on [O, 4. Then p/q is a best approximation to f
from R: if and only if p(x2)/q(x2> is a best approximation to f from
PRQ;,“.
COROLLARY 6.6. For a constant function k the following are equivalent :
i k is a best approximation to f from Ri,
ii k is a best approximation to f from Qi,
iiik is a best approximation to f from PQ,“, k > 0,
ivk is a best approximation to f from R Q,‘:,
v k is a best approximation to f from PRQ:,“, k > 0,
vi k is a best approximation to f from GQ,“, l>k>O,
vii k is a best approximation to f from GRQf,“, 1 > k > 0, and
 CIRCUIT APPROXIMATIONS 233

viii there are n + 2 successive points IXjI C Ia, bl suchthat

I(f- k)(xj)l = Ilf- kll lZnd (f- k)(xj) = -(f- k)(xj+I)*

COROLLARY 6.7. The best approximation operators associated with Qz,

m RQ;,“, PRQZ,“, GQ,“, or GRQS,“, respectively, is continuous at
;(g 0 on [a, b], for PQ,” or PRQ:,” and 1 > f > 0 on [a, bl for GQ,”
and GRQ,~,“) if and only if the best approximation to f cannot be represented
as a member of QrIt, PQ,“-i’, RQ,‘:::, PRQ,“,m_;‘,PQ,“-i’, or GRQZr---;,
respectively.
The proof for the above argument is not obvious but parallels the proof
in [12].

7. CONSTRUCTION OF APPROXIMATION

Since !JIRSRr(C) is not closed, there may not be a best approximation

to a,particular continuous function f. If we do know the best approxima-
tion, r,,, from the closure cl %RSRr(C) = RQ,” and E > 0 is given, then
we can construct an r E RR:(C) with no poles in the upper half plane for
which II%r - roll < E. Suppose that r,, = p/q and E > 0. A procedure is
as follows:
1. Let

6 < min{llx’” + Ill,clly(,” Ij~~) and 41 ‘4 - 6(x2” + 1).

Now q1 has degree 2n, no zeros in common with p, and is such that
IIP/4 -P/4111 < E*
2. Use the construction in the proof of Lemma 3.5 to find q2 and s2
so that
q1 = q$( x’) + x”s,‘( x2).

3. Using the comment after Lemma 3.4, find p2 and t, satisfying

P = 1)2q2 + x2t2s2.

4. Now
P2G2) + MX2)
r2 = E RR;(C)
q2(x2) + k2(x2)
and
Il%r, - roll < E.
234 DANIEL E. WULBERT

5. To rewrite r2 with no poles in the upper half plane, factor the

denominator and follow the procedure used in the proof of Lemma 3.6.
A similar procedure can be used for the construction of %PRSRr(C)
approximations.

ACKNOWLEDGMENTS

I thank professor J. W. Helton for introducing me to the families of rational functions in

circuit theory, and for interesting me in their connection to my work. I am grateful to Dr. Jeff
Allen who showed me that the gain functions, GQ,“, fit into this framework. In his
dissertation, Dr. Allen was the first to explicitly state a version of the characterization
theorem 5.11.

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