The surprising almost everywhere convergence of Fourier-Neumann series

The surprising almost everywhere convergence of Fourier-Neumann series∗ Óscar Ciaurri and Juan Luis Varona Dpto. de Matemáticas y Computación, Univ. de La Rioja, 26004 Logroño, Spain oscar.ciaurri@unirioja.es, jvarona@unirioja.es http://www.unirioja.es/dptos/dmc/jvarona Abstract For most orthogonal systems and their corresponding Fourier series, the study of the almost everywhere convergence for functions in Lp requires very complicated research, harder than in the case of the mean convergence. For instance, for trigonometric series, the almost everywhere convergence for functions in L2 is the celebrated Carleson theorem, proved in 1966 (and extended to Lp by Hunt in 1967). In this paper, we take the system p n = 0, 1, 2, . . . jnα (x) = 2(α + 2n + 1) Jα+2n+1 (x)x−α−1 , (with Jµ being the Bessel function of the first kind and of the order µ), which is orthonormal in L2 ((0, ∞), x2α+1 dx), and whose Fourier series are the so-called Fourier-Neumann series. We study the almost everywhere convergence of FourierNeumann series for functions in Lp ((0, ∞), x2α+1 dx) and we show that, surprisingly, the proof is relatively simple (inasmuch as the mean convergence has already been established). Keywords: Bessel functions, Fourier-Neumann series, almost everywhere convergence. 1 Introduction and main theorem Let Jµ denote the Bessel function of the first kind and of the order µ. It is well-known that, for α ≥ −1/2, we have Z ∞ δnm dx = , n, m = 0, 1, 2, . . . Jα+2n+1 (x)Jα+2m+1 (x) x 2(2n + α + 1) 0 (see [1, Ch. XIII, 13.41 (7), p. 404] and [1, Ch. XIII, 13.42 (1), p. 405]). Then, the system p n = 0, 1, 2, . . . jnα (x) = 2(α + 2n + 1) Jα+2n+1 (x)x−α−1 , is orthonormal in L2 ((0, ∞), dµα ) (L2 (dµα ) from now on), with dµα (x) = x2α+1 dx. For each suitable function f , take Snα f (x) = n X α cα k (f )jk (x), cα k (f ) k=0 ∗ THIS = Z ∞ 0 f (y)jkα (y) dµα (y), PAPER HAS BEEN PUBLISHED IN J. Comput. Appl. Math. 233 (2009), 663–666. 1 so Snα f denotes the n-th partial sum of its Fourier series with respect to the system {jnα }∞ n=0 , which is usually called the Fourier-Neumann series. Most orthogonal systems (the trigonometric system, Jacobi, Hermite and Laguerre orthogonal polynomials and functions, etc.) are complete in their corresponding L2 space. However, this does not happen with {jnα }∞ n=0 . To see this, let us consider the so-called modified Hankel transform Hα , that is Z ∞ Jα (xy) f (y)y 2α+1 dy, x > 0, (1) Hα f (x) = (xy)α 0 (defined for suitable functions). In the usual way, Hα can be extended to functions f ∈ L2 (dµα ): it becomes an isometry on L2 (dµα ), where Hα2 is the identity. Moreover, the Bessel functions and the Jacobi polynomials are related by means of Z ∞ Jα+2n+1 (t)Jα (xt) dt = xα Pn(α,0) (1 − 2x2 )χ[0,1] (x); (2) 0 see, for instance, [2, Ch. 8.11, (5), p. 47]. From this formula, Hα jnα is supported on 2 [0, 1], so consequently {jnα }∞ n=0 is not complete in L (dµα ). Furthermore, (2) allows α expressing Hα jn in terms of the Jacobi polynomials (which are a complete system). Then, by using that Hα is an isometry on L2 (dµα ), the subspace B2,α = span{jnα }∞ n=0 (closure in L2 (dµα )) can be identified with E2,α = {f ∈ L2 (dµα ) : Mα f = f }, where Mα is the multiplier defined by Hα (Mα f ) = χ[0,1] Hα f. Let us point out that, as Hα jnα is supported on [0, 1], we have Mα jnα = jnα so jnα ∈ E2,α indeed. The Lp (dµα )-mean convergence of the Fourier-Neumann series was studied by one of the authors in [3], and later extended in [4]. An important part of these papers is p devoted to identifying Bp,α = span{jnα }∞ n=0 (closure in L (dµα )). When p 6= 2, Hα can be defined on Lp (dµα ) under some circumstances, but it is not an isometry, and then the relation (2) can no longer be used. However, for a certain range of p’s (summed up in (4)), the extension of the multiplier Mα as a bounded operator from Lp (dµα ) into itself can be done in the usual way, by using suitable bounds of the Bessel functions. The operator Mα has several interesting properties, such as Mα2 f = Mα f and Z ∞ Z ∞ Mα f (y)g(y) dµα (y), (3) f (y)Mα g(y) dµα (y) = 0 0 ′ which is valid whenever f ∈ Lp (dµα ) and g ∈ Lp (dµα ) (with 1/p + 1/p′ = 1). Moreover, the space Ep,α = {f ∈ Lp (dµα ) : Mα f = f } can be defined, and some of its properties proved: Es,α ⊆ Er,α when s < r (the inclusion being continuous and ′ dense), the duality Ep,α = Ep′ ,α and, finally, Bp,α = Ep,α . The details can be found in [3, 5]. The goal of this paper is to analyze the almost everywhere convergence of the Fourier-Neumann series Snα f for functions f ∈ Lp (dµα ), with α ≥ −1/2. A partial study is done in [4], but now we are going to extend it, showing a further result. For most orthogonal systems and their corresponding Fourier series, the study of the almost everywhere convergence for functions in Lp is rather complicated, much more than that of the mean convergence. For instance, for trigonometric series, the almost everywhere convergence for functions in L2 was conjectured by Lusin in 1915, and proved by Carleson in 1966 (extended by Hunt in 1967 for Lp with 1 < p < ∞). However, we are going to see that the proofs of the almost everywhere convergence of the Fourier-Neumann series for functions in Lp (dµα ) are, surprisingly, relatively simple (supposing that the mean convergence has been previously established). 2 We will restrict our analysis to the “natural” interval p ∈ (p0 (α), p1 (α)) given by p0 (α) = 4(α + 1) 4(α + 1) <p< = p1 (α). 2α + 3 2α + 1 (4) In fact, the first requirement for having the partial sum of the Fourier-Neumann series in Lp (dµα ) (with 1 < p < ∞) is that jnα ∈ Lp (dµα ) for every n. By using well-known estimates for the Bessel functions (namely (6) and (7)), this is equivalent to p > p0 (α). p And, since the Fourier coefficients cα n (f ) must exist for every f ∈ L (dµα ), we must α p′ ′ have jn ∈ L (dµα ) for every n, where 1/p + 1/p = 1. This is equivalent to p < p1 (α). As we can see in [3], the range (4) is just the one for which the multiplier Mα : Lp (dµα ) → Lp (dµα ) is a bounded operator, which allows defining the subspaces Ep,α . However, the interval of mean convergence of the Fourier-Neumann series is not all of the range (4), but   4 , p0 (α) < p < min {4, p1 (α)} . (5) max 3 That is, if p satisfies (5) and f ∈ Ep,α , then Snα f → f when n → ∞ in the Lp (dµα )norm. Conversely, (5) is a necessary condition for the mean convergence. When f ∈ Lp (dµα ), the Fourier-Neumann series converges to Mα f (recall that Mα f = f for functions in Ep,α ). In [4] we prove that, if p satisfies (5), Snα f → f almost everywhere for every f ∈ Ep,α (and the convergence is to Mα f for f ∈ Lp (dµα )). Here, we extend this result by removing the condition 4/3 < p < 4 (which affects the case −1/2 ≤ α < 0) for the almost everywhere convergence. Thus, we have Theorem. Let us have α ≥ −1/2 and p satisfying (4). Then, Snα f → Mα f almost everywhere for every f ∈ Lp (dµα ). 2 Auxiliary results The Bessel functions satisfy the asymptotic formulas (see, for instance, [1, Ch. III, 3.1 (8), p. 40] and [1, Ch. VII, 7.21 (1), p. 199]) Jν (x) = xν 2ν Γ(ν + 1) + O(xν+2 ), x → 0+ , (6) 1/2 h i  νπ π  2 + O(x−1 ) , x → ∞. (7) − cos x − Jν (x) = πx 2 4 We will also use bounds with constants independent of the parameter ν of the Bessel function. These bounds are a consequence of the very precise estimates that appear in [6]. To be more precise, we will use the following bound that can be found in [4, 3]:  −1/4 |Jν (x)| ≤ Cx−1/4 |x − ν| + ν 1/3 , x ∈ (0, ∞), (8)  where C is a positive constant independent of ν. With this information, let us estimate kjnα kLp (dµα ) : p Lemma 1. Let α ≥ −1/2 and p > p0 (α). Then, {jnα }∞ n=0 ⊆ L (dµα ) and  −(α+1)+2(α+1)/p  , if p < 4, n  α kjn kLp (dµα ) ≤ C n−(α+1)/2 (log n)1/4 , if p = 4,   n−(5/6+α)+(6α+4)/(3p) , if p > 4, with C a positive constant independent of n. 3 Proof. The assertion that jnα ∈ Lp (dµα ) for every n = 0, 1, 2, . . . follows from (6) and (7). Then, estimates (8) show that kjnα kLp (dµα ) is bounded above by a constant times the right hand side. For a similar expression, see [7]. Now, let us note that, for x ∈ (0, ∞) fixed, the Bessel function |Jν (x)| has a huge decay when ν grows to ∞ (and consequently the same happens with |jnα (x)| when n → ∞). In fact, according to [1, Ch. III, 3.31 (1), p. 49], we have |Jν (x)| ≤ 2−ν xν , Γ(ν + 1) ν ≥ −1/2. (9) Then, we have Lemma 2. LetPα ≥ −1/2 and p with 1 < p < p1 (α). Then, for any f ∈ Lp (dµα ) the ∞ α Fourier series n=0 cα n (f )jn (x) converges absolutely for every x ∈ (0, ∞). (Note that we do not assert that this convergence is to f (x), not even almost everywhere.) Proof. Recall that cα n (f ) = Z ∞ 0 f (y)jnα (y)y 2α+1 dy. (10) Since p < p1 (α), it follows that p′ > p0 (α) (with 1/p + 1/p′ = 1). Then, from ′ Lemma 1, we have jnα ∈ Lp (dµα ) and, moreover, kjnα kLp′ (dµα ) ≤ Cnδ for some constant δ = δ(p, α). Thus, by Hölder’s inequality, δ α |cα n (f )| ≤ kf kLp (dµα ) kjn kLp′ (dµα ) ≤ Ckf kLp (dµα ) n . On the other hand, as a consequence of (9), we have p |jnα (x)| = 2(α + 2n + 1) |Jα+2n+1 (x)| x−α−1 p 2(α + 2n + 1) 2−(α+2n+1) x2n ≤ . Γ(α + 2n + 2) Therefore, α |cα n (f )jn (x)| ≤ Ckf kLp (dµα ) and the series 3 P∞ α α n=0 cn (f )jn (x) nδ+1/2 (x/2)2n Γ(α + 2n + 2) (11) converges absolutely. Proof of the theorem By Lemma 2, Snα f converges to some g pointwise when p satisfies (4). We want to prove that, almost everywhere, g = f if f ∈ Ep,α , or, more generally, g = Mα f if f ∈ Lp (dµα ). As established in the introduction, we know that, when p satisfies (5), Snα f converges to Mα f in the Lp (dµα )-norm; then, Snα f has a subsequence that converges to Mα f almost everywhere. Consequently g = Mα f and the convergence Snα f → Mα f almost everywhere is proved under the hypothesis (5). This is the argument used in [4]. Let us see how to remove the condition 4/3 < p < 4. For that, we are going to apply the summation process used in [5]. Thus, let us take λ0 S0α f + · · · + λn Snα f Rnα f = λ0 + · · · + λn with λk = 2(α + 2k + 2). Actually, as established in that paper, this method is equivalent to the one given by the Cesàro means of order 1, but the kernels that appear with Rnα are easier to handle, and consequently the use of Rnα is more convenient for 4 studying the uniform boundedness of the operators involved (and hence the mean convergence). In [5] it is proved that, when p satisfies (4) (i.e., without 4/3 < p < 4), Bp,α = Ep,α and Rnα f → f in the Lp (dµα )-norm for every f ∈ Ep,α . For general f ∈ Lp (dµα ), we always have Mα f ∈ Ep,α (because Mα2 f = Mα f ). Moreover, by using (3) and α α α Mα jkα = jkα , it follows that cα k (f ) = ck (Mα f ) for every k, and so Rn f = Rn (Mα f ). α α As Rn (Mα f ) converges in mean to Mα f , also Rn f converges in mean to Mα f . Then, there exists a subsequence of Rnα f that converges almost everywhere to Mα f . On the other hand, Rnα is a regular summation process. Then, by Lemma 2, given f ∈ Lp (dµα ) with p satisfying (4), we have that Rnα f (x) converges for every x ∈ (0, ∞) to the same function g(x) that is the pointwise limit of Snα f (x). Thus, for p satisfying (4), we have: a subsequence of Rnα f converges almost everywhere to Mα f , Snα f converges almost everywhere to g, and Rnα f converges almost everywhere to g. From these facts, g = Mα f and the theorem is proved. Open question. What happens for Snα f with f ∈ Lp (dµα ) and 1 < p ≤ p0 (α)? α Under these conditions, every cα k (f ) exists, so the partial sums Sn f are well defined. α Futhermore, Lemma 2 ensures that, pointwise, Sn f (x) converges to some function g(x). But, what is g(x)? Let us note that, when 1 < p ≤ p0 (α), the bounded multiplier Mα : Lp (dµα ) → Lp (dµα ) no longer exists. References [1] G. N. Watson, A Treatise on the Theory of Bessel Functions (2nd edition), Cambridge Univ. Press, Cambridge, 1944. [2] A. Erdélyi, W. Magnus, F. Oberhettinger, and F. Tricomi, Tables of Integral Transforms, Vol. II, McGraw-Hill, New York, 1954. [3] J. L. Varona, Fourier series of functions whose Hankel transform is supported on [0, 1], Constr. Approx. 10 (1994), 65–75. [4] Ó. Ciaurri, J. J. Guadalupe, M. Pérez, and J. L. Varona, Mean and almost everywhere convergence of Fourier-Neumann series, J. Math. Anal. Appl. 236 (1999), 125–147. [5] Ó. Ciaurri, K. Stempak, and J. L. Varona, Mean Cesàro-type summability of Fourier-Neumann series, Studia Sci. Math. Hung. 42 (2005), 413–430. [6] J. A. Barceló and A. Córdoba, Band-limited functions: Lp -convergence, Trans. Amer. Math. Soc. 313 (1989), 655–669. [7] K. Stempak, A weighted uniform Lp -estimate of Bessel functions: a note on a paper of Guo, Proc. Amer. Math. Soc. 128 (2000), 2943–2945. 5