A Paley-Wiener theorem for the Askey-Wilson function transform (original) (raw)

Paley–Wiener-Type Theorems for a Class of Integral Transforms

Journal of Mathematical Analysis and Applications, 2002

A characterization of weighted L2(I) spaces in terms of their images under various integral transformations is derived, where I is an interval (finite or infinite). This characterization is then used to derive Paley-Wiener-type theorems for these spaces. Unlike the classical Paley-Wiener theorem, our theorems use real variable techniques and do not require analytic continuation to the complex plane. The class of integral transformations considered is related to singular Sturm-Liouville boundary-value problems on a half line and on the whole line. 1 1991 Mathematics Subject Classifications. Primary 44A15, 34B24; Secondary 42B10, 33C45

Interpolation by weighted Paley–Wiener spaces associated with the Dunkl transform

Journal of Mathematical Analysis and Applications, 2012

Given α > − 1 2 , σ > 0 and 1 p < ∞, we study the interpolation problem in the space PW p,α σ of entire functions f : C → C of exponential type σ for which R | f (x)| p × |x| 2α+1 dx < ∞, with nodes of interpolation at s j /σ , j ∈ Z, where {s j : j ∈ N} is the increasing sequence of all positive roots of the Bessel function J α+1 (z) of order α + 1, and s j = −s − j for all j ∈ Z. We prove that if 4(α+1) 2α+3 := p 1 (α) < p < p 2 (α) := 4(α+1) 2α+1 , the interpolation problem f ∈ PW p,α σ and f σ −1 s j = c j for all j ∈ Z has a unique solution for every sequence {c j } of complex numbers satisfying j∈Z |c j | p (1+ | j|) 2α+1 < ∞, and that if p p 1 (α), the corresponding interpolation problem may not have a solution, and that the solution, if exists, is unique if and only if p p 2 (α). Finally, we show that R f (x) p |x| 2α+1 dx ∼ σ −2α−2 j∈Z f s j σ −1 p 1 + | j| 2α+1 , with the constant of equivalence depending only on p and α, holds for all entire functions f of exponential type σ if and only if p 1 (α) < p < p 2 (α).

Paley–Wiener theorem for the Weinstein transform and applications

Integral Transforms and Special Functions, 2017

In this paper our aim is to establish the Paley-Wiener Theorems for the Weinstein Transform. Furthermore, some applications are presents, in particular some properties for the generalized translation operator associated with the Weinstein operator are proved.

On the Paley–Wiener theorem in the Mellin transform setting

Journal of Approximation Theory, 2016

In this paper we establish a version of the Paley-Wiener theorem of Fourier analysis in the frame of the Mellin transform. We provide two different proofs, one involving complex analysis arguments, namely the Riemann surface of the logarithm and Cauchy theorems, and the other one employing a Bernstein inequality here derived for Mellin derivatives.

Asymptotic behaviours of a class of integral transforms in complex domains

Proceedings of the Edinburgh Mathematical Society, 1989

Zemanian [17] obtained abelian theorems for the Hankel and K-transforms of functions and then extended his results to the corresponding transforms of distributions in the sense of Schwartz [11]. Jones [6] has discussed at length asymptotic behaviours of transforms generalized in his sense. Following the technique of Zemanian many authors have obtained abelian theorems for more general transforms of functions and distributions in the sense of Schwartz. Mention may be made of the works of Joshi and Saxena [7], Lavoine and Misra [8] and Pathak [10]. However, these authors were confined to the transforms of real variables only.

A fresh approach to the Paley-Wiener theorem for Mellin transforms and the Mellin-Hardy spaces

Mathematische Nachrichten

Here we give a new approach to the Paley-Wiener theorem in a Mellin analysis setting which avoids the use of the Riemann surface of the logarithm and analytical branches and is based on new concepts of polar-analytic function in the Mellin setting and Mellin-Bernstein spaces. A notion of Hardy spaces in the Mellin setting is also given along with applications to exponential sampling formulas of optical physics.

Generalized Paley–Wiener Theorems

International Journal of Wavelets, Multiresolution and Information Processing, 2012

Non-harmonic Fourier transform is useful for the analysis of transient signals, where the integral kernel is from the boundary value of Möbius transform. In this note, we study the Paley–Wiener type extension theorems for the non-harmonic Fourier transform. Two extension theorems are established by using real variable techniques.

Askey–Wilson operator on entire functions of exponential type

Proceedings of the American Mathematical Society

In this paper, we first establish a series representation formula for the Askey-Wilson operator applied on entire functions of exponential type and then demonstrate its power in discovering summation formulas, some known and some new.