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Real Paley-Wiener theorems for the linear canonical Dunkl transform

Umamaheswari S, Sandeep Kumar Verma, Hatem Mejjaoli

TL;DR

The paper extends real Paley-Wiener theory to the linear canonical Dunkl transform (LCDT) by developing a Sobolev framework tailored to LCDT via the operator \Lambda_{k,M} and its square \Delta_{k,M}, establishing a real Paley-Wiener theorem on the Schwartz space, and characterizing LCDT spectral support through polynomial-domain and Boas-type conditions. Central tools include the LCDT \mathcal{D}_k^M, the linear canonical Dunkl operator \Lambda_{k,M}, and the associated Sobolev space \mathbf{W}^{s}_{k,M}(\mathbb{R}); the results link growth rates of LCDT-derived differential-difference operators to the location of the transform's support. Key contributions are (i) a real PW theorem for LCDT, (ii) a real PW theorem for polynomial-domain support, and (iii) a Boas-type Paley-Wiener theorem, expanding Paley-Wiener theory to a unified framework that encompasses scaling, rotation, and reflection symmetries via the LCDT. Collectively, these findings provide a robust analytical toolkit for LCDT-based harmonic analysis with potential applications in signal processing and mathematical physics where Dunkl-type symmetries arise.

Abstract

We examine the Sobolev space associated with the linear canonical Dunkl transform and explore some properties of the linear canonical Dunkl operators. Building on these results, we establish a real Paley-Wiener theorem for the linear canonical Dunkl transform. Further, we characterize the square-integrable function f whose linear canonical Dunkl transform of the function is supported in the polynomial domain. Finally, we develop the Boas-type Paley-Wiener theorem for the linear canonical Dunkl transform.

Real Paley-Wiener theorems for the linear canonical Dunkl transform

TL;DR

The paper extends real Paley-Wiener theory to the linear canonical Dunkl transform (LCDT) by developing a Sobolev framework tailored to LCDT via the operator \Lambda_{k,M} and its square \Delta_{k,M}, establishing a real Paley-Wiener theorem on the Schwartz space, and characterizing LCDT spectral support through polynomial-domain and Boas-type conditions. Central tools include the LCDT \mathcal{D}_k^M, the linear canonical Dunkl operator \Lambda_{k,M}, and the associated Sobolev space \mathbf{W}^{s}_{k,M}(\mathbb{R}); the results link growth rates of LCDT-derived differential-difference operators to the location of the transform's support. Key contributions are (i) a real PW theorem for LCDT, (ii) a real PW theorem for polynomial-domain support, and (iii) a Boas-type Paley-Wiener theorem, expanding Paley-Wiener theory to a unified framework that encompasses scaling, rotation, and reflection symmetries via the LCDT. Collectively, these findings provide a robust analytical toolkit for LCDT-based harmonic analysis with potential applications in signal processing and mathematical physics where Dunkl-type symmetries arise.

Abstract

We examine the Sobolev space associated with the linear canonical Dunkl transform and explore some properties of the linear canonical Dunkl operators. Building on these results, we establish a real Paley-Wiener theorem for the linear canonical Dunkl transform. Further, we characterize the square-integrable function f whose linear canonical Dunkl transform of the function is supported in the polynomial domain. Finally, we develop the Boas-type Paley-Wiener theorem for the linear canonical Dunkl transform.
Paper Structure (9 sections, 18 theorems, 121 equations)

This paper contains 9 sections, 18 theorems, 121 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Dunkl operator
  4. Linear canonical Dunkl operator
  5. Linear canonical Dunkl Sobolev space
  6. Paley-Wiener theorem for S(R)
  7. Real Paley-Wiener type theorems
  8. The linear canonical Dunkl transform of functions with polynomial domain support
  9. Boas-type Paley-Wiener theorem

Key Result

Theorem 1

For all $f \in \mathcal{ S(\mathbb{R})}$, the following limit exists where

Theorems & Definitions (33)

  • Theorem : Real Paley-Wiener theorem (Theorem \ref{['t:4.1']})
  • Theorem : Real Paley-Wiener theorem for non-convex domain (Theorem \ref{['t:5.1']})
  • Theorem : Boas type theorem (Theorem \ref{['t:5.6']})
  • Proposition 2.1
  • Proposition 2.2
  • Definition 3.1
  • Proposition 3.2
  • Proposition 3.3
  • proof
  • Remark 3.4
  • ...and 23 more