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Calderon's reproducing formula and extremal functions associated with the linear canonical Dunkl wavelet transform

Sandeep Kumar Verma, Umamaheswari S

TL;DR

This work advances harmonic analysis for systems with reflection symmetries by developing Calderón's reproducing formula in the setting of the linear canonical Dunkl continuous wavelet transform and by analyzing extremal approximation problems there. It introduces the Sobolev-type spaces $\mathbf{W}^s_{k,M}(\mathbb{R})$ and proves they are reproducing kernel Hilbert spaces with kernels $K_s$, enabling precise representation and evaluation of functions via reproducing kernels. Through Tikhonov regularization in these RKHS, the authors establish existence, representation, and convergence of extremal functions $f^*_{\rho,g}$ for LCDWT and $h_{\rho,g}^*$ for LCDT, highlighting stable best-approximation properties in this nonstandard transform framework. The results provide a rigorous foundation for best-approximation and reconstruction in LCD settings, with potential applications in optics, signal processing, and areas featuring Dunkl-type reflection symmetries.

Abstract

In this article, we undertake a two-fold investigation. First, we establish Calderons reproducing formula for the linear canonical Dunkl continuous wavelet transform. Further, we define the reproducing kernel linear canonical Dunkl Sobolev space and introduce a novel inner product associated with the continuous wavelet transform in this space. We then derive explicit formulas for the reproducing kernels and present several related results. In the second part, we investigate extremal functions associated with both the continuous wavelet and linear canonical Dunkl transform. In particular, we characterize the extremal functions, represent them in terms of the corresponding reproducing kernels, and establish structural properties relevant to their formulation.

Calderon's reproducing formula and extremal functions associated with the linear canonical Dunkl wavelet transform

TL;DR

This work advances harmonic analysis for systems with reflection symmetries by developing Calderón's reproducing formula in the setting of the linear canonical Dunkl continuous wavelet transform and by analyzing extremal approximation problems there. It introduces the Sobolev-type spaces and proves they are reproducing kernel Hilbert spaces with kernels , enabling precise representation and evaluation of functions via reproducing kernels. Through Tikhonov regularization in these RKHS, the authors establish existence, representation, and convergence of extremal functions for LCDWT and for LCDT, highlighting stable best-approximation properties in this nonstandard transform framework. The results provide a rigorous foundation for best-approximation and reconstruction in LCD settings, with potential applications in optics, signal processing, and areas featuring Dunkl-type reflection symmetries.

Abstract

In this article, we undertake a two-fold investigation. First, we establish Calderons reproducing formula for the linear canonical Dunkl continuous wavelet transform. Further, we define the reproducing kernel linear canonical Dunkl Sobolev space and introduce a novel inner product associated with the continuous wavelet transform in this space. We then derive explicit formulas for the reproducing kernels and present several related results. In the second part, we investigate extremal functions associated with both the continuous wavelet and linear canonical Dunkl transform. In particular, we characterize the extremal functions, represent them in terms of the corresponding reproducing kernels, and establish structural properties relevant to their formulation.
Paper Structure (9 sections, 21 theorems, 122 equations)

This paper contains 9 sections, 21 theorems, 122 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Linear Canonical Dunkl Transform
  4. Generalized translation and convolution operator for LCDT
  5. Linear Canonical Dunkl continuous Wavelet Transform
  6. Calderón's reproducing formula
  7. Extremal functions on linear canonical Dunkl Sobolev space
  8. The extremal function associated to LCDT and LCDWT
  9. Extremal function for LCDT

Key Result

Theorem 1

[see Theorem T: 4.12] Let $f\in L^2_k(\mathbb{R})$ and wavelet pair $(\psi,\varphi)$ such that $C^M_{\psi,\varphi} \neq 0$ and $D^M_k(\psi),D^M_k(\varphi)\in L^\infty_k(\mathbb{R}).$ Then for $0<\epsilon<\delta< \infty$ belongs to $L^2_k(\mathbb{R})$. Moreover,

Theorems & Definitions (48)

  • Theorem : A
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Proposition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Lemma 2.9
  • ...and 38 more