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

TL;DR

This paper develops a Calderon reproducing formula for the linear canonical Dunkl wavelet transform, introduces a new Sobolev space with a reproducing kernel, and characterizes extremal functions related to these transforms.

Contribution

It provides the first Calderon reproducing formula for the linear canonical Dunkl wavelet transform and characterizes extremal functions in this context.

Findings

Derived explicit formulas for reproducing kernels.

Established structural properties of extremal functions.

Defined a novel Sobolev space with an associated inner product.

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.