# Some Generalized $q$-Bessel Type Wavelets and Associated Transforms

## TL;DR

This paper introduces generalized $q$-Bessel wavelets and transforms within $q$-theory, extending classical Bessel wavelets to a broader context with proven reconstruction and Plancherel formulas.

## Contribution

It extends Bessel wavelets to generalized $q$-Bessel wavelets using a $(q,v)$-framework, providing new tools for wavelet analysis in $q$-theory.

## Key findings

- Development of generalized $q$-wavelets and transforms
- Proof of reconstruction formulas
- Establishment of Plancherel type formulas

## Abstract

In this paper wavelet functions are introduced in the context of $q$-theory. We precisely extend the case of Bessel and $q$-Bessel wavelets to the generalized $q$-Bessel wavelets. Starting from the $(q,v)$-extension ($v=(\alpha,\beta)$) of the $q$-case, associated generalized $q$-wavelets and generalized $q$-wavelet transforms are then developed for the new context. Reconstruction and Plancherel type formulas are proved.