On $q^2$-trigonometric functions and their $q^2$-Fourier transform

TL;DR

This paper introduces generalized $q^2$-trigonometric functions and develops a $q^2$-Fourier transform, establishing key properties like inversion and Plancherel theorems in the $q$-analog framework.

Contribution

It constructs new $q^2$-trigonometric functions and defines a $q^2$-Fourier transform, proving fundamental theorems analogous to classical Fourier analysis.

Findings

Defined generalized $q^2$-cosine, $q^2$-sine, and $q^2$-exponential functions.

Established $q$-analogues of inversion and Plancherel theorems.

Developed a $q^2$-Fourier transform with foundational properties.

Abstract

In this paper, we first construct generalized $q^2$-cosine, $q^2$-sine and $q^2$-exponential functions. We then use $q^2$-exponential function in order to define and investigate a $q^2$-Fourier transform. We establish $q$-analogues of inversion and Plancherel theorems.