, Volume 28, Issue 3, pp 443-461,
Open Access This content is freely available online to anyone, anywhere at any time.

The q-cosine Fourier transform and the q-heat equation

Abstract

The aim of this work is to establish in great detail The q-Fourier analysis related to the q-cosine. The wise reader will note that the considered q-cosine coincides with the one given by T.H. Koornwinder and S.F. Swarttouw. Through the q-cosine product formula, we define and analyze the properties of the q-even translation and the q-convolution. Adopting the Titchmarsh approach, we study the q-cosine Fourier transform and its inverse formula.

The second theme of this paper is an application of the q-Fourier analysis developed earlier. We extend the heat representation theory inaugurated by P.C. Rosenbloom and D.V. Widder to the q-analogue. We construct the q-solution source, the q-heat polynomials and solve the q-analytic Cauchy problem.

This research is supported by NPST Program of King Saud University, project number 10-MAT1293-02.
EDITORIAL NOTE: The article was accepted on 16 February, 2001, before the papers of the Ramanujan Journal were handled electronically. Unfortunately this article in its printed form was misplaced. The delay caused in publication is regretted.