J un 1 99 7 BASIC ANALOG OF FOURIER SERIES ON A q - QUADRATIC GRID

BASIC ANALOG OF FOURIER SERIES ON A q-QUADRATIC GRID

We prove orthogonality relations for some analogs of trigonometric functions on a q-quadratic grid and introduce the corresponding q-Fourier series. We also discuss several other properties of this basic trigonometric system and the q-Fourier series.

Basic Fourier series: convergence on and outside the q-linear grid

Basic Fourier expansions on q-quadratic and on q-linear grids were first considered in [8] and in [7], respectively. Recently, in [10], sufficient conditions for (uniform) convergence of the q-Fourier series in terms of basic trigonometric functions Sq and Cq , on a q-linear grid, were given. In [19] it was established an ”addition” theorem for the corresponding basic exponential function, bein...

4 J un 1 99 7 q ROTATING RATIONALLY

. C A ] 9 J ul 1 99 3 AN ANALOG OF THE FOURIER TRANSFORMATION FOR A q - HARMONIC OSCILLATOR

A q-version of the Fourier transformation and some of its properties are discussed. INTRODUCTION Models of q-harmonic oscillators are being developed in connection with quantum groups and their various applications (see, for example, Refs. [M], [Bi], [AS1], and [AS2]). For a complete correspondence with the quantum-mechanical oscillator problem, these models need an analog of the Fourier transf...

Quadratic Reciprocity and the Sign of the Gauss Sum via the Finite Weil Representation

We give new proofs of two basic results in number theory: The law of quadratic reciprocity and the sign of the Gauss sum. We show that these results are encoded in the relation between the discrete Fourier transform and the action of the Weyl element in the Weil representation modulo p, q and pq. 0. Introduction Two basic results due to Gauss are the quadratic reciprocity law and the sign of th...