Basic analog of Fourier series on a \(q\)-linear grid. (Q5958890)

Let \(f(x)\) be a function defined for \(-\infty<x<\infty\) and let \(0<q<1\). The authors define the symmetric \(q-\)difference operator by NEWLINE\[NEWLINE \delta f(x)=f(q^{1/2}x)-f(q^{-1/2}x) NEWLINE\]NEWLINE and consider the initial value problem NEWLINE\[NEWLINE \frac{\delta f(x)}{\delta x}=\lambda f(x),\quad f(0)=1. \tag{1}NEWLINE\]NEWLINE This is a \(q-\)difference analog of the differential initial value problem satisfied by the function \(e^{\lambda x}\). Then the authors naturally define an analog of the classical exponential function and the \(q-\)linear sine and cosine, \(S_{q}(z)\) and \(C_{q}(z)\) respectively. The functions \(S_{q}(z)\) and \(C_{q}(z)\) are linearly independent solutions of the \(q\)-linear initial value problem (1). In addition it is proved that these difference analogs of sine and cosine are orthogonal on a discrete set. Then the authors study Fourier expansions in series of these \(q\)-linear trigonometric functions and derive analytic bounds on the roots of \(S_{q}(z)\). This paper contains several misprints corrected by the authors in the erratum [J. Approximation Theory 113, No. 2, 326 (2001)].