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Basic analog of Fourier series on a \(q\)-linear grid. (English) Zbl 1077.33030

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 \[ \delta f(x)=f(q^{1/2}x)-f(q^{-1/2}x) \] and consider the initial value problem \[ \frac{\delta f(x)}{\delta x}=\lambda f(x),\quad f(0)=1. \tag{1} \] 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)].

MSC:

33D45 Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.)
39A13 Difference equations, scaling (\(q\)-differences)
42C30 Completeness of sets of functions in nontrigonometric harmonic analysis
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