Basic Fourier series on a \(q\)-linear grid: convergence theorems (Q852739)

Let \(f(x)\) be a function defined on \(-\infty<x<\infty\). For \(0<q<1\), the symmetric \(q\)-difference operator is defined by \(\delta f(x)=f(q^{1/2}x)-f(q^{-1/2}x)\). Then \(\frac{\delta f(x)}{\delta x}= \frac{f(q^{1/2}x)-f(q^{-1/2}x)}{x(q^{1/2}-q^{-1/2})}\). The \(q\)-linear initial value problem \(\frac{\delta f(x)}{\delta x}=\lambda\,f(x)\), \(f(0)=1\) , has two entire functions \(C_{q}(z)\) and \(S_{q}(z)\) as linear independent solutions, which are orthogonal on a discrete set. These functions are the \(q\)-analogues of the trigonometric functions and they are called \(q\)-linear trigonometric functions. The author studies Fourier series expansions with respect to these functions and gives sufficient conditions for pointwise convergence and for uniform convergence of such series. Some examples of \(q\)-Fourier series are also given and the corresponding questions about convergence are studied.