Abstract
We study Fourier–Bessel series on a q-linear grid, defined as expansions in complete q-orthogonal systems constructed with the third Jackson q-Bessel function, and obtain sufficient conditions for uniform convergence. The convergence results are illustrated with specific examples of expansions in q-Fourier–Bessel series.
Highlights
Based on the orthogonality relationJν Jνdt = 0, The research of L
In the papers [22,23,24], a theory of Fourier series on a q-linear grid was developed, using a q-analogue of the exponential function and the corresponding q-trigonometric functions introduced by Exton [30]
This paper investigates the most delicate convergence issues of the basic Fourier–Bessel series on a q-linear grid, based on the orthogonality relation (1.1), on mean convergence results [5,6], and on the localization of the zeros jnν [4]
Summary
In the papers [22,23,24], a theory of Fourier series on a q-linear grid was developed, using a q-analogue of the exponential function and the corresponding q-trigonometric functions introduced by Exton [30]. This was motivated by Bustoz–Suslov orthogonality and completeness results of q-quadratic Fourier series [21]. Our main contribution will be a result providing sufficient conditions for uniform convergence Since it was proved [2], under the same general conditions imposed by Hardy, that the above orthogonality relation characterizes the functions Jν(3)(z; q2), this is the most general Fourier theory based on functions q-orthogonal with respect to their own zeros. In the last section of the paper, two examples of basic Fourier–Bessel expansions are provided
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