A Whittaker-Shannon-Kotel'nikov sampling theorem related to the Dunkl transform on the real line is proved. To this end we state, in terms of Bessel functions, an orthonormal system which is complete in L 2 ((−1, 1), |x| 2α+1 dx). This orthonormal system is a generalization of the classical exponential system defining Fourier series.
Proved are weighted transplantation inequalities for Fourier-Bessel expansions. These extend known results on this subject by considering the largest possible range of parameters, allowing more weights and admitting a shift. The results are then used to produce a fairly general multiplier theorem with power weights for considered expansions. Also fractional… (More)
Let Jµ denote the Bessel function of order µ. The functions x −α/2−β/2−1/2 J α+β+2n+1 (x 1/2), n = 0, 1, 2,. .. , form an orthogonal system in L 2 ((0, ∞), x α+β dx) when α + β > −1. In this paper we analyze the range of p, α and β for which the Fourier series with respect to this system converges in the L p ((0, ∞), x α dx)-norm. Also, we describe the… (More)
An uncertainty inequality for the Fourier–Dunkl series, introduced by the authors in [ ´ This result is an extension of the classical uncertainty inequality for the Fourier series.
We relate the fractional powers of the discrete Laplacian with a standard time-fractional derivative in the sense of Liouville by encoding the iterative nature of the discrete operator through a time-fractional memory term.
In this paper, we develop a thorough analysis of the boundedness properties of the maximal operator for the Bochner-Riesz means related to the Fourier-Bessel expansions. For this operator, we study weighted and un-weighted inequalities in the spaces L p ((0, 1), x 2ν+1 dx). Moreover, weak and restricted weak type inequalities are obtained for the critical… (More)
We study an extension of the classical Paley-Wiener space structure, which is based on bilinear expansions of integral kernels into biorthogonal sequences of functions. The structure includes both sampling expansions and Fourier-Neumann type series as special cases. Concerning applications, several new results are obtained. From the Dunkl analogue of… (More)
For most orthogonal systems and their corresponding Fourier series, the study of the almost everywhere convergence for functions in L p requires very complicated research, harder than in the case of the mean convergence. For instance, for trigonometric series, the almost everywhere convergence for functions in L 2 is the celebrated Carleson theorem, proved… (More)