We prove a limit relation for the Dunkl–Bessel function of type BN with multiplicity parameters k1 on the roots ±ei and k2 on ±ei ± ej where k1 tends to infinity and the arguments are suitably scaled. It gives a good approximation in terms of the Dunkltype Bessel function of type AN−1 with multiplicity k2. For certain values of k2 an improved estimate is obtained from a corresponding limit rela...

In this article, we show that a finite dimensional Hilbert space can have an infinite Bessel sequence, but a normalized Bessel sequence in a finite dimensional Hilbert space must be of finite length. A relation between the dimension of a given finite dimensional Hilbert space and the bound of any finite normalized tight frame for the underlying space is obtained. Also some properties of the fra...

We report on an interferometry-based measurement of the phase and group velocities of optical Bessel beams, providing confirmation of their superluminal character in the non-diffractive region. The measurements were performed in free space with a continuous wave laser and femtosecond pulses for phase and group velocities respectively. The Bessel beams were produced using a conical mirror. Einst...

Abstract: We define q-analogues of Fourier-Bessel series, by means of complete qorthogonal systems constructed with the third Jackson q-Bessel function. Sufficient conditions for pointwise convergence of these series are obtained, in terms of a general convergence principle valid for other Fourier series on grids defined over numerable sets. The results are illustrated with specific examples of...

The discrete Painlevé III equation is investigated based on the bilinear formalism. It is shown that it admits the solutions expressed by the Casorati determinant whose entries are given by the discrete Bessel function. Moreover, based on the observation that these discrete Bessel functions are transformed to the q-Bessel functions by a simple variable transformation, we present a q-difference ...

In this paper we study spaces of Bessel potentials in n-dimensional Euclidean spaces. They are constructed on the basis of a rearrangement-invariant space (RIS) by using convolutions with Bessel-MacDonald kernels. Specifically, the treatment covers spaces of classical Bessel potentials. We establish two-sided estimates for the corresponding modulus of smoothness of order k ∈ N, ωk(f ; t), and d...

We derive completeness criteria for sequences of functions of the form f(xλn), where λn is the nth zero of a suitably chosen entire function. Using these criteria, we construct complete nonorthogonal systems of FourierBessel functions and their q-analogues, as well as other complete sets of qspecial functions. We discuss connections with uncertainty principles over q-linear grids and the comple...

A pair of quasi-de nite linear functionals fu0; u1g on the set of polynomials is called a coherent pair if their corresponding sequences of monic orthogonal polynomials fPng and fTng satisfy a relation Tn = P 0 n+1 n+ 1 nP 0 n n ; n 1; with n non-zero constants. We prove that if fu0; u1g is a coherent pair, then at least one of the functionals has to be classical, i.e. Hermite, Laguerre, Jacobi...

G-frames are natural generalizations of frames which cover many other recent generalizations of frames, e.g., bounded quasi-projectors, frames of subspaces, outer frames, oblique frames, pseudo-frames and a class of timefrequency localization operators. Moreover, it was shown that g-frames are equivalent to stable space splittings. In this paper, we study the stability of g-frames. We first pre...

The normalization of energy divergent Weber waves and finite energy Weber-Gauss beams is reported. The well-known Bessel and Mathieu waves are used to derive the integral relations between circular, elliptic, and parabolic waves and to present the Bessel and Mathieu wave decomposition of the Weber waves. The efficiency to approximate a Weber-Gauss beam as a finite superposition of Bessel-Gauss ...