The method of partial Fourier transform is used to find explicit formulas for two remarkable fundamental solutions for a generalized Tricomi operator. These fundamental solutions reflect clearly the mixed type of the Tricomi operator. In proving these results, we establish explicit formulas for Fourier transforms of some functions involving Bessel functions.

Purpose Paley's and Hardy's inequality are proved on a Hardy-type space for the Fourier–Dunkl expansions based complete orthonormal system of Dunkl kernels generalizing classical exponential defining Fourier series. Design/methodology/approach Although difficulties related to settings, techniques used by K. Sato were still efficient in this case establish inequalities which have expected simila...

We demonstrate a method for generation of beams of light with large angular momenta. The method utilizes whispering gallery mode resonators that transform a plane electromagnetic wave into high order Bessel beams. Interference pattern among the beams as well as shadow pictures induced by the beams are observed and studied.

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 L2((−1, 1), |x|2α+1 dx). This orthonormal system is a generalization of the classical exponential system defining Fourier series.

the theory of c-frames and c-bessel mappings are the generalizationsof the theory of frames and bessel sequences. in this paper, weobtain several equivalent conditions for dual of c-bessel mappings.we show that for a c-bessel mapping $f$, a retrievalformula with respect to a c-bessel mapping $g$ is satisfied if andonly if $g$ is sum of the canonical dual of $f$ with a c-besselmapping which  wea...

In this paper we study the duality of Bessel and g-Bessel sequences in Hilbert spaces. We show that a Bessel sequence is an inner summand of a frame and the sum of any Bessel sequence with Bessel bound less than one with a Parseval frame is a frame. Next we develop this results to the g-frame situation.

In this paper, we consider the non-homogeneous wave equation generated by Bessel operator. We prove existence and uniqueness of solution The representation is given. obtained a priori estimates in Sobolev type space. This problem was firstly considered work M. Assal [1] setting Bessel-Kingman hypergroups. technique used based on convolution theorem related estimates. Here, use different approac...

Bessel functions of matrix argument appeared as a subject of studies in the work of Herz [6]. One can find in the multivariate statistics literature some applications of these functions (see [11]). The main tool of Herz’s work was the Laplace transform and its inverse in the space of real symmetric matrices. He obtained several properties but a “good” differential system was lacking. This was t...