A number of new definite integrals involving Bessel functions are presented. These have been derived by finding new integral representations for the product of two Bessel functions of different order and argument in terms of the generalized hypergeometric function with subsequent reduction to special cases. Connection is made with Weber's second exponential integral and Laplace transforms of pr...

An overview of two types of beam solutions is presented, Gaussian beams and Bessel beams. Gaussian beams are examples of non-localized or diffracting beam solutions, and Bessel beams are example of localized, non-diffracting beam solutions. Gaussian beams stay bounded over a certain propagation range after which they diverge. Bessel beams are among a class of solutions to the wave equation that...

A two-parameter sequence of orthogonal polynomials {Pn(x;λ,t)}n≥0 with respect to the weight function xαe−λxρν(xt),α>−1,λ,t≥0,ρν(x)=2xν/2Kν(2x),x>0,ν≥0, where Kν(z) is modified Bessel function, investigated. The case λ=0 corresponds Prudnikov and t = 0 related Laguerre polynomials. special one-parameter {Pn(x;1−t,t)}n≥0,t∈[0,1] analysed as well.

In this dissertation, we study the Feichtinger Conjecture(FC), which has been shown to be equivalent to the celebrated Kadison-Singer Problem. The FC states that every norm-bounded below Bessel sequence in a Hilbert space can be partitioned into finitely many Riesz basic sequences. This study is divided into two parts. In the first part, we explore the FC in the setting of reproducing kernel Hi...

Using an extension of the Cox-Ingersoll-Ross [1] stochastic model of the short interest rate r, we study the convergence in law of the longterm return in order to make some approximations. We use the theory of Bessel processes and observe the convergence in law of the sequence (√ −2β3 δn ∫ nt 0 (Xu + δu 2β )du ) t≥0 with the X a generalized Besselsquare process with drift with stochastic revers...

The theory of localized frames is a recently introduced concept with broad implications to frame theory in general, as well as to the special cases of Gabor and wavelet frames. Using the new notion of a R-dual sequence associated with a Bessel sequence, we derive several duality principles concerning localization in abstract frame theory. As applications of our results we prove a duality princi...

We give two distinct approaches to finding bounds, as functions of the order ν, for the smallest real or purely imaginary zero of Bessel and some related functions. One approach is based on an old method due to Euler, Rayleigh, and others for evaluating the real zeros of the Bessel function Jν(x) when ν > −1. Here, among other things, we extend this method to get bounds for the two purely imagi...

In this paper the asymptotic behavior of an unstable integer-valued autoregressive model of order p (INAR(p)) is described. Under a natural assumption it is proved that the sequence of appropriately scaled random step functions formed from an unstable INAR(p) process converges weakly towards a squared Bessel process. We note that this limit behavior is quite different from that of familiar unst...

In this paper, we introduce the concept of dual frame of g-p-frame, and give the sufficient condition for a g-p-frame to have dual frames. Using operator theory and methods of functional analysis, we get some new properties of g-p-frame. In addition, we also characterize g-p-frame and g-q-Riesz bases by using analysis operator of g-p-Bessel sequence. c ©2017 All rights reserved.