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 systems of nonorthogonal Fourier-Bessel functions and their q-analogues, as well as other complete sets of q-special functions. The completeness of certain sets of q-Bessel functions is then used to prove that, if a func...

In this paper we deal with integrals whose integrand has a rapidly oscillating zeroorder Bessel function of the first kind with real parameters in its argument which can become large. We introduce and tabulate model integrals which depend on a single parameter, which can determine the behavior of the original integral near the zeros of the argument of the Bessel function. As an example of the u...

Nondiffracting beams are useful for alignment applications because the size of the beam does not change as the beam propagates. In this research we report a technique that allows for distance measurements with nondiffracting beams. With our approach a diffractive optical element is designed that generates two off-axis, tilted, nondiffracting Bessel function beams. These beams intersect at a des...

An efficient automatic quadrature procedure is developed for numerically computing the integrals 0 , where the function is smooth and nonoscillatory at infinity and is the Bessel functions of order ν =1,0 and 1/4. The procedure involves the use of an automatic integration scheme of modified FFT used for evaluating Fourier integrals and product type integration, and the modified W-transformation...

Indefinite inner product spaces of entire functions and functions analytic inside a disk are considered and their completeness studied. Spaces induced by the rotation invariant reproducing kernels in the form of the generalized hypergeometric function are completely characterized. A particular space generated by the modified Bessel function kernel is utilized to derive an analytic continuation ...

In this short note, by using mathematical induction and infinite product representations of the functions Jp : R → (−∞, 1] and Ip : R → [1,∞), defined by Jp(x) = 2Γ(p + 1)xJp(x) and Ip(x) = 2Γ(p + 1)xIp(x), an extension of Redheffer’s inequality for the function Jp and a Redheffer-type inequality for the function Ip are established. Here Jp and Ip, denotes the Bessel function and modified Besse...

The purpose of this paper is to present an algorithm for evaluating Hankel transform of the null and the first kind to the series of the Bessel function for Hankel transform of the first kind and Bessel and Struve function for Hankel transform of the null kind multiplied to the wavelet coefficients of the input function. It is both the exact analytical representation and the base for effective ...

We provide a proof of a conjecture in [2] on the existence and form of linear recursions for moments of powers of the Bessel function K0.

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...

We provide a proof of a conjecture in [2] on the existence and form of linear recursions for moments of powers of the Bessel function K0.