Related papers: Uniform convergence of Fourier-Bessel series on a …
We derive completeness criteria for sequences of functions of the form $% f(x\lambda_{n})$, where $\lambda_{n}$ is the $nth$ zero of a suitably chosen entire function. Using these criteria, we construct systems of nonorthogonal…
A q-type Holder condition on a function f is given in order to establish (uniform) convergence of the corresponding basic Fourier series S_q[f] to the function itself, on the set of points of the q-linear grid. Furthermore, by adding others…
We prove orthogonality relations for some analogs of trigonometric functions on a $q$-quadratic grid and introduce the corresponding $q$-Fourier series. We also discuss several other properties of this basic trigonometric system and the…
When dealing with Fourier expansions using the third Jackson (also known as Hahn-Exton) $q$-Bessel function, the corresponding positive zeros $j_{k\nu}$ and the "shifted" zeros, $qj_{k\nu}$, among others, play an essential role. Mixing…
n this paper, we present $q$-Bernoulli and $q$-Euler polynomials generated by the third Jackson $q$-Bessel function to construct new types of $q$-Lidstone expansion theorem. We prove that the entire function may be expanded in terms of…
We obtain a $q$-linear analogue of Gegenbauer's expansion of the plane wave. It is expanded in terms of the little $q$-Gegenbauer polynomials and the \textit{third} Jackson $q$-Bessel function. The result is obtained by using a method based…
We give a proof of the uniform convergence of Fourier series, using the methods of nonstandard analysis.
We derive a convergent expansion of the generalized hypergeometric function ${}_{p-1}F_p$ in terms of the Bessel functions ${}_{0}F_1$ that holds uniformly with respect to the argument in any horizontal strip of the complex plane. We…
In this work, we are interested by the $q$-Bessel Fourier transform with a new approach. Many important results of this $q$-integral transform are proved with a new constructive demonstrations and we establish in particular the associated…
Fourier series approximations of continuous but nonperiodic functions on an interval suffer the Gibbs phenomenon, which means there is a permanent oscillatory overshoot in the neighbourhoods of the endpoints. Fourier extensions circumvent…
In this work, we extend the analytic treatment of Bessel functions of large order and/or argument. We examine uniform asymptotic Bessel function expansions and show their accuracy and range of validity. Such situations arise in a variety of…
The conditions for convergence of square and rectangular Fejer means of functions on the infinite dimensional torus were obtained, also a generalization of the results for the case of abstract measure spaces was formulated.
We propose an algorithm for the orthogonal fast discrete spherical Bessel transform on an uniform grid. Our approach is based upon the spherical Bessel transform factorization into the two subsequent orthogonal transforms, namely the fast…
For the third q-Bessel function (first introduced by F.H. Jackson, later rediscovered by W.Hahn in a special case and by H. Exton) we derive Hansen-Lommel type orthogonality relations, which, by a symmetry, turn out to be equivalent to…
The Bessel function of the first kind $J_{N}\left(kx\right)$ is expanded in a Fourier-Legendre series, as is the modified Bessel functions of the first kind $I_{N}\left(kx\right)$. The purpose of these expansions in Legendre polynomials was…
In the present investigation our main aim is to give lower bounds for the ratio of some normalized $q$-Bessel functions and their sequences of partial sums. Especially, we consider Jackson's second and third $q$-Bessel functions and we…
In this note we consider the Fourier expansion of the Ferrers function P of the first kind. We determine its mode of convergence.
Several problems on Fourier series and trigonometric approximation on a hexagon and a triangle are studied. The results include Abel and Ces\`aro summability of Fourier series, degree of approximation and best approximation by trigonometric…
In the context of the Dunkl transform a complete orthogonal system arises in a very natural way. This paper studies the weighted norm convergence of the Fourier series expansion associated to this system. We establish conditions on the…
We define a q-analog of the modified Bessel and Bessel-Macdonald functions. As for the q-Bessel functions of Jackson there is a couple of functions of the both kind. They are arisen in the Harmonic analysis on quantum symmetric spaces…