Related papers: A new index transform with the square of Whittaker…
This article handles in a short manner a few Laplace transform pairs and some extensions to the basic equations are developed. They can be applied to a wide variety of functions in order to find the Laplace transform or its inverse when…
Classical integral representation of the Mellin type kernel in terms of the Laplace integral gives an idea to construct a new class of non-convolution (index) transforms. Particular examples give the Kontorovich-Lebedev-like transformation…
In this paper, we exploit the theory of convolution of index Whittaker transform for study of continuous and discrete Index Whittaker wavelet transform and discuss some of its basic properties. Certain boundedness, Plancherel as well as…
Notions of the orthogonality and convolution orthogonality are explored with the use of the Kontorovich-Lebedev transform and its convolution. New classes of the corresponding orthogonal polynomials and functions are investigated. Integral…
Identities between Whittaker and modified Bessel functions are derived for particular complex orders. Certain polynomials appear in such identities, which satisfy a fourth order differential equation (not of hypergeometric type), and they…
The Whittaker function and its diverse extensions have been actively investigated. Here we introduce an extension of the Whittaker function by using the known extended confluent hypergeometric function $\Phi_{p,v}$ and investigate some of…
The paper contains the inversion formula for the weighted spherical mean. The interest to reconstruction a function by its integral by sphere grews tremendously in the last six decades, stimulated by the spectrum of new problems and methods…
The Fourier transform is naturally defined for integrable functrions. Otherwise, it should be stipulated in which sense the Fourier transform is understood. We consider some class of radial and, generally saying, nonintegrable functions.…
Discrete analogs of the Lebedev-Skalskaya transforms are introduced and investigated. It involves series and integrals with respect to the kernels ${\rm Re} K_{\alpha+in}(x), {\rm Im} K_{\alpha+in}(x), x >0, n \in \mathbb{N}, |\alpha | <…
Several identities of the cosh-weighted finite Hilbert Transform and the Bertola-Katsevich-Tovbis inversion formulas are rederived by the Sokhotski-Plemelj formula and the Poincare-Bertrand formula. The explicit formulas are derived for the…
The present article is devoted to one example which related to the Salem function. The main attention is given to properties of one type of functions including items related to functional equations, graphs, the Lebesgue integral, etc.
We give a new formula for the rotation number (or Whitney index) of a smooth closed plane curve. This formula is obtained from the winding numbers associated with the regions and the crossing points of the curve. One difference with the…
Inversion of function sinc(x) is studied. New series and integral representations of branches of inverse function are obtained using Fourier analysis.
The inversion of nabla Laplace transform, corresponding to a causal sequence, is considered. Two classical methods, i.e., residual calculation method and partial fraction method are developed to perform the inverse nabla Laplace transform.…
Mathematical functions, which often appear in mathematical analysis, are referred to as special functions and have been studied over hundreds of years. Many books and dictionaries are available that describe their properties and serve as a…
We exhibit new biorthogonal sequences generated by index integrals of the squares of the modified Bessel functions and gamma functions. The composition orthogonality, involving differential operators is employed. Generalized Wilson…
We consider an integral transform given by $T_{\nu} f(s) := \pi \int_0^\infty rs J_{\nu}(r s)^2 f(r) \, dr$, where $J_{\nu}$ denotes the Bessel function of the first kind of order $\nu$. As shown by Walther (2002,…
Finding a computationally efficient algorithm for the inverse continuous wavelet transform is a fundamental topic in applications. In this paper, we show the convergence of the inverse wavelet transform.
Discrete analogs of the classical Kontorovich-Lebedev transforms are introduced and investigated. It involves series with the modified Bessel function or Macdonald function $K_{in}(x), x >0, n \in \mathbb{N}, i $ is the imaginary unit, and…
We define a scalar valued Fourier transform for functions on the Heisenberg group and establish some of its basic properties like inversion formula, Plancherel theorem and Riemann-Lebesgue lemma. We also restate certain well known theorems…