Related papers: A short study of generalized Dirichlet Integrals v…
We implement an efficient method of computation of two dimensional Fourier-type integrals based on approximation of the integrand by Gaussian radial basis functions, which constitute a standard tool in approximation theory. As a result, we…
A method for obtaining discretization formulas for the derivatives of a function is presented, which relies on a generalization of divided differences. These modified divided differences essentially correspond to a change of the dependent…
In this paper, we investigate fractional B splines and their connections with Fourier analysis, and establish connections with generalized Stirling-type numbers and distribution theory. Employing a generating function approach inspired by…
The main aim of the paper is to present a general version of the Fourier Tauberian theorem for monotone functions. This result, together with Berezin's inequality, allows us to obtain a refined version the Li-Yau estimate for the counting…
We investigate Bochner integrabilities of generalized Wiener functionals. We further formulate an It\^o formula for a diffusion in a distributional setting, and apply to investigate differentiability-index $s$ and integrability-index $p…
We provide an algorithm for computing semi-Fourier sequences for expressions constructed from arithmetic operations, exponentiations and integrations. The semi-Fourier sequence is a relaxed version of Fourier sequence for polynomials…
We provide a rigorous convergence proof demonstrating that the well-known semi-analytical Fourier cosine (COS) formula for the inverse Fourier transform of continuous probability distributions can be extended to discrete probability…
Inter alia, we present a Fourier series involving the generalised Stieltjes constants.
We introduce a class of integral theorems based on cyclic functions and Riemann sums approximating integrals. The Fourier integral theorem, derived as a combination of a transform and inverse transform, arises as a special case. The…
Tempered fractional derivatives originated from the tempered fractional diffusion equations (TFDEs) modeled on the whole space R (see [23]). For numerically solving TFDEs, two kinds of generalized Laguerre functions were defined and some…
In this paper, we consider a discrete version of iterated integrals by the naive (equally divided) Riemann sum. In particular, basic three formulas for usual iterated integrals are discritized. Moreover, we proved cyclic sum formulas for…
Integro-partial differential equations occur in many contexts in mathematical physics. Typical examples include time-dependent diffusion equations containing a parameter (e.g., the temperature) that depends on integrals of the unknown…
Inversion theorems of Wiener type are essential tools in analysis and number theory. We derive a weighted version of an inversion theorem of Wiener type for general Dirichlet series from that of Edwards from 1957, and we outline an…
We present a short overview of the recent results in the theory of diffusion and wave equations with generalised derivative operators. We give generic examples of such generalised diffusion and wave equations, which include time-fractional,…
New integral formulas involving the Meijer $G$-function are derived using recent results concerning distributional characterisations and distributional transformations in probability theory.
The Dirichlet form is a generalization of the Laplacian, heavily used in the study of many diffusion-like processes. In this paper we present a nonstandard representation theorem for the Dirichlet form, showing that the usual Dirichlet form…
In this paper, we introduce an algorithm that provides approximate solutions to semi-linear ordinary differential equations with highly oscillatory solutions, which, after an appropriate change of variables, can be rewritten as…
This article proposes a new approach in the treatment of the Hilbert transform and some cases of the Fourier transform whose improper integrals are principal values. This approach may be useful for teaching these issues to undergraduate…
We propose a discrete fractional random transform based on a generalization of the discrete fractional Fourier transform with an intrinsic randomness. Such discrete fractional random transform inheres excellent mathematical properties of…
The Postnikov character formula is used to express large portions of a Dirichlet character sum in terms of quadratic exponential sums. The quadratic sums are then computed using an analytic algorithm previously derived by the author. This…