Related papers: Cosine products, Fourier transforms, and random su…
The paper deals with continuous and compact mappings generated by the Fourier transform between distinguished function spaces on $\mathbb{R}^n$. The degree of compactness will be measured in terms of related entropy numbers. We are more…
By using computers to do experimental manipulations on Fourier series, we construct additional series with interesting properties. We construct several series whose sums remain unchanged when the $n^{th}$ term is multiplied by $\sin(n)/n$.…
We study right exact tensor products on the category of finitely presented functors. As our main technical tool, we use a multilinear version of the universal property of so-called Freyd categories. Furthermore, we compare our constructions…
We first review the introduction of star products in connection with deformations of Poisson brackets and the various cohomologies that are related to them. Then we concentrate on what we have called ``closed star products" and their…
It is shown that quasi all continuous functions on the unit circle have the property that, for many small subsets E of the circle, the partial sums of their Fourier series considered as functions restricted to E exhibit certain universality…
This note simplifies the proof of a recent result on the oscillation of the prime product in Martens Theorem, and provides a quantitative expression for the error term. In addition, the corresponding oscillation results for the finite sums…
This note presents a rigorous introduction to a selection of distributions along with their Fourier transforms, which are commonly encountered in signal processing and, in particular, magnetic resonance imaging (MRI). In contrast to many…
We introduce a finite version of free probability and show the link between recent results using polynomial convolutions and the traditional theory of free probability. One tool for accomplishing this is a seemingly new transformation that…
Taking the product of (2n+1)/(2n+2) raised to the power +1 or -1 according to the n-th term of the Thue-Morse sequence gives rise to an infinite product P while replacing (2n+1)/(2n+2) with (2n)/(2n+1) yields an infinite product Q, where P…
From a suitable integral representation of the Laplace transform of a positive semi-definite quadratic form of independent real random variables with not necessarily identical densities a univariate integral representation is derived for…
We present a family of explicit formulae for evaluating absolute moments of probability measures on $\mathbb{R}^d$ in terms of Fourier transforms. As to the space of probability measures possessing finite absolute moments of an arbitrary…
The wrapped normal distribution arises when a the density of a one-dimensional normal distribution is wrapped around the circle infinitely many times. At first look, evaluation of its probability density function appears tedious as an…
A construction of product measures is given for an arbitrary sequence of measure spaces via outer measure techniques without imposing any condition on the underlying measure spaces. This approach concludes finally the problem of the…
The finite and infinite harmonic sums form the general basis for the Mellin transforms of all individual functions $f_i(x)$ describing inclusive quantities such as coefficient and splitting functions which emerge in massless field theories.…
Infinite series of the type Sum{n=1,infinity}(alpha/2)_n_2F_1(-n, b; gamma; y)/(n n!) are investigated. Closed-form sums are obtained for alpha a positive integer alpha=1,2,3, ... The limiting case of b --> infinity, after y is replaced…
The twisted product of functions on $R^{2N}$ is extended to a $*$-algebra of tempered distributions which contains the rapidly decreasing smooth functions, the distributions of compact support, and all polynomials, and moreover is invariant…
Finite trigonometric sums occur in various branches of physics, mathematics, and their applications. These sums may contain various powers of one or more trigonometric functions. Sums with one trigonometric function are known, however sums…
Sums of independent random variables form the basis of many fundamental theorems in probability theory and statistics, and therefore, are well understood. The related problem of characterizing products of independent random variables seems…
First passage distributions of semi-Markov processes are of interest in fields such as reliability, survival analysis, and many others. The problem of finding or computing first passage distributions is, in general, quite challenging. We…
Using appropriate power series evaluations, we determine all moments of arbitrary positive powers of the arcsine. As consequences we evaluate several doubly infinite classes of power series involving central binomial coefficients and…