Related papers: Integral representation of functions on the circle
In this paper, we present a mixed-type integral-sum representation of the cylinder functions $\mathscr{C}_\mu(z)$, which holds for unrestricted complex values of the order $\mu$ and for any complex value of the variable $z$. Particular…
Every measurable function f on the circle can be represented as a sum of harmonics with positive spectrum, converging in measure. For convergence almost everywhere this is not true. We discuss several other subsets of Z for which one might…
We calculate the Fourier transform of a spherically symmetric exponential function. Our evaluation is much simpler than the known one. We use the polar coordinates and reduce the Fourier transform to the integral of a rational function of…
On any metric space, I provide an intrinsic characterization of those complex-valued functions which are uniform limits of Lipschitz functions. There are applications to function theory on complete Riemannian manifolds and, in particular,…
In the last decade there has been a growing interest in superoscillations in various fields of mathematics, physics and engineering. However, while in applications as optics the local oscillatory behaviour is the important property, some…
It is proved that harmonic functions are characterized by harmonicity of their spherical means, for which purpose the iterated spherical means are used. The similar characterization of solutions to the modified Helmholtz equation…
Series representations consisting of spherical harmonics are obtained for characteristic exponents and probability density functions of multivariate stable distributions under various conditions. A esult potentially applicable in a…
We give a technical overview of our exact-real implementation of various representations of the space of continuous unary real functions over the unit domain and a family of associated (partial) operations, including integration, range…
Several estimates for singular integrals, maximal functions and the spherical summation operator are given in the spaces $L^p_{\text{rad}}L^2_{\text{ang}}(\mathbb{R}^n)$, $n\geq 2$.
We will describe the Ziegler spectrum of the ring of entire complex valued functions
We study the algebraic and geometric properties of the integral closure of different rings of functions on a real algebraic variety : the regular functions and the continuous rational functions.
We survey a few classes of analytic functions on the disk that have real boundary values almost everywhere on the unit circle. We explore some of their properties, various decompositions, and some connections these functions make to…
We obtain rigorous results concerning the evaluation of integrals on the two sphere using complex methods. It is shown that for regular as well as singular functions which admit poles, the integral can be reduced to the calculation of…
We study real-valued valuations on the space of Lipschitz functions over the Euclidean unit sphere $S^{n-1}$. After introducing an appropriate notion of convergence, we show that continuous valuations are bounded on sets which are bounded…
A Herglotz function is a holomorphic map from the open complex unit disk into the closed complex right halfplane. A classical Herglotz function has an integral representation against a positive measure on the unit circle. We prove a free…
We give a general expression of spherical functions on $p$-adic homogeneous spaces of $G$, based on data of $G$ and functional equations of spherical functions. Then, we show a unified method to obtain functional equations of spherical…
The classical integral representation formulas for holomorphic functions defined on pseudoconvex domains in Stein manifolds play an important role in the constructive theory of functions of several complex variables. In this paper we…
Integral representations are derived for the parabolic cylinder functions $U(a,x)$, $V(a,x)$ and $W(a,x)$ and their derivatives. The new integrals will be used in numerical algorithms based on quadrature. They follow from contour integrals…
A complex-analytic structure within the unit disk of the complex plane is presented. It can be used to represent and analyze a large class of real functions. It is shown that any integrable real function can be obtained by means of the…
Let $\Omega\Subset\mathbb{C}^{n}$ be a domain with smooth boundary, $k\in\mathbb{N}$. It is shown that the integral of a holomorphic function in $L^1(\Omega)$ may be represented as the integral of this function against a smooth function…