On two new characterizations of Stieltjes transforms for distributions
type: Item
instance of: scholarly article
title: On two new characterizations of Stieltjes transforms for distributions
zbMATH Open document ID: 0605.46033
DOI: 10.1155/S0161171285000801
author: Sunil Kumar Sinha
published in: International Journal of Mathematics and Mathematical Sciences
publication date: 1985-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/46118
review text: For a function f, a complex number p, and a positive integer n, the transformations, dilation \(u_ n\) and exponential shift \(T^{-p}\) are respectively defined as: \[ u_ nf(t)=nf(nt)\quad and\quad T^{- p}f(t)=e^{-pt}f(t). \] Following the technique used by Gasztelyi for defining the Laplace transform, the present author has characterized the Stieltjes transform of distributions in the following two ways: I. If f is a Stieltjes transformable distribution (in a certain space), then the Stieltjes transform of f is defined as: \[ S[f](p)=(1/\phi (0))\lim_{j\to \infty}<u_ jT^{-p} f,\phi > \] where \(f(t)=S[\xi](p)\), \(0\leq t\leq R\) for every positive R, \(\phi\) is a test function in D, with \(\phi\) (0)\(\neq 0\) and \(\xi\) is another distribution. II. Another characterization of the Stieltjes transform of a distribution f is given as: \[ S[f](p)=<T^{-p} f,1> \] where \(f(t)=S[\xi](p)\), \(0\leq t\leq R\) for any positive R and \(\xi\) is another distribution. The standard theorem on analyticity, uniqueness and invertibility of Stieltjes transform using the above definitions are proved.
reviewed by: Lakshmi Shanker Dube
Mathematics Subject Classification ID: 46F12
Mathematics Subject Classification ID: 44A15
zbMATH DE Number: 3977698
zbMATH Keywords: dilation
zbMATH Keywords: exponential shift
zbMATH Keywords: Laplace transform
zbMATH Keywords: Stieltjes transform of distributions
zbMATH Keywords: analyticity
zbMATH Keywords: uniqueness
zbMATH Keywords: invertibility
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description: scientific article; zbMATH DE number 3977698