Characterization of Radon integrals as linear functionals
type: Item
instance of: scholarly article
title: Characterization of Radon integrals as linear functionals
zbMATH Open document ID: 1358.28010
published in: Journal of Mathematical Sciences (New York)
publication date: 2013-01-10T00:00:00Z
review text: Many mathematicians contributed to the theory of integration creating a variety of different techniques and approaches in this area and building integrals of different forms, types and structures. It is possible to classify this diversity of integrals inducing a definite order in their multiplicity. The main two classes of integrals are definite integrals, which are linear functionals, and indefinite integrals, which are linear operators. The further categorization of definite integrals constructed in [\textit{M. Burgin}, ``Directions and methods in integration'', in: Functional analysis and probability. New York, NY: Nova Science Publishers. 127--146 (2015)] contains three basic types of integration methodologies: constructive approaches and techniques, axiomatic approaches and techniques, and mixed approaches and techniques. An interesting and important problem in the area of integration is to find relations between constructive and axiomatic types of integrals. Namely, it is possible to ask two questions: what classes of linear functionals, which form the basis for axiomatic approaches, can be represented as constructive integrals, and how to axiomatically characterize constructive integrals in the class of all linear functionals. The paper under review makes a substantial advancement in the second direction, which originated with Jacques Hadamard and Maurice Fréchet. In his paper [C. R. Acad. Sci., Paris 136, 351--354 (1903; JFM 34.0419.06)], \textit{J. Hadamard} proved that it is possible to represent every linear bounded functional \(U(f)\) on the space \(C[a,b]\) of all continuous functions in the interval \([a,b]\) as a limit of the Cauchy-Riemann integrals where \[ U(f)=\lim_{n\to\infty}\int^b_a f(x) H_n(x)\mathrm dx \] In his turn, \textit{M. Fréchet} proved [Trans. Am. Math. Soc. 5, 493--499 (1904; JFM 35.0389.03)] that any bounded linear functional \(U(f)\) on the space \(C^n[a,b]\) can be represented in the form \[ U(f)=\sum_{i=0}^{n-1}A_jf^{(j)}(x)+\li…
Mathematics Subject Classification ID: 28C05
zbMATH DE Number: 6124257
author: V. K. Zakharov
author: A. V. Mikhalev
author: Timofey V. Rodionov
reviewed by: Mark Burgin
MaRDI profile type: Publication
full work available at URL: s10958-012-0913-1
OpenAlex ID: W2038712559
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DOI: 10.1007/S10958-012-0913-1
description: scientific article; zbMATH DE number 6124257