Explicit expressions and computational methods for the Fortet–Mourier distance of positive measures to finite weighted sums of Dirac measures

IF 0.9 3区数学 Q2 MATHEMATICS Journal of Approximation Theory Pub Date : 2023-10-01 DOI:10.1016/j.jat.2023.105947

Sander C. Hille , Esmée S. Theewis

引用次数: 0

Abstract

Explicit expressions and computational approaches are given for the Fortet–Mourier distance between a positively weighted sum of Dirac measures on a metric space and a positive finite Borel measure. Explicit expressions are given for the distance to a single Dirac measure. For the case of a sum of several Dirac measures one needs to resort to a computational approach. In particular, two algorithms are given to compute the Fortet–Mourier norm of a molecular measure, i.e. a finite weighted sum of Dirac measures. It is discussed how one of these can be modified to allow computation of the dual bounded Lipschitz (or Dudley) norm of such measures.

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Dirac测度有限加权和正测度的Fortet-Mourier距离的显式表达式和计算方法

给出了度量空间上Dirac测度的正加权和与正有限Borel测度之间的Fortet–Mourier距离的显式表达式和计算方法。给出了到单个Dirac测度的距离的显式表达式。对于几个狄拉克测度之和的情况，需要采用计算方法。特别地，给出了两种算法来计算分子测度的Fortet–Mourier范数，即Dirac测度的有限加权和。讨论了如何修改其中一个，以允许计算此类测度的对偶有界Lipschitz（或Dudley）范数。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Journal of Approximation Theory 数学-数学

CiteScore

1.90

自引率

11.10%

发文量

审稿时长

6-12 weeks

期刊介绍： The Journal of Approximation Theory is devoted to advances in pure and applied approximation theory and related areas. These areas include, among others: • Classical approximation • Abstract approximation • Constructive approximation • Degree of approximation • Fourier expansions • Interpolation of operators • General orthogonal systems • Interpolation and quadratures • Multivariate approximation • Orthogonal polynomials • Padé approximation • Rational approximation • Spline functions of one and several variables • Approximation by radial basis functions in Euclidean spaces, on spheres, and on more general manifolds • Special functions with strong connections to classical harmonic analysis, orthogonal polynomial, and approximation theory (as opposed to combinatorics, number theory, representation theory, generating functions, formal theory, and so forth) • Approximation theoretic aspects of real or complex function theory, function theory, difference or differential equations, function spaces, or harmonic analysis • Wavelet Theory and its applications in signal and image processing, and in differential equations with special emphasis on connections between wavelet theory and elements of approximation theory (such as approximation orders, Besov and Sobolev spaces, and so forth) • Gabor (Weyl-Heisenberg) expansions and sampling theory.

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