Authors: Janković, Slobodanka 
Merkle, Milan
Affiliations: Mathematical Institute of the Serbian Academy of Sciences and Arts 
Title: A mean value theorem for systems of integrals
Journal: Journal of Mathematical Analysis and Applications
Volume: 342
Issue: 1
First page: 334
Last page: 339
Issue Date: 1-Jun-2008
Rank: M21
ISSN: 0022-247X
DOI: 10.1016/j.jmaa.2007.12.012
More than a century ago, G. Kowalewski stated that for each n continuous functions on a compact interval [a, b], there exists an n-point quadrature rule (with respect to Lebesgue measure on [a, b]), which is exact for given functions. Here we generalize this result to continuous functions with an arbitrary positive and finite measure on an arbitrary interval. The proof relies on a new version of Carathéodory's convex hull theorem, that we also prove in the paper. As an application, we give a discrete representation of second order characteristics for a family of continuous functions of a single random variable.
Keywords: Carathéodory's convex hull theorem | Correlation | Covariance | Quadrature rules
Publisher: Elsevier
Project: Ministry of Science and Environmental Protection of Serbia, project number 144021

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