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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

A simple proof of the Hobby-Rice theorem


Author: Allan Pinkus
Journal: Proc. Amer. Math. Soc. 60 (1976), 82-84
MSC: Primary 41A65
MathSciNet review: 0425470
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Abstract: This paper presents a simple proof of the following theorem due to Hobby and Rice.

Theorem. Let $ \{ {\varphi _i}(x)\} _{i = 1}^n$ be n real functions in $ {L^1}(d\mu ;[0,1])$, where $ \mu $ is a finite, nonatomic, real measure. Then there exist $ \{ {\xi _i}\} _{i = 1}^r,r \leqslant n,0 = {\xi _0} < {\xi _1} < \cdots < {\xi _r} < {\xi _{r + 1}} = 1$ such that

$\displaystyle \sum\limits_{j = 1}^{r + 1} {{{( - 1)}^j}\int_{{\xi _{j - 1}}}^{{\xi _j}} {{\varphi _i}(x)\;d\mu (x) = 0,\quad i = 1, \ldots ,n.}}$

A matrix version of the above theorem is also proven. These results are of importance in the study of $ {L^1}$-approximation.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1976-0425470-0
PII: S 0002-9939(1976)0425470-0
Keywords: $ {L^1}$-approximation
Article copyright: © Copyright 1976 American Mathematical Society