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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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A simple proof of the Hobby-Rice theorem
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by Allan Pinkus PDF
Proc. Amer. Math. Soc. 60 (1976), 82-84 Request permission

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 \[ \sum \limits _{j = 1}^{r + 1} {{{( - 1)}^j}\int _{{\xi _{j - 1}}}^{{\xi _j}} {{\varphi _i}(x)\;d\mu (x) = 0,\quad i = 1, \ldots ,n.}}\] 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
  • © Copyright 1976 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 60 (1976), 82-84
  • MSC: Primary 41A65
  • DOI: https://doi.org/10.1090/S0002-9939-1976-0425470-0
  • MathSciNet review: 0425470