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A Simple Proof of the Hobby-Rice Theorem

Allan Pinkus
Proceedings of the American Mathematical Society
Vol. 60, No. 1 (Oct., 1976), pp. 82-84
DOI: 10.2307/2041117
Stable URL: http://www.jstor.org/stable/2041117
Page Count: 3
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A Simple Proof of the Hobby-Rice Theorem
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Abstract

This paper presents a simple proof of the following theorem due to Hobby and Rice. THEOREM. Let $\{\varphi_i(x)\}^n_{i = 1}$ be $n$ real functions in $L^1(d\mu; \lbrack 0, 1 \rbrack)$, where $\mu$ is a finite, nonatomic, real measure. Then there exist $\{\xi_i\}^r_{i = 1}, r \leqslant n, 0 = \xi_0 < \xi_1 < \cdots < \xi_r < \xi_{r + 1} = 1$ such that $$\sum_{j=1} ^{r + 1} (-1)^j \int^{\xi_j}_{\xi_{j - 1}} \varphi_i(x)d\mu(x) = 0, \quad i = 1,\ldots, n.$$ 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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