A simple proof of the Hobby-Rice theorem

Author:
Allan Pinkus

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

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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* \[ \sum \limits _{j = 1}^{r + 1} {{{( - 1)}^j}\int _{{\xi _{j - 1}}}^{{\xi _j}} {{\varphi _i}(x)\;d\mu (x) = 0,\quad i = 1, \ldots ,n.}}\] 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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Keywords:
<IMG WIDTH="28" HEIGHT="23" ALIGN="BOTTOM" BORDER="0" SRC="images/img1.gif" ALT="${L^1}$">-approximation

Article copyright:
© Copyright 1976
American Mathematical Society