Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9939-1976-0425470-0
MathSciNet review: 0425470
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 41A65

Retrieve articles in all journals with MSC: 41A65


Additional Information

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