A simple proof of the Hobby-Rice theorem

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.