A simple proof of the Hobby-Rice theorem

Author:
Allan Pinkus

Journal:
Proc. Amer. Math. Soc. **60** (1976), 82-84

MSC:
Primary 41A65

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* *be n real functions in* , where *is a finite, nonatomic, real measure. Then there exist* *such that*

A matrix version of the above theorem is also proven. These results are of importance in the study of -approximation.

**[1]**E. W. Cheney,*Applications of fixed-point theorems to approximation theory*, Theory of approximation, with applications (Proc. Conf., Univ. Calgary, Calgary, Alta., 1975; dedicated to the memory of Eckard Schmidt), Academic Press, New York, 1976, pp. 1–8. MR**0417655****[2]**Charles R. Hobby and John R. Rice,*A moment problem in 𝐿₁ approximation*, Proc. Amer. Math. Soc.**16**(1965), 665–670. MR**0178292**, 10.1090/S0002-9939-1965-0178292-5**[3]**Samuel Karlin and William J. Studden,*Tchebycheff systems: With applications in analysis and statistics*, Pure and Applied Mathematics, Vol. XV, Interscience Publishers John Wiley & Sons, New York-London-Sydney, 1966. MR**0204922****[4]**M. G. Kreĭn,*The ideas of P. L. Čebyšev and A. A. Markov in the theory of limiting values of integrals and their further development*, Uspehi Matem. Nauk (N.S.)**6**(1951), no. 4 (44), 3–120 (Russian). MR**0044591****[5]**C. A. Micchelli and A. Pinkus,*On n-widths in*, IBM Research Report #5478, 1975.**[6]**L. Nirenberg,*Topics in nonlinear functional analysis*, Courant Institute of Mathematical Sciences, New York University, New York, 1974. With a chapter by E. Zehnder; Notes by R. A. Artino; Lecture Notes, 1973–1974. MR**0488102**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1976-0425470-0

Keywords:
-approximation

Article copyright:
© Copyright 1976
American Mathematical Society