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* *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, edited by A. G. Law and B. N. Sahney, Academic Press, New York, 1976, pp. 1-8. MR**0417655 (54:5705)****[2]**C. R. Hobby and J. R. Rice,*A moment problem in**approximation*, Proc. Amer. Math. Soc.**16**(1965), 665-670. MR**31**#2550. MR**0178292 (31:2550)****[3]**S. Karlin and W. J. Studden,*Tchebycheff systems*:*With applications in analysis and statistics*, Interscience, New York, 1966. MR**34**#4757. MR**0204922 (34:4757)****[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 Mat. Nauk**6**(1951), no. 4 (44), 3-120; English transl., Amer. Math. Soc. Transl. (2)**12**(1959), 1-122. MR**13**, 445;**22**#3947a. MR**0044591 (13:445c)****[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 Inst. of Math. Sciences, New York, 1974. MR**0488102 (58:7672)**

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

Retrieve articles in all journals with MSC: 41A65

Additional Information

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

Keywords:
-approximation

Article copyright:
© Copyright 1976
American Mathematical Society