Discrete rational approximation

Author:
Jerry M. Wolfe

Journal:
Math. Comp. **29** (1975), 540-548

MSC:
Primary 41A50; Secondary 65D15

MathSciNet review:
0377377

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Abstract: In this paper, the problem of approximating a function defined on a finite subset of the real line by a family of generalized rational functions whose numerator and denominator spaces satisfy the Haar conditions on some closed interval [*a, b*] containing the finite set is considered. The pointwise closure of the family restricted to the finite set is explicitly determined. The representation obtained is used to analyze the convergence of best approximations on discrete subsets of [*a, b*] to best approximations over the whole interval (as the discrete subsets become dense) in the case that the function approximated is continuous on [*a, b*] and the rational family consists of quotients of algebraic polynomials. It is found that the convergence is uniform over [*a, b*] if the function approximated is a so-called normal point. Only norms with are employed.

**[1]**H. Werner,*Die Bedeutung der Normalität bei rationaler Tschebyscheff-Approximation*, Computing (Arch. Elektron. Rechnen)**2**(1967), 34–52 (German, with English summary). MR**0212466****[2]**MICHAEL E. BURKE,*Discrete Nonlinear and Simultaneous Approximation*, Thesis, University of Oregon, 1971.**[3]**E. W. Cheney and A. A. Goldstein,*Mean-square approximation by generalized rational functions*, Math. Z.**95**(1967), 232–241. MR**0219952****[4]**Charles B. Dunham,*Degeneracy in mean rational approximation*, J. Approximation Theory**4**(1971), 225–229. MR**0282110**

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DOI:
http://dx.doi.org/10.1090/S0025-5718-1975-0377377-2

Article copyright:
© Copyright 1975
American Mathematical Society