A unified approach to uniform real approximation by polynomials with linear restrictions

Author:
Bruce L. Chalmers

Journal:
Trans. Amer. Math. Soc. **166** (1972), 309-316

MSC:
Primary 41A50

DOI:
https://doi.org/10.1090/S0002-9947-1972-0294962-0

MathSciNet review:
0294962

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Abstract | References | Similar Articles | Additional Information

Abstract: Problems concerning approximation of real-valued continuous functions of a real variable by polynomials of degree smaller than *n* with various linear restrictions have been studied by several authors. This paper is an attempt to provide a unified approach to these problems. In particular, the notion of restricted derivatives approximation is seen to fit into the theory and includes as special cases the notions of monotone approximation and restricted range approximation. Also bounded coefficients approximation, -interpolator approximation, and polynomial approximation with interpolation fit into our scheme.

**[1]**K. Atkinson and A. Sharma,*A partial characterization of poised Hermite-Birkhoff interpolation problems*, SIAM J. Numer. Anal.**6**(1969), 230-235. MR**41**#9419. MR**0264828 (41:9419)****[2]**C. B. Dunham,*Chebyshev approximation with respect to a weight function*, J. Approximation Theory**2**(1969), 223-232. MR**40**#6137. MR**0252922 (40:6137)****[3]**David Ferguson,*The question of uniqueness for G. D. Birkhoff interpolation problems*, J. Approximation Theory**2**(1969), 1-28. MR**40**#599. MR**0247331 (40:599)****[4]**M. A. Griesel,*Uniform approximation with respect to a general error function*, Doctoral Dissertation, University of California, Riverside, Calif., 1971.**[5]**H. L. Loeb, D. G. Moursund, L. L. Schumaker and G. D. Taylor,*Uniform generalized weight function polynomial approximation with interpolation*, SIAM J. Numer. Anal.**6**(1969), 284-293. MR**40**#3679. MR**0250441 (40:3679)****[6]**G. G. Lorentz and K. L. Zeller,*Monotone approximation by algebraic polynomials*, Trans. Amer. Math. Soc.**149**(1970), 1-18. MR**0285843 (44:3060)****[7]**R. A. Lorentz,*Uniqueness of approximation by monotone polynomials*(to appear).**[8]**D. G. Moursund,*Chebyshev approximation using a generalized weight function*, SIAM J. Numer. Anal.**3**(1966), 435-450. MR**34**#4771. MR**0204936 (34:4771)****[9]**T. J. Rivlin and H. S. Shapiro,*A unified approach to certain problems of approximation and minimization*, SIAM J. Numer. Anal.**9**(1961), 670-699. MR**24**#A3462. MR**0133636 (24:A3462)****[10]**J. A. Roulier,*Monotone approximation of certain classes of continuous functions*, J. Approximation Theory**1**(1968), 319-324. MR**38**#4875. MR**0236580 (38:4875)****[11]**J. A. Roulier and G. D. Taylor,*Uniform approximation by polynomials having bounded coefficients*, Abh. Math. Sem. Univ. Hamburg**36**(1971), 126-135. MR**0336177 (49:953)****[12]**I. J. Schoenberg,*On Hermite-Birkhoff interpolation*, J. Math. Anal. Appl.**16**(1966), 538-543. MR**34**#3160. MR**0203307 (34:3160)****[13]**O. Shisha,*Monotone approximation*, Pacific J. Math.**15**(1965), 667-671. MR**32**#2802. MR**0185334 (32:2802)****[14]**G. D. Taylor,*On approximation by polynomials having restricted ranges*, SIAM J. Numer. Anal.**5**(1968), 258-268. MR**40**#6142. MR**0252927 (40:6142)****[15]**-,*Approximation by functions having restricted ranges*. III, J. Math. Anal. Appl.**27**(1969), 241-248. MR**41**#2261. MR**0257611 (41:2261)**

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DOI:
https://doi.org/10.1090/S0002-9947-1972-0294962-0

Keywords:
Approximation with linear restrictions,
polynomials of best approximation,
extremal sets,
uniqueness of best approximation,
generalized Haar condition

Article copyright:
© Copyright 1972
American Mathematical Society