Inverse theorem for best polynomial approximation in

Authors:
Z. Ditzian, D. Jiang and D. Leviatan

Journal:
Proc. Amer. Math. Soc. **120** (1994), 151-155

MSC:
Primary 41A25; Secondary 41A10, 41A17, 41A27

MathSciNet review:
1160297

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Abstract | References | Similar Articles | Additional Information

Abstract: A direct theorem for best polynomial approximation of a function in , has recently been established. Here we present a matching inverse theorem. In particular, we obtain as a corollary the equivalence for between and . The present result complements the known direct and inverse theorem for best polynomial approximation in . Analogous results for approximating periodic functions by trigonometric polynomials in , are known.

**[1]**R. A. DeVore, D. Leviatan, and X. M. Yu,*Polynomial approximation in*, Constr. Approx.**8**(1992), 187-201. MR**1152876 (93f:41011)****[2]**Z. Ditzian and V. Totik,*Moduli of smoothness*, Springer-Verlag, New York, 1987. MR**914149 (89h:41002)****[3]**Z. Ditzian and D. Jiang,*Approximation of functions by polynomials in*, Canad. J. Math.**44**(1992), 924-940. MR**1186473 (93h:41016)****[4]**T. Erdélyi, A. Máté, and P. Nevai,*Inequalities for generalized non-negative polynomials*, Constr. Approx.**8**(1992), 241-255. MR**1152881 (93e:41020)****[5]**E. Hille, G. Szegö, and J. D. Tamarkin,*On some generalizations of a theorem of A. Markoff*, Duke Math. J.**3**(1937), 729-739. MR**1546027****[6]**P. Nevai,*Bernstein's inequality in**for*, J. Approx. Theory**27**(1979), 239-243. MR**555623 (80m:41009)****[7]**J. Peetre,*A remark on Sobolev spaces. The case*, J. Approx. Theory**13**(1975), 218-228. MR**0374900 (51:11096)****[8]**G. T. Tachev,*Direct theorem for best algebraic approximation in*, Math. Balkanica**4**(1990), 381-390. MR**1170258 (93h:41027)****[9]**-,*A converse theorem for the algebraic approximation in*, Serdica**17**(1991), 161-166. MR**1148311 (93b:41020)**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1994-1160297-7

Keywords:
Inverse theorems,
best polynomial approximation,
spaces,

Article copyright:
© Copyright 1994
American Mathematical Society