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A Characterization of Smoothness in Terms of Approximation by Algebraic Polynomials in LP

✍ Scribed by V.A. Operstein

Publisher
Elsevier Science
Year
1995
Tongue
English
Weight
307 KB
Volume
81
Category
Article
ISSN
0021-9045

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✦ Synopsis

We prove direct and inverse theorems for the classical modulus of smoothness and approximation by algebraic polynomials in (L_{p}[-1,1]). These theorems contain the well-known theorems of A. Timan, V. Dzyadyk, G. Freud, and Yu. Brudnyi as special cases when (p=x). They also provide a characterization of the spaces (\operatorname{Lip}(\alpha, p)) (Lipschitz spaces in (L_{p}) ) for (0<\alpha<\alpha, 1 \leqslant p \leqslant \alpha). ' 1495 Academic Press. Inc.

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The best polynomial approximation is closely related to the Ditzian᎐Totik modulus of smoothness. In 1988, Z. Ditzian and V. Totik gave some equivalences between them and the class of Besov-type spaces B p with 1 F p F ϱ and ␣, s 1 F s F ϱ. We extend these equivalences to the similar Besov-type space