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Let G be a finite simple connected domain in the complex plane C, bounded by a Carleson curve Γ := ∂G. In this work the direct and inverse theorems of approximation theory by the algebraic polynomials in the weighted generalized grand Smirnov classes εp),θ(G,ω) and , 1 < p < ∞, in the term of the rth, r = 1, 2,..., mean modulus of smoothness are proved. As a corollary the constructive characterizations of the weighted generalized grand Lipschitz classes are obtained.
Abstract
We give a theorem of Vijayaraghavan type for summability methods for double sequences, which allows a conclusion from boundedness in a mean and a one-sided Tauberian condition to the boundedness of the sequence itself. We apply the result to certain power series methods for double sequences improving a recent Tauberian result by S. Baron and the author [4].
he purpose of this paper is to give the direct and inverse theorem for pointwise approximation by Bernstein type operators.
Abstract
An asymmetric operator of generalized translation is introduced in this paper. Using this operator, we define a generalized modulus of smoothness and prove direct and inverse theorems of approximation theory for it.
May, C. P. , Saturation and inverse theorems for combinations of a class of exponential-type operators, Canad. J. Math. , 28 (1976), no. 6, 1224–1250. MR 55 #8640 May C. P
Indiana Univ. Math. J. 27 127 – 142 10.1512/iumj.1978.27.27011 . [2] Becker , M. 1979 An elementary proof of the inverse theorem
165 – 166 10.1006/jath.1994.1120 . [2] Ditzian , Z. 1979 A global inverse theorem for combinations of Bernstein polynomials J. Approx
, C. P. , Saturation and inverse theorems for combinations of a class of exporieritial- type operators , Canad. J. Math. 28 ( 1976 ), 1224 – 1250 . MR 55 #8640 [10
References [1] Berens , H. Lorentz , G. G. 1972 Inverse theorems for Bernstein polynomials Indiana
Nathanson, M. B. and Tenenbaum, G., Inverse theorems and the number of sums and products, in: Structure theory of set addition, Astérisque 258 (Deshouillers et al., ed.), SMF (Paris, 1999), pp. 195-204. MR 2000h :11110
