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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 εp),θ(G,ω), 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.

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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].

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he purpose of this paper is to give the direct and inverse theorem for pointwise approximation by Bernstein type operators.

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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.

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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

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Indiana Univ. Math. J. 27 127 – 142 10.1512/iumj.1978.27.27011 . [2] Becker , M. 1979 An elementary proof of the inverse theorem

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165 – 166 10.1006/jath.1994.1120 . [2] Ditzian , Z. 1979 A global inverse theorem for combinations of Bernstein polynomials J. Approx

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, 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

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References [1] Berens , H. Lorentz , G. G. 1972 Inverse theorems for Bernstein polynomials Indiana

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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

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