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Approximation Springer Verlag Berlin . [6] Draganov , B. R. 2002 A new modulus of smoothness for trigonometric polynomial approximation East J

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We study the L p-saturation for the linear combination of Bernstein-Kantorovich operators. As a result we obtain the saturation class by using K-functional as well as some modulus of smoothness.

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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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. [13] Ditzian , Z. 1999 A modulus of smoothness on the unit sphere J. d’Anal. Math. 79 189 – 200 10.1007/BF02788240

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Abstract.

We prove some new direct and converse results on simultaneous approximation by the combinations of Bernstein–Kantorovich operators using the Ditzian–Totik modulus of smoothness.

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The paper provides a detailed study of inequalities of complete moduli of smoothness of functions with transformed Fourier series by moduli of smoothness of initial functions. Upper and lower estimates of the norms and best approximations of the functions with transformed Fourier series by the best approximations of initial functions are also obtained.

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In this paper we give the direct, inverse and equivalence theorems for generalization Meyer-König and Zeller type operators in the space L p (1 ≦ p ) with Ditzian-Totik modulus.

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We construct a new kind of rational operator which can be used to approximate functions with endpoints singularities by algebric weights in [−1,1], and establish new direct and converse results involving higher modulus of smoothness and a very general class of step functions, which cannot be obtained by weighted polynomial approximation. Our results also improve related results of Della Vecchia [5].

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In this paper we study I-approximation properties of certain class of linear positive operators. The two main tools used in this paper are I-convergence and Ditzian-Totik modulus of smoothness. Furthermore, we define q-Lupaş-Durrmeyer operators and give local and global approximation results for such 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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