Degree of Simultaneous Coconvex Polynomial Approximation
نویسنده
چکیده
Let f 2 C 1 ?1; 1] change its convexity nitely many times in the interval, say s times, at Y s : ?1 < y s < < y 1 < 1. We estimate the degree of simultaneous approximation of f and its derivative by polynomials of degree n, which change convexity exactly at the points Y s , and their derivatives. We show that provided n is suuciently large, depending on the location of the points Y s , the rate of approximation can be estimated by C (s)=n times the second Ditzian{Totik modulus of smoothness of f 0. This should be compared to a recent paper by the authors together with I. A. Shevchuk where f is merely assumed to be continuous and estimates of coconvex approximation are given by means of the third Ditzian{Totik modulus of smoothness. However, no simultaneous approximation is given there.
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There are two kinds of estimates of the degree of approximation of continuous functions on [−1, 1] by algebraic polynomials, Nikolskii-type pointwise estimates and Jackson-type uniform estimates, involving either ordinary moduli of smoothness, or the DitzianTotik (DT) ones, or the recent estimates involving the weighted DTmoduli of smoothness. The purpose of this paper is to complete the table ...
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