Degree of Simultaneous Coconvex Polynomial Approximation |
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Authors: | K Kopotun D Leviatan |
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Institution: | 1. Department of Mathematics, Vanderbilt University, Nashville, TN, 37240, USA 2. School of Mathematical Sciences, Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv, 69978, Israel
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Abstract: | $ {\rm Let}\ f\ \epsilon\ {C^{1}}-1,1] $ change its convexity finitely many times in the interval, say s times, at ${\rm at}\ {Y_{s}}\:\ -1\ <\ y_{s}\ <\ \dots\ < y_{1}\ < 1 $ . We estimate the degree of simultaneous approximation of ? 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 sufficiently 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 ?′. This should be compared to a recent paper by the authors together with I. A. Shevchuk where ? 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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