Uniform estimates for polynomial approximation in domains with corners |
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Authors: | F.G. Abdullayev I.A. Shevchuk |
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Affiliation: | aDepartment of Mathematics, Faculty of Arts and Science, Mersin University, 33343 Mersin, Turkey;bNational Taras Shevchenko University of Kyiv, Faculty of Mechanics and Mathematics, 01033 Kyiv, Ukraine |
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Abstract: | Let be a domain with a Jordan boundary ∂G, consisting of l smooth curves Γj, such that {zj}Γj-1∩Γj≠, j=1,…,l, where Γ0Γl. Denote by αjπ, 0<αj2, the angles at zj's between the curves Γj-1 and Γj, exterior with respect to G. Let Φ be a conformal mapping of the exterior of onto the exterior of the unit disk, normed by Φ′(∞)>0. We assume that there is a neighborhood U of , such that , wherez≠zj if αj1. Set gGsup{|g(z)|:zG}. Then we prove Theorem. Let and 0βr. If a function f is analytic in G and f(r)βG<+∞, then for each nlr there is an algebraic polynomial Pn of degree <n, such that |
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