Comparison of the best uniform approximations of analytic functions in the disk and on its boundary |
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Authors: | A A Pekarskii |
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Institution: | (1) Belarusian State Technological University, ul. Sverdlova 13A, Minsk, 220030, Belarus |
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Abstract: | Denote by C A the set of functions that are analytic in the disk |z| < 1 and continuous on its closure |z| ≤ 1; let ? n , n = 0, 1, 2, ..., be the set of rational functions of degree at most n. Denote by R n (f) (R n (f) A ) the best uniform approximation of a function f ∈ C A on the circle |z| = 1 (in the disk |z| ≤ 1) by the set ? n . The following equality is proved for any n ≥ 1: sup{R n (f) A /R n (f): f ∈ C A ? ? n } = 2. We also consider a similar problem of comparing the best approximations of functions in C A by polynomials and trigonometric polynomials. |
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