On a result by geronimus

In 1935, Ya.L. Geronimus found the best integral approximation on the period [ −π,π ) of the function sin( n + 1) t − 2 q sin nt, q ∈ ℝ, by the subspace of trigonometric polynomials of degree at most n − 1. This result is an integral analog of the known theorem by E.I. Zolotarev (1868). At present,...

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Bibliographic Details
Published in:Proceedings of the Steklov Institute of Mathematics 2011-07, Vol.273 (Suppl 1), p.37-48
Main Authors: Babenko, A. G., Kryakin, Yu. V., Yudin, V. A.
Format: Article
Language:English
Subjects:
Mathematics
Mathematics and Statistics
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Summary:In 1935, Ya.L. Geronimus found the best integral approximation on the period [ −π,π ) of the function sin( n + 1) t − 2 q sin nt, q ∈ ℝ, by the subspace of trigonometric polynomials of degree at most n − 1. This result is an integral analog of the known theorem by E.I. Zolotarev (1868). At present, there are several methods of proving this fact. We propose one more variant of the proof. In the case | q | ≥ 1, we apply the (2 π/n )-periodization and the fact that the function | sin nt | is orthogonal to the harmonic cos t on the period. In the case | q | < 1, we use the duality relations for Chebyshev’s theorem (1859) on a rational function least deviating from zero on a closed interval with respect to the uniform metric.
ISSN:0081-5438
1531-8605
DOI:10.1134/S008154381105004X