Exact values of Kolmogorov widths of classes of Poisson integrals
- Autores: A. S. Serdyuk, V. V. Bodenchuk
- Localización: Journal of approximation theory, ISSN 0021-9045, Vol. 173, Nº 1, 2013, págs. 89-109
- Idioma: inglés
- DOI: 10.1016/j.jat.2013.05.002
- Enlaces
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Resumen
We prove that the Poisson kernel Pq,£] (t) = �ò¡Û k =1 qk cos �Û kt .
£]£k 2 �ß , q . (0, 1), £] . R, satisfies Kushpel¡¦s condition Cy,2n beginning with a number nq where nq is the smallest number n . 9, for which the following inequality is satisfied:
43 10(1 . q) q ¡Ô n + 160 57(n .
¡Ô n) q (1 . q)2 .
�Þ 1 2 + 2q (1 + q2)(1 . q) �â�Þ 1 . q 1 + q �â 4 1.q2 .
As a consequence, for all n . nq we obtain lower bounds for Kolmogorov widths in the space C of classes Cq £],¡Û of Poisson integrals of functions that belong to the unit ball in the space L¡Û. The obtained estimates coincide with the best uniform approximations by trigonometric polynomials for these classes. As a result, we obtain exact values for widths of classes Cq £],¡Û and show that subspaces of trigonometric polynomials of order n . 1 are optimal for widths of dimension 2n.
