The best linear approximation methods for classes of functions analytic in a disk and exact values of their $$$n$$$-widths

S.B. Vakarchuk; M.B. Vakarchuk

doi:10.15421/242522

The best linear approximation methods for classes of functions analytic in a disk and exact values of their $$$n$$$-widths

S.B. Vakarchuk (Alfred Nobel University), https://orcid.org/0000-0002-2562-8844
M.B. Vakarchuk (Oles Honchar Dnipro National University), https://orcid.org/0000-0001-8606-9057

Abstract

The first results on calculation the exact values of the Kolmogorov $$$n$$$-widths of classes of functions analytic in a disk are associated with the names of K.I. Babenko (upper bound, 1958), V.M. Tikhomirov (lower bound, 1960) and L.V. Taikov (1967, 1977). The article traces the evolution of the appearance and development emergence of more general classes of analytic functions in a disk and the construction of the best linear approximation methods for them. Particular attention is paid to the classes formed by the averaged moduli of continuity and majorants. New exact results are presented that continue this theme in the linear normed spaces $$$\widetilde{\mathcal{B}}(p,q,\lambda)$$$, $$$0<p<q<\infty$$$, $$$\min(q,\lambda) \geqslant 1$$$.

Keywords

class of functions; moduli of continuity; majorant; $$$n$$$-width; the best linear approximation method

MSC 2020

Pri 30E10, Sec 41A45, 41A46, 41A10, 41A44, 41A50

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