On extremal subspaces for widths of classes of convolutions

N.V. Parfinovich

doi:10.15421/241708

On extremal subspaces for widths of classes of convolutions

N.V. Parfinovich (Oles Honchar Dnipropetrovsk National University)

Abstract

We obtained the exact values of the best $$$L_1$$$-approximations of the classes $$$K*F$$$ ($$$r\in \mathbb{N}$$$) of periodic functions $$$K*f$$$ such that $$$f$$$ belongs to a given rearrangement-invariant set $$$F$$$ and $$$K$$$ is $$$2\pi$$$-periodic, not increasing oscillation, kernel, by subspaces of generalized polynomial splines with nodes at points $$$2k\pi / n$$$ ($$$n\in \mathbb{N}$$$, $$$k\in \mathbb{Z}$$$). It is shown that these subspaces are extremal for the Kolmogorov widths of the corresponding functional classes.

Keywords

best approximation; width; periodic function; convolution; spline

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