Uniform approximation of periodical functions by trigonometric sums of a special type

TL;DR

This paper investigates the approximation capabilities of special trigonometric sums for periodic functions, demonstrating they can outperform classical sums like Fourier and Zygmund in uniform approximation under certain conditions.

Contribution

It introduces and analyzes the approximation properties of U_{n,p}^ ext{ψ} sums, establishing their optimality and superiority over traditional sums for specific function classes.

Findings

U_{n,p}^ ext{ψ} sums achieve higher order uniform approximation than Fourier sums.

The sums provide the order of the best uniform approximation within certain parameter ranges.

The paper solves the Kolmogorov-Nikol'skii problem for these sums in a general setting.

Abstract

The approximation properties of the trigonometric sums U_{n,p}^\psi of a special type are investigated on the classes C^\psi_{\beta, \infty} of (\psi,\beta)-differentiable (in the sense of Stepanets) periodical functions. The solution of Kolmogorov-Nikol'skii problem in a sufficiently general case is found as a result of consistency between the parameters of approximating sums and approximated classes. It is shown that, in some important cases the sums under consideration provide higher order of approximation in the uniform metric on the classes C^\psi_{\beta, \infty} than Fourier sums, Zygmund sums and de la Valle Poussin sums do. The range of parameters within the limits of it the sums U_{n,p}^\psi supply the order of the best uniform approximation on the classes C^\psi_{\beta, \infty} is indicated.