The best approximation by trigonometric polynomials of classes of convolutions generated by some linear combinations of periodic kernels

TL;DR

This paper investigates approximation properties of trigonometric polynomials for classes of convolutions generated by linear combinations of periodic kernels, establishing conditions under which certain approximation bounds hold.

Contribution

It proves the Nagy condition for linear combinations of Poisson kernels for sufficiently large n and constructs examples of linear combinations of Bernoulli and conjugate Poisson kernels satisfying Nikolsky but not Nagy conditions.

Findings

Nagy condition holds for Poisson kernel combinations when n is large enough.

Existence of kernel combinations satisfying Nikolsky but not Nagy conditions.

Results extend understanding of approximation properties of convolution classes.

Abstract

For arbitrary nontrivial linear combinations of a finite number of Poisson kernels, the fulfillment of the Nagy condition is established for all numbers n, starting from some number. It is also proved for any n the existence of linear combinations of m Bernoulli kernels and linear combinations of m conjugate Poisson kernels that satisfy the Nikolsky condition and at the same time do not satisfy the Nagy condition.