A remark on the trigonometric system

TL;DR

This paper investigates a new property of the trigonometric system, demonstrating its optimal convergence behavior on certain classes of functions depending on the decay rate of Fourier coefficients.

Contribution

It introduces a novel property of the trigonometric system and establishes its optimality in convergence order for specific decay rates of Fourier coefficients.

Findings

Trigonometric system exhibits optimal convergence for very slow decay rates.

Trigonometric system exhibits optimal convergence for very fast decay rates.

Convergence behavior depends critically on the decay rate of Fourier coefficients.

Abstract

We present a new property of the trigonometric system arranged in a natural order. It is shown that the sequence of subspaces of trigonometric polynomials is optimal in the sense of order of convergence on convolution classes K*Up in Lq for any 1<p,q<\infty just in the cases of "very slow" or "very fast" rate of decay of Fourier coefficients of K.