Some Remarks on the Best Approximation Rate of Certain Trigonometric Series

TL;DR

This paper provides a comprehensive analysis of the best approximation rates for certain trigonometric series within complex continuous functions, introducing a new generalized condition that extends previous quasimonotonicity concepts.

Contribution

It introduces a new generalized condition extending $O$-regularly varying quasimonotonicity, enabling a complete characterization of approximation rates for specific trigonometric series.

Findings

Established a complete approximation rate characterization under the new condition

Generalized the concept of quasimonotonicity for broader applicability

Provided an application illustrating the theoretical results

Abstract

The main object of the present paper is to give a complete result regarding the best approximation rate of certain trigonometric series in general complex valued continuous function space under a new condition which gives an essential generalization to $O$-regularly varying quasimonotonicity. An application is present in Section 3.