Trigonometric series with noninteger harmonics

TL;DR

This paper establishes necessary and sufficient conditions for the uniform convergence of trigonometric series with noninteger harmonic exponents, extending classical results to more general sequences and noninteger powers.

Contribution

It provides a comprehensive analysis of convergence criteria for trigonometric series with noninteger harmonics, a novel extension of classical harmonic analysis results.

Findings

Derived necessary and sufficient conditions for uniform convergence

Extended classical convergence results to noninteger harmonic exponents

Analyzed series with general positive coefficient sequences

Abstract

Let $\{c_k\}$ be a nonincreasing sequence of positive numbers (more general classes of sequences are also considered), and $\alpha>0$ be not an integer. We find necessary and sufficient conditions for the uniform convergence of the series $\sum_k c_k\sin k^{\alpha}x$ and $\sum_k c_k\cos k^{\alpha}x$ on the real line and its bounded subsets.