Modulus of continuity for superlacunar trigonometric series and continuity of Gaussian stationary random processes

TL;DR

This paper explores the relationship between Fourier coefficients and the modulus of continuity in lacunar trigonometric series, and applies these findings to determine conditions for the continuity of Gaussian stationary processes.

Contribution

It establishes bilateral relations between Fourier coefficients and continuity measures, and provides new conditions for the continuity of Gaussian stationary processes.

Findings

Derived bilateral relations between Fourier coefficients and modulus of continuity.

Established conditions for continuity and discontinuity of Gaussian stationary processes.

Connected properties of lacunar series with stochastic process continuity.

Abstract

We deduce bilateral interrelations between Fourier coefficients for lacunar trigonometric series and modulus of their continuity. We obtain also as an application some conditions for continuity and discontinuity for Gaussian periodic stationary random centered processes.