On the existence of universal series by trigonometric system

TL;DR

This paper proves the existence of a universal trigonometric series that can approximate functions in weighted $L^1$ spaces with arbitrary small exceptions, using a specific convergence condition on the coefficients.

Contribution

It introduces a new class of universal trigonometric series with coefficients satisfying a weighted square-summability condition, extending the concept of universality in weighted function spaces.

Findings

Existence of a universal series with specific coefficient conditions

Construction of weighted functions approximating arbitrary functions

Series is universal with respect to rearrangements in weighted $L^1$ spaces

Abstract

In this paper we prove the following: let $\omega(t)$ be a continuous function, increasing in $[0,\infty)$ and $\omega(+0)=0$. Then there exists a series of the form$\sum_{k=-\infty}^\infty C_ke^{ikx}$ with $\sum_{k=-\infty}^\infty C^2_k \omega(|C_k|)<\infty$, $C_{-k}=\bar{C}_k$, with the following property: for each $\epsilon>0$ a weighted function $\mu(x), 0<\mu(x) \le1,| \{x\in[0,2\pi]: \mu(x)\not =1 \}| <\epsilon $ can be constructed, so that the series is universal in the weighted space $L_\mu^1[0,2\pi]$ with respect to rearrangements.