Paper

Elements (functions) that are universal with respect to a minimal system

Grigoryan, Martin, Avetisyan, Zhirayr, Ruzhansky, Michael

2023

Overview

We call an element \\(U\\) conditionally universal for a sequential convergence space \\(\\mathbf{\\Omega}\\) with respect to a minimal system \\(\\{\\varphi_n\\}_{n=1}^\\infty\\) in a continuously and densely embedded Banach space \\(\\mathcal{X}\\hookrightarrow\\mathbf{\\Omega}\\) if the partial sums of its phase-modified Fourier series is dense in \\(\\mathbf{\\Omega}\\). We will call the element \\(U\\) almost universal if the change of phases (signs) needs to be performed only on a thin subset of Fourier coefficients. In this paper we prove the existence of an almost universal element under certain assumptions on the system \\(\\{\\varphi_n\\}_{n=1}^\\infty\\). We will call a function \\(U\\) asymptotically conditionally universal in a space \\(L^1(\\mathcal{M})\\) if the partial sums of its phase-modified Fourier series is dense in \\(L^1(F_m)\\) for an ever-growing sequence of subsets \\(F_m\\subset\\mathcal{M}\\) with asymptotically null complement. Here we prove the existence of such functions \\(U\\) under certain assumptions on the system \\(\\{\\varphi_n\\}_{n=1}^\\infty\\). Moreover, we show that every integrable function can be slightly modified to yield such a function \\(U\\). In particular, we establish the existence of almost universal functions for \\(L^p([0,1])\\), \\(p\\in(0,1)\\), and asymptotically conditionally universal functions for \\(L^1([0,1])\\), with respect to the trigonometric system.

• Publisher: Cornell University Library, arXiv.org
• Subject: Asymptotic properties / Banach spaces / Fourier series / Mathematical functions