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

Abstract

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.