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.