It is proved that if the increasing sequence kn n=0..∞ n=0 of nonnegative integers has density greater than 1/2 and D is an arbitrary simply connected subregion of CRthen the system of Hermite associated functions Gkn(z) n=0..∞ is complete in the space H(D) of complex functions holomorphic in D.

We prove a new type of universality theorem for the Riemann zeta-function and other $L$-functions (which are universal in the sense of Voronin’s theorem). In contrast to previous universality theorems for the zeta-function or its various generalizations, here the approximating shifts are taken from the orbit of an ergodic transformation on the real line.

Let $E={\bigcup }_{n=1}^{\infty }{I}_n$ be the union of infinitely many disjoint closed intervals where $I_n=[a_n$, $b_n]$, $0<a_1<b_1<a_2<b_2<\cdots <b_n<\cdots$, ${lim}_{n\to \infty }{b}_n=\infty .$ Let $\alpha \left(t\right)$ be a nonnegative function and ${\left\{{\lambda }_n\right\}}_{n=1}^{\infty }$ a sequence of distinct complex numbers. In this paper, a theorem on the completeness of the system $\left\{t^{{\lambda }_n}{log}^{m_n}t\right\}$ in $C_0\left(E\right)$ is obtained where $C_0\left(E\right)$ is the weighted Banach space consists of complex functions continuous on $E$ with $f\left(t\right){\mathrm{e}}^{-\alpha \left(t\right)}$ vanishing at infinity.

Denote by spanf₁,f₂,... the collection of all finite linear combinations of the functions f₁,f₂,... over ℝ. The principal result of the paper is the following. Theorem (Full Clarkson-Erdős-Schwartz Theorem). Suppose ${\left({\lambda }_j\right)}_{j=1}^{\infty }$ is a sequence of distinct positive numbers. Then $span1,x^{\lambda ₁},x^{\lambda ₂},...$ is dense in C[0,1] if and only if ${\sum }_{j=1}^{\infty }\left({\lambda }_j\right)/({\lambda }_j²+1)=\infty$. Moreover, if ${\sum }_{j=1}^{\infty }\left({\lambda }_j\right)/({\lambda }_j²+1)<\infty$, then every function from the C[0,1] closure of $span1,x^{\lambda ₁},x^{\lambda ₂},...$ can be represented as an analytic function on z ∈ ℂ ∖ (-∞, 0]: |z| < 1 restricted to (0,1). This result improves an earlier result...