On universal formal power series

Abstract

The point source of this work is Seleznev's theorem which asserts the existence of a power series which satisfies universal approximation properties in C * . The paper deals with a strengthened version of this result. We establish a double approximation theorem on formal power series using a weighted backward shift operator. Moreover we give strong conditions that guarantee the existence of common universal series of an uncountable family of weighted backward shift with respect to the simultaneous approximation. Finally we obtain results on admissible growth of universal formal power series. We especially prove that you cannot control the defect of analyticity of such a series even if there exist universal series in the well-known intersection of formal Gevrey classes.

Polynomial approximations and universality

“…In the last theorem, we can require the approximation of partial sums for l = 0 alone, and by Mergelyan's Theorem we can assume that h ∈ A(K). This gives a special case of Theorem 2.8 of [6]. 2.…”

Common universal restrictions of power series

“…Let us consider the set Proof. It suffices to use the continuity of the partial sums and the proof works as in [9] Lemma 3.5 for instance.…”

Pointwise universal trigonometric series

“…Those interested in the subject ought to look into [1], which together with a systematic approach to the theory of universal series provides a large survey part and extensive list of references on the subject. We would also like to mention papers [2,4,5,6,14,17,18], which are not in the list of references in [1] and the papers [3,7,8,9,10] dealing specifically with universal trigonometric series. We start by mentioning two old results on universal series.…”