Pointwise universal trigonometric series

Abstract

A series S a = ∞ n=−∞ a n z n is called a pointwise universal trigonometric series if for any f ∈ C (T), there exists a strictly increasing sequence {n k } k∈N of positive integers such that n k j=−n k a j z j converges to f (z) pointwise on T. We find growth conditions on coefficients allowing and forbidding the existence of a pointwise universal trigonometric series. For instance, if |a n | = O (e |n| ln −1−ε |n| ) as |n| → ∞ for some ε > 0, then the series S a cannot be pointwise universal. On the other hand, there exists a pointwise universal trigonometric series S a with |a n | = O (e |n| ln −1 |n| ) as |n| → ∞.

Generic boundary behaviour of Taylor series in Hardy and Bergman spaces

“…The estimate combined with a result of Shkarin (see [28]) implies that for f ∈ A p at most one continuous (pointwise) limit function can exist on each nontrivial subarc of T. We shall show, in contrast, that maximal sets of limit functions generically exist on metrically large subsets of T. A trigonometric (or power) series on T is called universal in the sense of Menshov if each measurable function g : T → C is the almost everywhere limit of a subsequence of the partial sums (see e.g. [20]).…”