Pointwise universal trigonometric series

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:360

Article

Shkarin, S.

关键词: Universal series;    Trigonometric series;    Power series;   

DOI  :  10.1016/j.jmaa.2009.07.004

来源: Elsevier

【 摘 要 】

A series S(alpha) = Sigma(infinity)(n)=-infinity 0a(n)z(n) is called a pointwise universal trigonometric series if for any f is an element of C(T), there exists a strictly increasing sequence (n(k))(k is an element of N) of positive integers such that Sigma(nk)(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|[n(-1-epsilon) |n|) as |n|) -> infinity for some epsilon > 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| -> infinity. (C) 2009 Elsevier Inc. All rights reserved.

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