A Dichotomy Principle for Universal Series

Farmaki, Vassiliki; Nestoridis, Vassili

doi:10.4064/ba56-2-1

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2009

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“…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.…”

Section: Introductionmentioning

confidence: 99%

Pointwise universal trigonometric series

Shkarin

2009

Journal of Mathematical Analysis and Applications

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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| → ∞.

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