Pointwise universal trigonometric series

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 ∑ j = − n k 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 ( | 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 ( | n | ln − 1 | n | ) as | n | → ∞ .