Generalization of Zygmund's theorem on the smoothness of the sum of trigonometric series

We consider the trigonometric series Sigma(m epsilon z) c(m)(eimx), where {c(m) : m is an element of Z} is a sequence of complex numbers such that Sigma(m is an element of z) vertical bar cm vertical bar < infinity. Then the trigonometric series converges absolutely and uniformly. We denote by f(x) its sum, which is clearly continuous. We give sufficient conditions in terms of certain means of {c(m)} to ensure that f (x) belongs to one of the Zygmund classes Zyg(alpha) and zyg(alpha), where 0 < alpha <= 2. Our theorems generalize the corresponding result of Zygmund [2] given in the special case alpha = 1. Our proof is essentially different from that of Zygmund. We establish two lemmas which reveal interesting interrelations between the order of magnitude of certain initial means and that of certain tail means of the sequence {c(m)}.