ACTA issues

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

Lívia Krizsán, Ferenc Móricz

Acta Sci. Math. (Szeged) 78:3-4(2012), 459-468

Abstract. We consider the trigonometric series $\sum_{m\in{\msbm Z}} c_m e^{imx}$, where $\{c_m: m\in{\msbm Z}\} $ is a sequence of complex numbers such that $\sum_{m\in{\msbm Z}} |c_m| < \infty.$ 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 $\mathop{\rm Zyg} (\alpha )$ and zyg$(\alpha )$, where $0< \alpha\le 2$. 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\} $.

AMS Subject Classification (1991): 26A16, 42A16

Keyword(s): trigonometric series, absolute convergence, Lipschitz classes $\mathop{\rm Lip} (\alpha )$ and lip$(\alpha )$, $0< \alpha\le 1$, Zygmund classes $\mathop{\rm Zyg} (\alpha )$ and zyg$(\alpha )$, $0< \alpha\le 2$

Received November 21, 2011, and in revised form February 24, 2012. (Registered under 65/2011.)