On a theorem of Marcinkiewicz and Zygmund for Taylor series

Abstract

LetK be the class of trigonometric series of power type, i.e. Taylor series $\sum\nolimits_{n = 0}^\infty {c_n z^n } $ for z=e^ix, whose partial sums for all x in E, where E is a nondenumerable subset of [0, 2π), lie on a finite number of circles (a priori depending on x) in the complex plane. The main result of this paper is that for every member of the class K, there exist a complex number ω, |ω|=1, and two positive integers $\nu , \kappa , \nu < \kappa $ , such that for the coefficients c_n we have: $c_{\mu + \lambda \left( {\kappa - \nu } \right)} = c_\mu \omega ^\lambda , \mu = \nu ,\nu + 1, \ldots , \kappa - 1, \lambda = 1,2,3, \ldots $ . Thus, every member of the class K has (with minor modifications) a representation of the form: $P(x)\sum\nolimits_{n = 0}^\infty {e^{iknx} ,} $ where P(x) is a suitable trigonometric polynomial and k a positive integer. The proof is elementary but rather long. This result is closely related to a theorem of Marcinkiewicz and Zygmund on the circular structure of the set of limit points of the sequence of partial sums of (C, 1) summable Taylor series.