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.

Averages over smooth measures on smooth compact hypersurfaces in Rⁿ are studied. With assumptions on the decay of the Fourier transform of the measure we obtain mixed norm estimates for these means, for example L^p estimates of multiparameter maximal functions over compact hypersurfaces.