On a theorem of Marcinkiewicz and Zygmund for Taylor series

E. S. Katsoprinakis Department of Mathematics, University of Crete

TBD mathscidoc:1701.332696

Arkiv for Matematik, 27, (1), 105-126, 1988.1
Let$K$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.
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  title={On a theorem of Marcinkiewicz and Zygmund for Taylor series},
  author={E. S. Katsoprinakis},
  booktitle={Arkiv for Matematik},
E. S. Katsoprinakis. On a theorem of Marcinkiewicz and Zygmund for Taylor series. 1988. Vol. 27. In Arkiv for Matematik. pp.105-126. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203529219056505.
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