On a theorem of Marcinkiewicz and Zygmund for Taylor series. (English) Zbl 0676.42004

Summary: Let K be the class of trigonometric series of power type, i.e. Taylor series \(\sum^{\infty}_{n=0}c_ nz^ n\) for \(z=e^{ix}\), whose partial sums for all x in E, where E is a nondenumerable subset of [0,2\(\pi)\), 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 \(\omega\), \(| \omega | =1\), and two positive integers \(\nu,\kappa,\nu <\kappa\), such that for the coefficients \(c_ n\) we have: \(c_{\mu +\lambda (\kappa -\nu)}=c_{\mu}\omega^{\lambda},\) \(\mu =\nu,\nu +1,...,\kappa -1,\) \(\lambda =1,2,3,..\). Thus, every member of the class K has (with minor modifications) a representation of the form: \(P(x)\sum^{\infty}_{n=0}e^{iknx},\) 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.


42A16 Fourier coefficients, Fourier series of functions with special properties, special Fourier series
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[1] Kahane, J.-P., Sur la structure circulaire des ensembles de points limites des sommes partielles d’une série de Taylor,Acta Sci. Math. (Szeged) 45 (1983), 247–251. · Zbl 0528.30004
[2] Marcinkiewicz, J. andZygmund, A., On the behavior of trigonometric series and power series,Trans. Amer. Math. Soc. 50 (1941), 407–453. · JFM 67.0225.02
[3] Zygmund, A.,Trigonometric series, 2nd ed. reprinted, I, II. Cambridge University Press, 1979. · JFM 58.0296.09
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