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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.

MSC:

42A16 Fourier coefficients, Fourier series of functions with special properties, special Fourier series
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References:

[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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