## Partial sums of Taylor series on a circle.(English)Zbl 0701.30003

We characterize the power series $$\sum^{\infty}_{n=0}c_ nz^ n$$ with the geometric property that, for sufficiently many points z, $$| z| =1$$, a circle C(z) contains infinitely many partial sums. We show that $$\sum^{\infty}_{n=0}c_ nz^ n$$ is a rational function of special type; more precisely, there are $$t\in {\mathbb{R}}$$ and $$n_ 0$$, such that, the sequence $$c_ ne^{int}$$, $$n\geq n_ 0$$, is periodic. This result answers in the affirmative a question of J.-P. Kahane and furnishes stronger versions of the main results obtained by the first author [Ark. Mat. 27, No.1, 105-126 (1989; Zbl 0676.42004)]. We are led to consider special families of circles C(z) with center g(z) and investigate the possibility for a polynomial R(z) to satisfy R(z)$$\in C(z)$$ for infinitely many z, $$| z| =1$$. These polynomials are related to the partial sums of the Taylor expansion of the center function g(z). We also give necessary and sufficient conditions for the existence of infinitely many such polynomials R(z).
Reviewer: E.S.Katsoprinakis

### MSC:

 30B10 Power series (including lacunary series) in one complex variable 30B40 Analytic continuation of functions of one complex variable 30C10 Polynomials and rational functions of one complex variable

Zbl 0676.42004
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### References:

 [1] J.-P. KAHANE, Sur la structure circulaire des ensembles de points limites des sommes partielles d’une série de Taylor, Acta Sci. Math. (Szeged), 45, n° 1-4 (1983), 247-251. · Zbl 0528.30004 [2] E.S. KATSOPRINAKIS, Characterization of power series with partial sums on a finite number of circles (in Greek), Ph. D. thesis 1988, Dept. of Mathematics, University of Crete, Iraklion, Greece. [3] E.S. KATSOPRINAKIS, On a theorem of Marcinkiewicz and Zygmund for Taylor series, Arkiv for Matematik, to appear. · Zbl 0676.42004 [4] J. MARCINKIEWICZ and A. ZYGMUND, On the behavior of trigonometric series and power series, T.A.M.S., 50 (1941), 407-453. · JFM 67.0225.02 [5] A. ZYGMUND, Trigonometric series, 2nd edition, reprinted, Vol. I. II, Cambridge : Cambridge University Press, 1979. · Zbl 0367.42001
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