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


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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