Partial sums of Taylor series on a circle
Annales de l'Institut Fourier, Volume 39 (1989) no. 3, pp. 715-736.

We characterize the power series n=0 c n z 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 n=0 c n z n is a rational function of special type; more precisely, there are t and n 0 , such that, the sequence c n e int , nn 0 , is periodic. This result answers in the affirmative a question of J.-P. Kahane and furnishes stronger versions of the main results of [Katsoprinakis, Arkiv for Matematik]. 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)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).

Ayant comme prototype la famille des cercles définis par trois sommes partielles d’une série de puissances, nous sommes amenés à considérer des cercles C(z), |z|=1, comme suit:

Soient B, A, Q trois polynômes et λ>0 un nombre entier, tels que, A(0)Q(0)0, degA< deg Q et λ> deg B; nous supposerons de plus que les polynômes A et Q n’ont pas de facteurs communs. Nous désignons par C(z), |z|=1, le cercle avec centre B(z)+z λ A(z)Q(z) -1 et rayon |z λ A(z)Q(z) -1 |. Étant donné P un polynôme, nous disons que R(z)=B(z)+z λ P(z) est une “continuation”, si R(z)C(z) pour une infinité de points z, |z|=1.

Nous démontrons que toute continuation est une somme de Taylor de B(z)+z λ A(z)Q(z) 1 . De plus, le nombre des continuations est infini, si et seulement si, le polynôme Q est un facteur non constant d’un polynôme de la forme 1-(e iθ z) ρ , avec θR et ρ>0.

Les résultats précédents répondent par l’affirmative à une question de J.-P. Kahane et impliquent des versions plus fortes du résultat principal de [Katsoprinakis, Arkiv for Matematik].

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     title = {Partial sums of {Taylor} series on a circle},
     journal = {Annales de l'Institut Fourier},
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     publisher = {Institut Fourier},
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Katsoprinakis, E. S.; Nestoridis, V. N. Partial sums of Taylor series on a circle. Annales de l'Institut Fourier, Volume 39 (1989) no. 3, pp. 715-736. doi : 10.5802/aif.1184. https://aif.centre-mersenne.org/articles/10.5802/aif.1184/

[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. | MR | Zbl

[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

[4] J. Marcinkiewicz and A. Zygmund, On the behavior of Trigonometric series and power series, T.A.M.S., 50 (1941), 407-453. | JFM | MR | Zbl

[5] A. Zygmund, Trigonometric Series, 2nd edition, reprinted, Vol. I. II, Cambridge : Cambridge University Press, 1979. | Zbl

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