Hostname: page-component-586b7cd67f-tf8b9 Total loading time: 0 Render date: 2024-11-22T04:50:04.360Z Has data issue: false hasContentIssue false
  • English
  • Français

A Lower Bound for the Number of Negative Zeros of Power Series

Published online by Cambridge University Press:  20 November 2018

W. Gawronski
W. Gawronski*
Affiliation:
Abteilung für Mathematik Universität Ulm Oberer EselbergD7900 Ulm/DonauGermany
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper we are concerned with power series of the type

1

which admit unique analytic extension onto a domain containing the negative real axis. Our primary object is to establish a general theorem giving a lower estimate for the number of different zeros of (1) on the negative real axis. W. Jurkat and A. Peyerimhoff showed that for a certain class of coefficient functions a(z) the number of negative zeros of (1) is closely related to the behaviour of a(z) at z = 0. In particular they proved the following theorem [4, p. 219, Theorem 4].

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 22 , Issue 1 , 01 March 1979 , pp. 47 - 52
Copyright
Copyright © Canadian Mathematical Society 1979

References

1. Butzer, P. L. and Nessel, R. J., Fourier Analysis and Approximation, vol. I, Academic Press, New York, London, 1971.Google Scholar
2. Gawronski, W. and Peyerimhoff, A., On the zeros of power series with rational coefficients, Arch. Math., XXIX (1977), Fasc. 2, 173-186.Google Scholar
3. Hille, E., Analytic Function Theory, vol. II, Blaisdell Pub. Comp., Waltham Mass., Toronto, London, 1962.Google Scholar
4. Jurkat, W. B. and Peyerimhoff, A., On power series with negative zeros, Tôhoku Math. J., vol. 24, no. 2, 207-221.Google Scholar
5. Lindelöf, E., Le calcul des résidus, Chelsea Pub. Comp., New York, 1947.Google Scholar
6. Peyerimhoff, A., On the zeros of power series, Mich. Math. J., vol. 13 (1966), 193-214.Google Scholar
7. Polya, G., On the zeros of an integral function represented by Fourier's Integral, Messenger of Math., 3 (1923), 185-188.Google Scholar
8. Polya, G. und Szegö, G., Aufgaben und Lehrsàtze aus der Analysis I, Springer Verlag, Berlin, Heidelberg, New York, 1970.Google Scholar