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On Interpolating Blaschke Products and Blaschke-Oscillatory Equations

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Constructive Approximation Aims and scope

Abstract

This research is partially a continuation of a 2007 paper by the author. Growth estimates for generalized logarithmic derivatives of Blaschke products are provided under the assumption that the zero sequences are either uniformly separated or exponential. Such Blaschke products are known as interpolating Blaschke products. The growth estimates are then proven to be sharp in a rather strong sense. The sharpness discussion yields a solution to an open problem posed by E. Fricain and J. Mashreghi in 2008. Finally, several aspects are pointed out to illustrate that interpolating Blaschke products appear naturally in studying the oscillation of solutions of a differential equation f″+A(z)f=0, where A(z) is analytic in the unit disc. In particular, a unit disc analogue of a 1988 result due to S. Bank on prescribed zero sequences for entire solutions is obtained, and a more careful analysis of a 1955 example due to B. Schwarz on the case \(A(z)=\frac{1+4\gamma^{2}}{(1-z^{2})^{2}}\) reveals that an infinite zero sequence is always a union of two exponential sequences.

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Authors and Affiliations

  1. Department of Physics and Mathematics, University of Eastern Finland, P.O. Box 111, 80101, Joensuu, Finland

    Janne Heittokangas

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  1. Janne Heittokangas
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Correspondence to Janne Heittokangas.

Additional information

Communicated by Stephan Ruscheweyh.

The research reported in this paper was supported in part by the Academy of Finland #121281.

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Heittokangas, J. On Interpolating Blaschke Products and Blaschke-Oscillatory Equations. Constr Approx 34, 1–21 (2011). https://doi.org/10.1007/s00365-010-9100-0

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  • DOI: https://doi.org/10.1007/s00365-010-9100-0

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