Hostname: page-component-745bb68f8f-lrblm Total loading time: 0 Render date: 2025-02-11T18:36:22.681Z Has data issue: false hasContentIssue false
  • English
  • Français

Peaking and interpolation by complex polynomials

Part of: Miscellaneous topics of analysis in the complex domain Geometric function theory

Published online by Cambridge University Press:  05 May 2021

Thomas H. MacGregor  and
Michael P. Sterner
Thomas H. MacGregor
Affiliation:
State University of New York at Albany, Professor Emeritus, 60 S. Washington St., Athens, NY12015, USA
Michael P. Sterner
Affiliation:
Department of Biology, Chemistry, and Mathematics, University of Montevallo, Station 6493, Harman Hall, Montevallo, AL35115, USA (sternerm@montevallo.edu)
Rights & Permissions [Opens in a new window]

Abstract

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.

Classical results about peaking from complex interpolation theory are extended to polynomials on a closed disk, and on the complement of its interior. New results are obtained concerning interpolation by univalent polynomials on a Jordan domain whose boundary satisfies certain smoothness conditions.

Keywords

peakingunivalent functionsinterpolationpolynomials

MSC classification

Primary: 30E05: Moment problems, interpolation problems 30C10: Polynomials
Secondary: 30C20: Conformal mappings of special domains
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 64 , Issue 2 , May 2021 , pp. 129 - 138
Check for updates
Copyright
Copyright © The Author(s), 2021. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society

References

Brickman, L., MacGregor, T. H. and Wilken, D. R., Convex hulls of some classical families of univalent functions, Trans. Am. Math. Soc. 156 (1971), 91–107.CrossRefGoogle Scholar
Cantor, D. J. and Phelphs, R. R., An elementary interpolation theorem, Proc. Am. Math. Soc. 16 (1965), 523–525.CrossRefGoogle Scholar
Clunie, J. G., Hallenbeck, D. J. and MacGregor, T. H., A peaking and interpolation problem for univalent functions, J. Math Anal. Appl. 111 (1985), 559–570.10.1016/0022-247X(85)90235-5CrossRefGoogle Scholar
Davis, P. J., Interpolation and approximation (Ginn (Blaisdell), Waltham, MA, 1963).Google Scholar
Duren, P. L., Univalent functions (Springer-Verlag, New York, 1983).Google Scholar
Goodman, A. W., Curvature under an analytic transformation, J. London Math. Soc. 43 (1968), 527–533.10.1112/jlms/s1-43.1.527CrossRefGoogle Scholar
Jones, W. B. and Ruscheweyh, S., Blaschke product interpolation and its application to the design of digital filters, Constr. Approx. 3 (1987), 405–409.10.1007/BF01890578CrossRefGoogle Scholar
MacGregor, T. H. and Tepper, D. E., Finite boundary interpolation by univalent functions, J. Math. Anal. Appl. 52 (1988), 315–321.Google Scholar
Pólya, G. and Szegö, G., Problems and theorems in analysis II (Springer-Verlag, Berlin, 1976).10.1007/978-1-4757-6292-1CrossRefGoogle Scholar
Pommerenke, Ch., Boundary behaviour of conformal maps (Springer-Verlag, New York, 1992).10.1007/978-3-662-02770-7CrossRefGoogle Scholar
Savin, V. V., On the moduli of Riemann surfaces, Soviet Math. Dokl. 12 (1971), 267–270.Google Scholar
Sheil-Small, T., Complex polynomials, Cambridge Studies in Advanced Mathematics, Volume 75 (Cambridge University Press, Cambridge, 2002).10.1017/CBO9780511543074CrossRefGoogle Scholar
Younis, R., Interpolation by a finite Blaschke product, Proc. Am. Math. Soc. 78 (1980), 451–452.Google Scholar