Hostname: page-component-78c5997874-fbnjt Total loading time: 0 Render date: 2024-11-20T03:54:17.411Z Has data issue: false hasContentIssue false
  • English
  • Français

Extreme Pick-Nevanlinna Interpolants

Published online by Cambridge University Press:  20 November 2018

Stephen D. Fisher  and
Dmitry Khavinson
Stephen D. Fisher
Affiliation:
Department of Mathematics, Northwestern University
Dmitry Khavinson
Affiliation:
Department of Mathematics, University of Arkansas
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.

Following the investigations of $\text{B}$. Abrahamse $[1]$, F. Forelli $[11]$, M. Heins $[14]$ and others, we continue the study of the Pick-Nevanlinna interpolation problem in multiply-connected planar domains. One major focus is on the problem of characterizing the extreme points of the convex set of interpolants of a fixed data set. Several other related problems are discussed.

Keywords

30D5030D99
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 51 , Issue 5 , 01 October 1999 , pp. 977 - 995
Copyright
Copyright © Canadian Mathematical Society 1999

References

[1] Abrahamse, M. B., The Pick interpolation theoremfor finitely connected domains. MichiganMath. J. 26(1979), 195–203.Google Scholar
[2] Cole, B., Lewis, K. and Wermer, J., Pick conditions on a uniformalgebra and vonNeumann inequalities. J. Funct. Anal. 107(1992), 235–254.Google Scholar
[3] Cole, B., Lewis, K. and Wermer, J., A characterization of Pick bodies. J. LondonMath. Soc. (2) 48(1993), 316–328.Google Scholar
[4] Cole, B., Lewis, K. and Wermer, J., Pick interpolation, von Neumann inequalities and hyperconvex sets. Preprint.Google Scholar
[5] Delsarte, P., Genin, Y. and Kamp, Y., On the role of the Pick-Nevanlinna problem in circuit and system theory. Internat. J. Circuit Theory Appl. 9(1981), 117–187.Google Scholar
[6] Duren, P. L., The Theory of Hp Spaces. Academic Press, New York, 1970.Google Scholar
[7] Edwards, R. E., Functional Analysis. Holt, Rinehart and Winston, New York, 1965.Google Scholar
[8] Fisher, S. D., Function Theory on Planar Domains. JohnWiley and Sons, New York, 1983.Google Scholar
[9] Fisher, S. D., Pick-Nevanlinna interpolation on finitely-connected domains. Studia Math. 103(1992), 265–273.Google Scholar
[10] Fisher, S. D. and Micchelli, C. A., Optimal sampling of holomorphic functions II. Math. Ann. 273(1985), 131–147.Google Scholar
[11] Forelli, F., The extreme points of some classes of holomorphic functions. Duke Math J. 46(1979), 763–772.Google Scholar
[12] Forster, O., Lectures on Riemann Surfaces. Graduate Texts in Math., 81 Springer-Verlag, New York, 1981.Google Scholar
[13] Garabedian, P., Schwarz's lemma and the Szegö kernel function. Trans. Amer.Math. Soc. 67(1949), 1–35.Google Scholar
[14] Heins, M., Extreme Pick-Nevanlinna interpolating functions. J. Math. Kyoto Univ. 25(1985), 757–766.Google Scholar
[15] Heins, M., A lemma on positive harmonic functions. Ann. of Math. 52(1950), 568–573.Google Scholar
[16] Heins, M., Carathéodory Bodies. Ann. Acad. Sci. Fenn. Ser. A I Math. 2(1976), 203–232.Google Scholar
[17] Heins, M., Extreme normalized analytic functions with positive real part. Ann. Acad. Sci. Fenn. Ser. A I Math. 10(1985), 239–245.Google Scholar
[18] Khavinson, D., On removal of periods of conjugate functions in multiply connected domains. Michigan Math. J. 31(1984), 371–379.Google Scholar
[19] Ya. Khavinson, S., Foundations of the Theory of Extremal Problems for Bounded Analytic Functions with Additional Conditions. Amer. Math. Soc. Transl. 129(1986), 63–112.Google Scholar
[20] Marshall, D. E., An elementary proof of the Pick-Nevanlinna interpolation theorem. Michigan Math. J. 21(1975), 219–223.Google Scholar
[21] Nevanlinna, R., Über beschränkte Funktionen die in gegebenen Punkten vorgeschriebenWerte annahmen. Ann. Acad. Sci. Fenn. (1) 13(1919).Google Scholar
[22] Royden, H. L., The boundary values of analytic and harmonic functions. Math. Z. 78(1962), 1–24.Google Scholar