# Pick-Nevanlinna interpolation on finitely-connected domains

Studia Mathematica (1992)

• Volume: 103, Issue: 3, page 265-273
• ISSN: 0039-3223

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## Abstract

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Let Ω be a domain in the complex plane bounded by m+1 disjoint, analytic simple closed curves and let ${z}_{0},...,{z}_{n}$ be n+1 distinct points in Ω. We show that for each (n+1)-tuple $\left({w}_{0},...,{w}_{n}\right)$ of complex numbers, there is a unique analytic function B such that: (a) B is continuous on the closure of Ω and has constant modulus on each component of the boundary of Ω; (b) B has n or fewer zeros in Ω; and (c) $B\left({z}_{j}\right)={w}_{j}$, 0 ≤ j ≤ n.

## How to cite

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Fisher, Stephen. "Pick-Nevanlinna interpolation on finitely-connected domains." Studia Mathematica 103.3 (1992): 265-273. <http://eudml.org/doc/215949>.

@article{Fisher1992,
abstract = {Let Ω be a domain in the complex plane bounded by m+1 disjoint, analytic simple closed curves and let $z_0,...,z_n$ be n+1 distinct points in Ω. We show that for each (n+1)-tuple $(w_0,...,w_n)$ of complex numbers, there is a unique analytic function B such that: (a) B is continuous on the closure of Ω and has constant modulus on each component of the boundary of Ω; (b) B has n or fewer zeros in Ω; and (c) $B(z_j) = w_j$, 0 ≤ j ≤ n.},
author = {Fisher, Stephen},
journal = {Studia Mathematica},
keywords = {Pick-Nevanlinna interpolation},
language = {eng},
number = {3},
pages = {265-273},
title = {Pick-Nevanlinna interpolation on finitely-connected domains},
url = {http://eudml.org/doc/215949},
volume = {103},
year = {1992},
}

TY - JOUR
AU - Fisher, Stephen
TI - Pick-Nevanlinna interpolation on finitely-connected domains
JO - Studia Mathematica
PY - 1992
VL - 103
IS - 3
SP - 265
EP - 273
AB - Let Ω be a domain in the complex plane bounded by m+1 disjoint, analytic simple closed curves and let $z_0,...,z_n$ be n+1 distinct points in Ω. We show that for each (n+1)-tuple $(w_0,...,w_n)$ of complex numbers, there is a unique analytic function B such that: (a) B is continuous on the closure of Ω and has constant modulus on each component of the boundary of Ω; (b) B has n or fewer zeros in Ω; and (c) $B(z_j) = w_j$, 0 ≤ j ≤ n.
LA - eng
KW - Pick-Nevanlinna interpolation
UR - http://eudml.org/doc/215949
ER -

## References

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1. [A] L. Ahlfors, Conformal Invariants: Topics in Geometric Function Theory, McGraw-Hill, New York 1973. Zbl0272.30012
2. [CW] R. Coifman and G. Weiss, A kernel associated with certain multiply connected domains and its application to factorization theorems, Studia Math. 28 (1966), 31-68. Zbl0149.03202
3. [D] Ph. Delsarte, Y. Genin and Y. Kamp, The Pick-Nevanlinna problem, Internat. J. Circuit Theory Appl. 9 (1981), 177-187. Zbl0458.94048
4. [F] S. D. Fisher, Function Theory on Planar Domains, Wiley, New York 1983.
5. [FM1] S. D. Fisher and C. A. Micchelli, n-widths of sets of analytic functions, Duke Math. J. 47 (1980), 789-801.
6. [FM2] S. D. Fisher and C. A. Micchelli, Optimal sampling of holomorphic functions II, Math. Ann. 273 (1985), 131-147. Zbl0561.30008
7. [G] P. Garabedian, Schwarz's lemma and the Szegö kernel function, Trans. Amer. Math. Soc. 67 (1949), 1-35. Zbl0035.05402
8. [M] J. Munkres, Elements of Algebraic Topology, Addison-Wesley, Menlo Park, Calif., 1984. Zbl0673.55001

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