A converse to a theorem of Adamyan, Arov and Krein

Authors:
J. Agler and N. J. Young

Journal:
J. Amer. Math. Soc. **12** (1999), 305-333

MSC (1991):
Primary 46E22; Secondary 47B38

DOI:
https://doi.org/10.1090/S0894-0347-99-00291-X

MathSciNet review:
1643649

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: A well known theorem of Akhiezer, Adamyan, Arov and Krein gives a criterion (in terms of the signature of a certain Hermitian matrix) for interpolation by a meromorphic function in the unit disc with at most $m$ poles subject to an $L^\infty$-norm bound on the unit circle. One can view this theorem as an assertion about the Hardy space $H^2$ of analytic functions on the disc and its reproducing kernel. A similar assertion makes sense (though it is not usually true) for an *arbitrary* Hilbert space of functions. One can therefore ask for which spaces the assertion *is* true. We answer this question by showing that it holds precisely for a class of spaces closely related to $H^2$.

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Additional Information

**J. Agler**

Affiliation:
Department of Mathematics, University of California at San Diego, La Jolla, California 92093

MR Author ID:
216240

Email:
jagler@ucsd.edu

**N. J. Young**

Affiliation:
Department of Mathematics, University of Newcastle, Newcastle upon Tyne NE1 7RU, England

Email:
N.J.Young@ncl.ac.uk

Keywords:
Interpolation,
reproducing kernel,
multiplier,
Pick’s theorem,
Adamyan-Arov-Krein theorem,
Akhiezer’s theorem

Received by editor(s):
May 28, 1997

Additional Notes:
J. Agler’s research was supported by an NSF grant in Modern Analysis.

Article copyright:
© Copyright 1999
American Mathematical Society