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Journal of the American Mathematical Society
Journal of the American Mathematical Society
ISSN 1088-6834(online) ISSN 0894-0347(print)

 

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
MathSciNet review: 1643649
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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
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

DOI: http://dx.doi.org/10.1090/S0894-0347-99-00291-X
PII: S 0894-0347(99)00291-X
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