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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Loewner's theorem for kernels
having a finite number of negative squares


Authors: D. Alpay and J. Rovnyak
Journal: Proc. Amer. Math. Soc. 127 (1999), 1109-1117
MSC (1991): Primary 30E05, 47A57; Secondary 46C20, 47B50
MathSciNet review: 1473653
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Abstract | References | Similar Articles | Additional Information

Abstract: By a theorem of Loewner, a continuously differentiable real-valued function on a real interval whose difference quotient is a nonnegative kernel is the restriction of a holomorphic function which has nonnegative imaginary part in the upper half-plane and is holomorphic across the interval. An analogous result is obtained when the difference-quotient kernel has a finite number of negative squares.


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Additional Information

D. Alpay
Affiliation: Department of Mathematics \ Ben-Gurion University of the Negev \ P. O. Box 653 \ 84105 Beer-Sheva, Israel
Email: dany@math.bgu.ac.il

J. Rovnyak
Affiliation: Department of Mathematics \ University of Virginia \ Charlottesville, Virginia 22903-3199
Email: rovnyak@Virginia.EDU

DOI: http://dx.doi.org/10.1090/S0002-9939-99-04618-3
PII: S 0002-9939(99)04618-3
Keywords: Loewner, L\"owner, Pontryagin space, reproducing kernel, negative squares, Pick, Schur, Nevanlinna.
Received by editor(s): July 25, 1997
Additional Notes: The second author was supported by the National Science Foundation under DMS–9501304.
Communicated by: Theodore W. Gamelin
Article copyright: © Copyright 1999 American Mathematical Society