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Loewner's theorem for kernels having a finite number of negative squares
Author(s):
D.
Alpay;
J.
Rovnyak
Journal:
Proc. Amer. Math. Soc.
127
(1999),
1109-1117.
MSC (1991):
Primary 30E05, 47A57;
Secondary 46C20, 47B50
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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.
References:
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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:
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
Copyright of article:
Copyright
1999,
American Mathematical Society
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