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Hankel vector moment sequences and the non-tangential regularity at infinity of two variable Pick functions

Authors: Jim Agler and John E. McCarthy
Journal: Trans. Amer. Math. Soc. 366 (2014), 1379-1411
MSC (2010): Primary 32A70, 46E22
Published electronically: September 19, 2013
MathSciNet review: 3145735
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Abstract: A Pick function of $ d$ variables is a holomorphic map from $ \Pi ^d$ to $ \Pi $, where $ \Pi $ is the upper halfplane. Some Pick functions of one variable have an asymptotic expansion at infinity, a power series $ \sum _{n=1}^\infty \rho _n z^{-n}$ with real numbers $ \rho _n$ that gives an asymptotic expansion on non-tangential approach regions to infinity. In 1921 H. Hamburger characterized which sequences $ \{ \rho _n\} $ can occur. We give an extension of Hamburger's results to Pick functions of two variables.

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Additional Information

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

John E. McCarthy
Affiliation: Department of Mathematics, Washington University, St. Louis, Missouri 63130

Received by editor(s): January 17, 2012
Published electronically: September 19, 2013
Additional Notes: The first author was partially supported by National Science Foundation Grants DMS 0801259 and DMS 1068830
The second author was partially supported by National Science Foundation Grants DMS 0966845 and DMS 1300280
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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