The truncated complex -moment problem

Authors:
Raúl Curto and Lawrence A. Fialkow

Journal:
Trans. Amer. Math. Soc. **352** (2000), 2825-2855

MSC (2000):
Primary 47A57, 44A60, 30E05; Secondary 15A57, 15-04, 47N40, 47A20

DOI:
https://doi.org/10.1090/S0002-9947-00-02472-7

Published electronically:
February 28, 2000

MathSciNet review:
1661305

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Abstract | References | Similar Articles | Additional Information

Let denote a sequence of complex numbers ( ), and let denote a closed subset of the complex plane . The Truncated Complex -Moment Problem for entails determining whether there exists a positive Borel measure on such that ( ) and . For a semi-algebraic set determined by a collection of complex polynomials , we characterize the existence of a finitely atomic representing measure with the fewest possible atoms in terms of positivity and extension properties of the moment matrix and the localizing matrices . We prove that there exists a -atomic representing measure for supported in if and only if and there is some rank-preserving extension for which , where or .

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Additional Information

**Raúl Curto**

Affiliation:
Department of Mathematics, The University of Iowa, Iowa City, Iowa 52242

Email:
curto@math.uiowa.edu

**Lawrence A. Fialkow**

Affiliation:
Department of Mathematics and Computer Science, State University of New York, New Paltz, New York 12561

Email:
fialkow@mcs.newpaltz.edu

DOI:
https://doi.org/10.1090/S0002-9947-00-02472-7

Keywords:
Truncated complex moment problem,
moment matrix extension,
flat extensions of positive matrices,
semi-algebraic sets,
localizing matrix

Received by editor(s):
May 14, 1998

Published electronically:
February 28, 2000

Additional Notes:
Research partially supported by NSF grants. The second-named author was also partially supported by the State University of New York at New Paltz Research and Creative Projects Award Program.

Dedicated:
Dedicated to Professor Aaron D. Fialkow on the occasion of his eighty-seventh birthday

Article copyright:
© Copyright 2000
American Mathematical Society