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
Full-text PDF
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