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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Trigonometric moment problems for
arbitrary finite subsets of $\mathbf{Z}^{n}$

Author: Jean-Pierre Gabardo
Journal: Trans. Amer. Math. Soc. 350 (1998), 4473-4498
MSC (1991): Primary 42A70, 44A60
MathSciNet review: 1443194
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Abstract: We consider finite subsets $\Lambda \subset \mathbf{Z}^{n}$ satisfying the extension property, i.e. the property that every collection $\{c_{\mathbf{k}}\}_{\mathbf{k} \in \Lambda - \Lambda }$ of complex numbers which is positive-definite on $\Lambda $ is the restriction to $\Lambda - \Lambda $ of the Fourier coefficients of some positive measure on $\mathbf{T}^{n}$. A simple algebraic condition on the set of trigonometric polynomials with non-zero coefficients restricted to $\Lambda $ is shown to imply the failure of the extension property for $\Lambda $. This condition is used to characterize the one-dimensional sets satisfying the extension property and to provide many examples of sets failing to satisfy it in higher dimensions. Another condition, in terms of unitary matrices, is investigated and is shown to be equivalent to the extension property. New two-dimensional examples of sets satisfying the extension property are given as well as explicit examples of collections for which the extension property fails.

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Additional Information

Jean-Pierre Gabardo
Affiliation: Department of Mathematics and Statistics McMaster University Hamilton, Ontario, L8S 4K1 Canada

Keywords: Tight frames, evaluation polynomials, representing measures, positive-definite, extension problem
Received by editor(s): June 15, 1996
Additional Notes: The author was supported by NSERC grant OGP0036564
Article copyright: © Copyright 1998 American Mathematical Society