Trigonometric moment problems for

arbitrary finite subsets of

Author:
Jean-Pierre Gabardo

Journal:
Trans. Amer. Math. Soc. **350** (1998), 4473-4498

MSC (1991):
Primary 42A70, 44A60

DOI:
https://doi.org/10.1090/S0002-9947-98-02091-1

MathSciNet review:
1443194

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We consider finite subsets satisfying the extension property, i.e. the property that every collection of complex numbers which is positive-definite on is the restriction to of the Fourier coefficients of some positive measure on . A simple algebraic condition on the set of trigonometric polynomials with non-zero coefficients restricted to is shown to imply the failure of the extension property for . 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

Email:
gabardo@mcmail.cis.mcmaster.ca

DOI:
https://doi.org/10.1090/S0002-9947-98-02091-1

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