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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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Trigonometric moment problems for arbitrary finite subsets of $\mathbb Z^n$
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by Jean-Pierre Gabardo PDF
Trans. Amer. Math. Soc. 350 (1998), 4473-4498 Request permission

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
  • MR Author ID: 269511
  • Email: gabardo@mcmail.cis.mcmaster.ca
  • Received by editor(s): June 15, 1996
  • Additional Notes: The author was supported by NSERC grant OGP0036564
  • © Copyright 1998 American Mathematical Society
  • 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