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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

The proof of Tchakaloff's Theorem


Authors: Christian Bayer and Josef Teichmann
Journal: Proc. Amer. Math. Soc. 134 (2006), 3035-3040
MSC (2000): Primary 65D32, 52A21
Published electronically: May 4, 2006
MathSciNet review: 2231629
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Abstract: We provide a simple proof of Tchakaloff's Theorem on the existence of cubature formulas of degree $ m$ for Borel measures with moments up to order $ m$. The result improves known results for non-compact support, since we do not need conditions on $ (m+1)$st moments. In fact, we reduce the classical assertion of Tchakaloff's Theorem to a well-known statement going back to F. Riesz.


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Additional Information

Christian Bayer
Affiliation: Technical University of Vienna, e105, Wiedner Hauptstrasse 8-10, A-1040 Wien, Austria
Email: cbayer@fam.tuwien.ac.at

Josef Teichmann
Affiliation: Technical University of Vienna, e105, Wiedner Hauptstrasse 8-10, A-1040 Wien, Austria
Email: jteichma@fam.tuwien.ac.at

DOI: http://dx.doi.org/10.1090/S0002-9939-06-08249-9
PII: S 0002-9939(06)08249-9
Keywords: Quadrature, cubature, truncated moment problem, Tchakaloff's Theorem
Received by editor(s): March 7, 2005
Received by editor(s) in revised form: May 3, 2005
Published electronically: May 4, 2006
Additional Notes: The authors are grateful to Professor Peter Gruber for mentioning the word “Stützebene” in the right moment. The first author acknowledges the support from FWF-Wissenschaftskolleg “Differential Equations”\ W 800-N05. The second author acknowledges the support from the RTN network HPRN-CT-2002-00281 and from the FWF grant Z-36.
Communicated by: Andreas Seeger
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.