The proof of Tchakaloff’s Theorem
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- by Christian Bayer and Josef Teichmann
- Proc. Amer. Math. Soc. 134 (2006), 3035-3040
- DOI: https://doi.org/10.1090/S0002-9939-06-08249-9
- Published electronically: May 4, 2006
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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.References
- Raúl E. Curto and Lawrence A. Fialkow, A duality proof of Tchakaloff’s theorem, J. Math. Anal. Appl. 269 (2002), no. 2, 519–532. MR 1907129, DOI 10.1016/S0022-247X(02)00034-3
- Hans Föllmer and Alexander Schied, Stochastic finance, De Gruyter Studies in Mathematics, vol. 27, Walter de Gruyter & Co., Berlin, 2002. An introduction in discrete time. MR 1925197, DOI 10.1515/9783110198065
- S. Karlin and L. S. Shapley, Geometry of moment spaces, Mem. Amer. Math. Soc. 12 (1953), 93. MR 59329
- J. H. B. Kemperman, On the sharpness of Tchebycheff type inequalities. I, II, III, Indag. Math. 27 (1956), 554–571; 572–587; 588–601. Nederl. Akad. Wetensch. Proc. Ser. A 68. MR 0190964
- J. H. B. Kemperman, Geometry of the moment problem, Moments in mathematics (San Antonio, Tex., 1987) Proc. Sympos. Appl. Math., vol. 37, Amer. Math. Soc., Providence, RI, 1987, pp. 16–53. MR 921083, DOI 10.1090/psapm/037/921083
- Mihai Putinar, A note on Tchakaloff’s theorem, Proc. Amer. Math. Soc. 125 (1997), no. 8, 2409–2414. MR 1389533, DOI 10.1090/S0002-9939-97-03862-8
- Bruce Reznick, Sums of even powers of real linear forms, Mem. Amer. Math. Soc. 96 (1992), no. 463, viii+155. MR 1096187, DOI 10.1090/memo/0463
- Frigyes Riesz, Sur certains systèmes singuliers d’équations intégrales, Ann. Sci. École Norm. Sup. (3), 15, 69–81, 1911.
- R. Tyrrell Rockafellar, Convex analysis, Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, N.J., 1970. MR 0274683, DOI 10.1515/9781400873173
- Vladimir Tchakaloff, Formules de cubatures mécaniques à coefficients non négatifs, Bull. Sci. Math. (2) 81 (1957), 123–134 (French). MR 94632
Bibliographic 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
- MR Author ID: 654648
- Email: jteichma@fam.tuwien.ac.at
- 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
- © Copyright 2006
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 134 (2006), 3035-3040
- MSC (2000): Primary 65D32, 52A21
- DOI: https://doi.org/10.1090/S0002-9939-06-08249-9
- MathSciNet review: 2231629