no. 7-8
Functional Analysis
A Striktpositivstellensatz for measurable functions
[Un Striktpositivstellensatz pour les fonctions mesurables]

Présenté par : Michel Raynaud

Putinar, Mihai1
1 Mathematics Department, University of California, Santa Barbara, CA 93106, USA
Comptes Rendus. Mathématique, Tome 347 (2009) no. 7-8, pp. 381-384.
est référencé par Corrigendum to the Note “A Striktpositivstellensatz for measurable functions” [C. R. Acad. Sci. Paris, Ser. I 347 (7–8) (2009) 381–384]

La décomposition dans une somme de carrés ponderés d'une fonction de Borel positive sur un ensemble mesurable est obtenue grace au théorème spectral pour les systemes commutatifs des opérateurs autoadjoints. Un résultat similaire, obtenu pour les pôlynomes ou certaines fonctions rationelles a été fortement exploité au cours des dernières douze années pour l'optimisation non-lineaire et non-convexe.

A weighted sums of squares decomposition of positive Borel measurable functions on a bounded Borel subset of the Euclidean space is obtained via duality from the spectral theorem for tuples of commuting self-adjoint operators. The analogous result for polynomials or certain rational functions was amply exploited during the last decade in a variety of applications.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2009.02.010
Putinar, Mihai 1

1 Mathematics Department, University of California, Santa Barbara, CA 93106, USA 
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Putinar, Mihai. A Striktpositivstellensatz for measurable functions. Comptes Rendus. Mathématique, Tome 347 (2009) no. 7-8, pp. 381-384. doi : 10.1016/j.crma.2009.02.010. http://www.numdam.org/articles/10.1016/j.crma.2009.02.010/

[1] Bertsimas, D.; Popescu, I. Optimal inequalities in probability theory: a convex optimization approach, SIAM J. Optim., Volume 15 (2005) no. 3, pp. 780-804

[2] Helton, J.W.; Putinar, M. Positive polynomials in scalar and matrix variables, the spectral theorem and optimization, Operator Theory, Structured Matrices, and Dilations, Theta, Bucharest, 2007, pp. 229-306

[3] Positive Polynomials in Control (Henrion, D.; Garulli, A., eds.), Lecture Notes in Control and Information Sciences, Springer, Berlin, 2005

[4] Kakutani, S. Ein Beweis des Satzes von M. Eidelheit über konvexe Mengen, Proc. Imp. Acad. Tokyo, Volume 13 (1937), pp. 93-94

[5] Köthe, G. Topological Vector Spaces. I, Springer, Berlin, 1969

[6] Krein, M.G.; Rutman, M.A. Linear operators leaving invariant a cone in a Banach space, Uspekhi Mat. Nauk, Volume 23 (1948), pp. 3-95 (in Russian)

[7] Lasserre, J.B. Optimisation globale et théorie des moments, C. R. Acad. Sci. Paris Sér. I, Volume 331 (2000), pp. 929-934

[8] Lasserre, J.B. Global optimization with polynomials and the problem of moments, SIAM J. Optim., Volume 11 (2001), pp. 796-817

[9] Prestel, A.; Delzell, C. Positive Polynomials, Springer, Berlin, 2001

[10] Putinar, M. Positive polynomials on compact semi-algebraic sets, Indiana Univ. Math. J., Volume 42 (1993), pp. 969-984

[11] Emerging Applications of Algebraic Geometry (Putinar, M.; Sullivant, S., eds.), IMA Series in Applied Mathematics, Springer, Berlin, 2009

[12] Riesz, M. Sur le problème des moments. Troisième Note, Ark. Mat. Fys., Volume 16 (1923), pp. 1-52

[13] Riesz, F.; Sz.-Nagy, B. Functional Analysis, Dover Books on Advanced Mathematics, Dover Publications, Inc., New York, 1990 (Transl. from the 2nd French ed. by Leo F. Boron. Reprint of the 1955 orig. publ. by Ungar Publ. Co.)

[14] Schmüdgen, K. The K-moment problem for compact semi-algebraic sets, Math. Ann., Volume 289 (1991), pp. 203-206

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☆ Partially supported by the National Science Foundation-USA.