The moment problem for continuous linear functionals
Author:
Jean B. Lasserre
Journal:
Trans. Amer. Math. Soc. 365 (2013), 24892504
MSC (2010):
Primary 44A60, 13B25, 14P10, 30C10
Published electronically:
October 4, 2012
MathSciNet review:
3020106
Fulltext PDF
Abstract 
References 
Similar Articles 
Additional Information
Abstract: Given a closed (and not necessarily compact) basic semialgebraic set , we solve the moment problem for continuous linear functionals. Namely, we introduce a weighted norm on , and show that the closures of the preordering and quadratic module (associated with the generators of ) is the cone of polynomials nonnegative on . We also prove that and solve the moment problem for continuous linear functionals and completely characterize those continuous linear functionals nonnegative on and (hence on ). When has a nonempty interior, we also provide in explicit form a canonical projection for any polynomial , on the (degreetruncated) preordering or quadratic module. Remarkably, the support of is very sparse and does not depend on ! This enables us to provide an explicit Positivstellensatz on . And last but not least, we provide a simple characterization of polynomials nonnegative on , which is crucial in proving the above results.
 1.
Robert
B. Ash, Real analysis and probability, Academic Press, New
YorkLondon, 1972. Probability and Mathematical Statistics, No. 11. MR 0435320
(55 #8280)
 2.
Christian
Berg, Jens
Peter Reus Christensen, and Paul
Ressel, Positive definite functions on abelian semigroups,
Math. Ann. 223 (1976), no. 3, 253–274. MR 0420150
(54 #8165)
 3.
Christian
Berg, The multidimensional moment problem and semigroups,
Moments in mathematics (San Antonio, Tex., 1987) Proc. Sympos. Appl.
Math., vol. 37, Amer. Math. Soc., Providence, RI, 1987,
pp. 110–124. MR 921086
(89k:44013), 10.1090/psapm/037/921086
 4.
Grigoriy
Blekherman, There are significantly more nonnegative polynomials
than sums of squares, Israel J. Math. 153 (2006),
355–380. MR 2254649
(2007f:14062), 10.1007/BF02771790
 5.
J. Cimpric, M. Marshall, T. Netzer, Closures of quadratic modules, Israel J. Math., to appear.
 6.
M. Ghasemi, S. Kuhlmann, E. Samei, The moment problem for continuous positive semidefinite linear functionals, arXiv:1010.279v3, November 2010.
 7.
Didier
Henrion, JeanBernard
Lasserre, and Johan
Löfberg, GloptiPoly 3: moments, optimization and semidefinite
programming, Optim. Methods Softw. 24 (2009),
no. 45, 761–779. MR 2554910
(2010i:90086), 10.1080/10556780802699201
 8.
S.
Kuhlmann and M.
Marshall, Positivity, sums of squares and the
multidimensional moment problem, Trans. Amer.
Math. Soc. 354 (2002), no. 11, 4285–4301 (electronic). MR 1926876
(2003j:14078), 10.1090/S0002994702030751
 9.
S.
Kuhlmann, M.
Marshall, and N.
Schwartz, Positivity, sums of squares and the multidimensional
moment problem. II, Adv. Geom. 5 (2005), no. 4,
583–606. MR 2174483
(2006i:14064), 10.1515/advg.2005.5.4.583
 10.
Jean
B. Lasserre and Tim
Netzer, SOS approximations of nonnegative polynomials via simple
high degree perturbations, Math. Z. 256 (2007),
no. 1, 99–112. MR 2282261
(2008a:12002), 10.1007/s0020900600618
 11.
Jean
B. Lasserre, Sufficient conditions for a real polynomial to be a
sum of squares, Arch. Math. (Basel) 89 (2007),
no. 5, 390–398. MR 2363689
(2008k:11041), 10.1007/s000130072251y
 12.
Victoria
Powers and Claus
Scheiderer, The moment problem for noncompact semialgebraic
sets, Adv. Geom. 1 (2001), no. 1, 71–88.
MR
1823953 (2002c:14086), 10.1515/advg.2001.005
 13.
Mihai
Putinar, Positive polynomials on compact semialgebraic sets,
Indiana Univ. Math. J. 42 (1993), no. 3,
969–984. MR 1254128
(95h:47014), 10.1512/iumj.1993.42.42045
 14.
Claus
Scheiderer, Positivity and sums of squares: a guide to recent
results, Emerging applications of algebraic geometry, IMA Vol. Math.
Appl., vol. 149, Springer, New York, 2009, pp. 271–324. MR 2500469
(2010h:14092), 10.1007/9780387096865_8
 15.
Konrad
Schmüdgen, The 𝐾moment problem for compact
semialgebraic sets, Math. Ann. 289 (1991),
no. 2, 203–206. MR 1092173
(92b:44011), 10.1007/BF01446568
 16.
K.
Schmüdgen, Positive cones in enveloping algebras, Rep.
Math. Phys. 14 (1978), no. 3, 385–404. MR 530471
(80g:17006), 10.1016/00344877(78)900083
 17.
Lieven
Vandenberghe and Stephen
Boyd, Semidefinite programming, SIAM Rev. 38
(1996), no. 1, 49–95. MR 1379041
(96m:90005), 10.1137/1038003
 1.
 R. Ash. Real Analysis and Probability, Academic Press, Inc., Boston (1972). MR 0435320 (55:8280)
 2.
 C. Berg, J.P.R. Christensen and P. Ressel, Positive definite functions on Abelian semigroups. Math. Ann. 223, 253274 (1976). MR 0420150 (54:8165)
 3.
 C. Berg, The multidimensional moment problem and semigroups. Proc. Symp. Appl. Math. 37, 110124 (1987). MR 921086 (89k:44013)
 4.
 G. Blekherman, There are significantly more nonnegative polynomials than sums of squares, Isr. J. Math. 153, 355380 (2006). MR 2254649 (2007f:14062)
 5.
 J. Cimpric, M. Marshall, T. Netzer, Closures of quadratic modules, Israel J. Math., to appear.
 6.
 M. Ghasemi, S. Kuhlmann, E. Samei, The moment problem for continuous positive semidefinite linear functionals, arXiv:1010.279v3, November 2010.
 7.
 D. Henrion, J.B. Lasserre and J. Lofberg, GloptiPoly 3: moments, optimization and semidefinite programming, Optim. Methods and Software 24, 761779 (2009). MR 2554910 (2010i:90086)
 8.
 S. Kuhlmann, M. Marshall, Positivity sums of squares and the multidimensional moment problem, Trans. Amer. Math. Soc. 354, 42854301 (2002). MR 1926876 (2003j:14078)
 9.
 S. Kuhlmann, M. Marshall, N. Schwartz, Positivity sums of squares and the multidimensional moment problem II, Adv. Geom. 5, 583606 (2005). MR 2174483 (2006i:14064)
 10.
 J.B. Lasserre and T. Netzer, SOS approximations of nonnegative polynomials via simple high degree perturbations, Math. Z. 256, 99112 (2006). MR 2282261 (2008a:12002)
 11.
 J.B. Lasserre, Sufficient conditions for a real polynomial to be a sum of squares, Arch. Math. 89, 390398. (2007) MR 2363689 (2008k:11041)
 12.
 V. Powers, C. Scheiderer, The moment problem for noncompact semialgebraic sets, Adv. Geom. 1, 7188 (2001). MR 1823953 (2002c:14086)
 13.
 M. Putinar, Positive polynomials on compact sets, Ind. Univ. Math. J. 42, 969984 (1993). MR 1254128 (95h:47014)
 14.
 C. Scheiderer, Positivity and sums of squares: A guide to recent results. In: Emerging Applications of Algebraic Geometry (M. Putinar, S. Sullivant, eds.), IMA Volumes Math. Appl. 149, Springer, 2009, pp. 271324. MR 2500469 (2010h:14092)
 15.
 K. Schmüdgen, The moment problem for compact semialgebraic sets, Math. Ann. 289, 203206 (1991). MR 1092173 (92b:44011)
 16.
 K. Schmüdgen, Positive cones in enveloping algebras, Rep. Math. Physics 14, 385404 (1978). MR 530471 (80g:17006)
 17.
 L. Vandenberghe and S. Boyd, Semidefinite programming, SIAM Rev. 38, 4995 (1996). MR 1379041 (96m:90005)
Similar Articles
Retrieve articles in Transactions of the American Mathematical Society
with MSC (2010):
44A60,
13B25,
14P10,
30C10
Retrieve articles in all journals
with MSC (2010):
44A60,
13B25,
14P10,
30C10
Additional Information
Jean B. Lasserre
Affiliation:
LAASCNRS and Institute of Mathematics, University of Toulouse, LAAS, 7 avenue du Colonel Roche, 31077 Toulouse Cédex 4, France
Email:
lasserre@laas.fr
DOI:
http://dx.doi.org/10.1090/S000299472012057011
Keywords:
Moment problems,
real algebraic geometry,
positive polynomials,
semialgebraic sets
Received by editor(s):
July 19, 2011
Received by editor(s) in revised form:
September 5, 2011, and September 7, 2011
Published electronically:
October 4, 2012
Article copyright:
© Copyright 2012
American Mathematical Society
