A generalization of the moment problem to a complex measure space and an approximation technique using backward moments

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April 2011, 30(1): 187-207. doi: 10.3934/dcds.2011.30.187

A generalization of the moment problem to a complex measure space and an approximation technique using backward moments

1.	Department of Mathematical Sciences, KAIST, 335 Gwahangno, Yuseong-gu, Daejeon, 305-701, South Korea

Received March 2010 Revised September 2010 Published February 2011

Abstract
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One traditionally considers positive measures in the moment problem. However, this restriction makes its theory and application limited. The main purpose of this paper is to generalize it to deal with complex measures. More precisely, the theory of the truncated moment problem is extended to include complex measures. This extended theory provides considerable flexibility in its applications. In fact, we also develop an approximation technique based on control of moments. The key idea is to use the heat equation as a link that connects the generalized moment problem and this approximation technique. The backward moment of a measure is introduced as the moment of a solution to the heat equation at a backward time and then used to approximate the given measure. This approximation gives a geometric convergence order as the number of moments under control increases. Numerical examples are given that show the properties of approximation technique.

Keywords: age of a heat distribution, long time asymptotics, truncated moment problem., Heat equation.

Mathematics Subject Classification: Primary: 35K05, 30E05; Secondary: 33F0.

Citation: Yong-Jung Kim. A generalization of the moment problem to a complex measure space and an approximation technique using backward moments. Discrete & Continuous Dynamical Systems - A, 2011, 30 (1) : 187-207. doi: 10.3934/dcds.2011.30.187

References:

[1]	N. I. Akhiezer, "The Classical Moment Problem and Some Related Questions in Analysis,", Hafner, (1965). Google Scholar
[2]	A. Atzmon, A moment problem for positive measures on the unit disc,, Pacific J. Math. \textbf{59} (1975), 59 (1975), 317. Google Scholar
[3]	C. Berg, The multidimensional moment problem and semigroups,, Moments in mathematics, 37 (1987), 110. Google Scholar
[4]	B. P. Boas, The Stieltjes moment problem for functions of bounded variation,, Bull. Amer. Math. Soc., 45 (1939), 399. doi: 10.1090/S0002-9904-1939-06992-9. Google Scholar
[5]	J. A. Carrillo, A. Jüngel, P. A. Markowich, G. Toscani and A. Unterreiter, Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities,, Monatsh. Math., 133 (2001), 1. doi: 10.1007/s006050170032. Google Scholar
[6]	J. Chung, E. Kim and Y.-J. Kim, Asymptotic agreement of moments and higher order contraction in the Burgers equation,, J. Differential Equations, 248 (2010), 2417. doi: 10.1016/j.jde.2010.01.006. Google Scholar
[7]	R. E. Curto and L. A. Fialkow, Recursiveness, positivity, and truncated moment problems,, Houston J. Math., 17 (1991), 603. Google Scholar
[8]	R. E. Curto and L. A. Fialkow, Solution of the truncated complex moment problem for flat data,, Mem. Amer. Math. Soc., 119 (1996). Google Scholar
[9]	R. E. Curto and L. A. Fialkow, Flat extensions of positive moment matrices: Recursively generated relations,, Mem. Amer. Math. Soc., 136 (1998). Google Scholar
[10]	R. E. Curto and L. A. Fialkow, Truncated $K$-moment problems in several variables,, J. Operator Theory, 54 (2005), 189. Google Scholar
[11]	R. E. Curto and L. A. Fialkow, An analogue of the Riesz-Haviland theorem for the truncated moment problem,, J. Funct. Anal., 255 (2008), 2709. doi: 10.1016/j.jfa.2008.09.003. Google Scholar
[12]	J. Denzler and R. McCann, Fast diffusion to self-similarity: Complete spectrum, long time asymptotics, and numerology,, Arch. Ration. Mech. Anal., 175 (2005), 301. doi: 10.1007/s00205-004-0336-3. Google Scholar
[13]	A. J. Duran, The Stieltjes moments problem for rapidly decreasing functions,, Proc. Amer. Math. Soc., 107 (1989), 731. Google Scholar
[14]	J. Duoandikoetxea and E. Zuazua, Moments, masses de Dirac et decomposition de fonctions,, C. R. Acad. Sci. Paris Ser. I Math., 315 (1992), 693. Google Scholar
[15]	L. Fialkow, Positivity, extensions and the truncated complex moment problem,, in, 185 (1993), 133. Google Scholar
[16]	L. Fialkow, Truncated multivariable moment problems with finite variety,, J. Operator Theory, 60 (2008), 343. Google Scholar
[17]	Y.-J. Kim and R. J. McCann, Potential theory and optimal convergence rates in fast nonlinear diffusion,, J. Math. Pures Appl., 86 (2006), 42. doi: 10.1016/j.matpur.2006.01.002. Google Scholar
[18]	Y.-J. Kim and W.-M. Ni, On the rate of convergence and asymptotic profile of solutions to the viscous Burgers equation,, Indiana Univ. Math. J., 51 (2002), 727. doi: 10.1512/iumj.2002.51.2247. Google Scholar
[19]	Y.-J Kim and W.-M. Ni, Higher order approximations in the heat equation and the truncated moment problem,, SIAM J. Math. Anal., 40 (2009), 2241. doi: 10.1137/08071778X. Google Scholar
[20]	H. J. Landau, Classical background of the moment problem,, Moments in mathematics, 37 (1987), 1. Google Scholar
[21]	J. Philip, Estimates of the age of a heat distribution,, Ark. Mat., 7 (1968), 351. doi: 10.1007/BF02591028. Google Scholar
[22]	M. Putinar, A two-dimensional moment problem,, J. Funct. Anal., 80 (1988), 1. doi: 10.1016/0022-1236(88)90060-2. Google Scholar
[23]	M. Putinar and F.-H. Vasilescu, Solving moment problems by dimensional extension,, Ann. of Math. (2), 149 (1999), 1087. doi: 10.2307/121083. Google Scholar
[24]	J. A. Shohat and J. D. Tamarkin, "The Problem of Moments,", American Mathematical Society Mathematical surveys, II (1943). Google Scholar
[25]	J. Stochel and F. H. Szafraniec, The complex moment problem and subnormality: A polar decomposition approach,, J. Funct. Anal., 159 (1998), 432. doi: 10.1006/jfan.1998.3284. Google Scholar
[26]	T. Yanagisawa, Asymptotic behavior of solutions to the viscous Burgers equation,, Osaka J. Math., 44 (2007), 99. Google Scholar
[27]	T. P. Witelski and A. J. Bernoff, Self-similar asymptotics for linear and nonlinear diffusion equations,, Stud. Appl. Math., 100 (1998), 153. doi: 10.1111/1467-9590.00074. Google Scholar

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References:

[1]	N. I. Akhiezer, "The Classical Moment Problem and Some Related Questions in Analysis,", Hafner, (1965). Google Scholar
[2]	A. Atzmon, A moment problem for positive measures on the unit disc,, Pacific J. Math. \textbf{59} (1975), 59 (1975), 317. Google Scholar
[3]	C. Berg, The multidimensional moment problem and semigroups,, Moments in mathematics, 37 (1987), 110. Google Scholar
[4]	B. P. Boas, The Stieltjes moment problem for functions of bounded variation,, Bull. Amer. Math. Soc., 45 (1939), 399. doi: 10.1090/S0002-9904-1939-06992-9. Google Scholar
[5]	J. A. Carrillo, A. Jüngel, P. A. Markowich, G. Toscani and A. Unterreiter, Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities,, Monatsh. Math., 133 (2001), 1. doi: 10.1007/s006050170032. Google Scholar
[6]	J. Chung, E. Kim and Y.-J. Kim, Asymptotic agreement of moments and higher order contraction in the Burgers equation,, J. Differential Equations, 248 (2010), 2417. doi: 10.1016/j.jde.2010.01.006. Google Scholar
[7]	R. E. Curto and L. A. Fialkow, Recursiveness, positivity, and truncated moment problems,, Houston J. Math., 17 (1991), 603. Google Scholar
[8]	R. E. Curto and L. A. Fialkow, Solution of the truncated complex moment problem for flat data,, Mem. Amer. Math. Soc., 119 (1996). Google Scholar
[9]	R. E. Curto and L. A. Fialkow, Flat extensions of positive moment matrices: Recursively generated relations,, Mem. Amer. Math. Soc., 136 (1998). Google Scholar
[10]	R. E. Curto and L. A. Fialkow, Truncated $K$-moment problems in several variables,, J. Operator Theory, 54 (2005), 189. Google Scholar
[11]	R. E. Curto and L. A. Fialkow, An analogue of the Riesz-Haviland theorem for the truncated moment problem,, J. Funct. Anal., 255 (2008), 2709. doi: 10.1016/j.jfa.2008.09.003. Google Scholar
[12]	J. Denzler and R. McCann, Fast diffusion to self-similarity: Complete spectrum, long time asymptotics, and numerology,, Arch. Ration. Mech. Anal., 175 (2005), 301. doi: 10.1007/s00205-004-0336-3. Google Scholar
[13]	A. J. Duran, The Stieltjes moments problem for rapidly decreasing functions,, Proc. Amer. Math. Soc., 107 (1989), 731. Google Scholar
[14]	J. Duoandikoetxea and E. Zuazua, Moments, masses de Dirac et decomposition de fonctions,, C. R. Acad. Sci. Paris Ser. I Math., 315 (1992), 693. Google Scholar
[15]	L. Fialkow, Positivity, extensions and the truncated complex moment problem,, in, 185 (1993), 133. Google Scholar
[16]	L. Fialkow, Truncated multivariable moment problems with finite variety,, J. Operator Theory, 60 (2008), 343. Google Scholar
[17]	Y.-J. Kim and R. J. McCann, Potential theory and optimal convergence rates in fast nonlinear diffusion,, J. Math. Pures Appl., 86 (2006), 42. doi: 10.1016/j.matpur.2006.01.002. Google Scholar
[18]	Y.-J. Kim and W.-M. Ni, On the rate of convergence and asymptotic profile of solutions to the viscous Burgers equation,, Indiana Univ. Math. J., 51 (2002), 727. doi: 10.1512/iumj.2002.51.2247. Google Scholar
[19]	Y.-J Kim and W.-M. Ni, Higher order approximations in the heat equation and the truncated moment problem,, SIAM J. Math. Anal., 40 (2009), 2241. doi: 10.1137/08071778X. Google Scholar
[20]	H. J. Landau, Classical background of the moment problem,, Moments in mathematics, 37 (1987), 1. Google Scholar
[21]	J. Philip, Estimates of the age of a heat distribution,, Ark. Mat., 7 (1968), 351. doi: 10.1007/BF02591028. Google Scholar
[22]	M. Putinar, A two-dimensional moment problem,, J. Funct. Anal., 80 (1988), 1. doi: 10.1016/0022-1236(88)90060-2. Google Scholar
[23]	M. Putinar and F.-H. Vasilescu, Solving moment problems by dimensional extension,, Ann. of Math. (2), 149 (1999), 1087. doi: 10.2307/121083. Google Scholar
[24]	J. A. Shohat and J. D. Tamarkin, "The Problem of Moments,", American Mathematical Society Mathematical surveys, II (1943). Google Scholar
[25]	J. Stochel and F. H. Szafraniec, The complex moment problem and subnormality: A polar decomposition approach,, J. Funct. Anal., 159 (1998), 432. doi: 10.1006/jfan.1998.3284. Google Scholar
[26]	T. Yanagisawa, Asymptotic behavior of solutions to the viscous Burgers equation,, Osaka J. Math., 44 (2007), 99. Google Scholar
[27]	T. P. Witelski and A. J. Bernoff, Self-similar asymptotics for linear and nonlinear diffusion equations,, Stud. Appl. Math., 100 (1998), 153. doi: 10.1111/1467-9590.00074. Google Scholar

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