# American Institute of Mathematical Sciences

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

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.
Citation: Yong-Jung Kim. A generalization of the moment problem to a complex measure space and an approximation technique using backward moments. Discrete and Continuous Dynamical Systems, 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, New York 1965 [2] A. Atzmon, A moment problem for positive measures on the unit disc, Pacific J. Math. 59 (1975), 317-325. [3] C. Berg, The multidimensional moment problem and semigroups, Moments in mathematics, Proc. Sympos. Appl. Math., 37 (1987), 110-124. [4] B. P. Boas, The Stieltjes moment problem for functions of bounded variation, Bull. Amer. Math. Soc., 45 (1939), 399-404. doi: 10.1090/S0002-9904-1939-06992-9. [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), 1-82. doi: 10.1007/s006050170032. [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), 2417-2434. doi: 10.1016/j.jde.2010.01.006. [7] R. E. Curto and L. A. Fialkow, Recursiveness, positivity, and truncated moment problems, Houston J. Math., 17 (1991), 603-635. [8] R. E. Curto and L. A. Fialkow, Solution of the truncated complex moment problem for flat data, Mem. Amer. Math. Soc., 119 (1996), x+52 pp. [9] R. E. Curto and L. A. Fialkow, Flat extensions of positive moment matrices: Recursively generated relations, Mem. Amer. Math. Soc., 136 (1998), x+56 pp. [10] R. E. Curto and L. A. Fialkow, Truncated $K$-moment problems in several variables, J. Operator Theory, 54 (2005), 189-226. [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), 2709-2731. doi: 10.1016/j.jfa.2008.09.003. [12] J. Denzler and R. McCann, Fast diffusion to self-similarity: Complete spectrum, long time asymptotics, and numerology, Arch. Ration. Mech. Anal., 175 (2005) 301-342. doi: 10.1007/s00205-004-0336-3. [13] A. J. Duran, The Stieltjes moments problem for rapidly decreasing functions, Proc. Amer. Math. Soc., 107 (1989), 731-741. [14] J. Duoandikoetxea and E. Zuazua, Moments, masses de Dirac et decomposition de fonctions, C. R. Acad. Sci. Paris Ser. I Math., 315 (1992), 693-698. [15] L. Fialkow, Positivity, extensions and the truncated complex moment problem, in "Multivariable operator theory (Seattle, WA, 1993)," 133-150, Contemp. Math., 185, Amer. Math. Soc., Providence, RI, 1995. [16] L. Fialkow, Truncated multivariable moment problems with finite variety, J. Operator Theory, 60 (2008), 343-377. [17] Y.-J. Kim and R. J. McCann, Potential theory and optimal convergence rates in fast nonlinear diffusion, J. Math. Pures Appl., 86 (2006), 42-67. doi: 10.1016/j.matpur.2006.01.002. [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), 727-752. doi: 10.1512/iumj.2002.51.2247. [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), 2241-2261. doi: 10.1137/08071778X. [20] H. J. Landau, Classical background of the moment problem, Moments in mathematics, Proc. Sympos. Appl. Math., 37 (1987), 1-15. [21] J. Philip, Estimates of the age of a heat distribution, Ark. Mat., 7 (1968), 351-358. doi: 10.1007/BF02591028. [22] M. Putinar, A two-dimensional moment problem, J. Funct. Anal., 80 (1988), 1-8. doi: 10.1016/0022-1236(88)90060-2. [23] M. Putinar and F.-H. Vasilescu, Solving moment problems by dimensional extension, Ann. of Math. (2), 149 (1999), 1087-1107. doi: 10.2307/121083. [24] J. A. Shohat and J. D. Tamarkin, "The Problem of Moments," American Mathematical Society Mathematical surveys, vol. II, American Mathematical Society, New York, 1943. xiv+140 pp. [25] J. Stochel and F. H. Szafraniec, The complex moment problem and subnormality: A polar decomposition approach, J. Funct. Anal., 159 (1998), 432-491. doi: 10.1006/jfan.1998.3284. [26] T. Yanagisawa, Asymptotic behavior of solutions to the viscous Burgers equation, Osaka J. Math., 44 (2007), 99-119. [27] T. P. Witelski and A. J. Bernoff, Self-similar asymptotics for linear and nonlinear diffusion equations, Stud. Appl. Math., 100 (1998), 153-193. doi: 10.1111/1467-9590.00074.

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##### References:
 [1] N. I. Akhiezer, "The Classical Moment Problem and Some Related Questions in Analysis," Hafner, New York 1965 [2] A. Atzmon, A moment problem for positive measures on the unit disc, Pacific J. Math. 59 (1975), 317-325. [3] C. Berg, The multidimensional moment problem and semigroups, Moments in mathematics, Proc. Sympos. Appl. Math., 37 (1987), 110-124. [4] B. P. Boas, The Stieltjes moment problem for functions of bounded variation, Bull. Amer. Math. Soc., 45 (1939), 399-404. doi: 10.1090/S0002-9904-1939-06992-9. [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), 1-82. doi: 10.1007/s006050170032. [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), 2417-2434. doi: 10.1016/j.jde.2010.01.006. [7] R. E. Curto and L. A. Fialkow, Recursiveness, positivity, and truncated moment problems, Houston J. Math., 17 (1991), 603-635. [8] R. E. Curto and L. A. Fialkow, Solution of the truncated complex moment problem for flat data, Mem. Amer. Math. Soc., 119 (1996), x+52 pp. [9] R. E. Curto and L. A. Fialkow, Flat extensions of positive moment matrices: Recursively generated relations, Mem. Amer. Math. Soc., 136 (1998), x+56 pp. [10] R. E. Curto and L. A. Fialkow, Truncated $K$-moment problems in several variables, J. Operator Theory, 54 (2005), 189-226. [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), 2709-2731. doi: 10.1016/j.jfa.2008.09.003. [12] J. Denzler and R. McCann, Fast diffusion to self-similarity: Complete spectrum, long time asymptotics, and numerology, Arch. Ration. Mech. Anal., 175 (2005) 301-342. doi: 10.1007/s00205-004-0336-3. [13] A. J. Duran, The Stieltjes moments problem for rapidly decreasing functions, Proc. Amer. Math. Soc., 107 (1989), 731-741. [14] J. Duoandikoetxea and E. Zuazua, Moments, masses de Dirac et decomposition de fonctions, C. R. Acad. Sci. Paris Ser. I Math., 315 (1992), 693-698. [15] L. Fialkow, Positivity, extensions and the truncated complex moment problem, in "Multivariable operator theory (Seattle, WA, 1993)," 133-150, Contemp. Math., 185, Amer. Math. Soc., Providence, RI, 1995. [16] L. Fialkow, Truncated multivariable moment problems with finite variety, J. Operator Theory, 60 (2008), 343-377. [17] Y.-J. Kim and R. J. McCann, Potential theory and optimal convergence rates in fast nonlinear diffusion, J. Math. Pures Appl., 86 (2006), 42-67. doi: 10.1016/j.matpur.2006.01.002. [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), 727-752. doi: 10.1512/iumj.2002.51.2247. [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), 2241-2261. doi: 10.1137/08071778X. [20] H. J. Landau, Classical background of the moment problem, Moments in mathematics, Proc. Sympos. Appl. Math., 37 (1987), 1-15. [21] J. Philip, Estimates of the age of a heat distribution, Ark. Mat., 7 (1968), 351-358. doi: 10.1007/BF02591028. [22] M. Putinar, A two-dimensional moment problem, J. Funct. Anal., 80 (1988), 1-8. doi: 10.1016/0022-1236(88)90060-2. [23] M. Putinar and F.-H. Vasilescu, Solving moment problems by dimensional extension, Ann. of Math. (2), 149 (1999), 1087-1107. doi: 10.2307/121083. [24] J. A. Shohat and J. D. Tamarkin, "The Problem of Moments," American Mathematical Society Mathematical surveys, vol. II, American Mathematical Society, New York, 1943. xiv+140 pp. [25] J. Stochel and F. H. Szafraniec, The complex moment problem and subnormality: A polar decomposition approach, J. Funct. Anal., 159 (1998), 432-491. doi: 10.1006/jfan.1998.3284. [26] T. Yanagisawa, Asymptotic behavior of solutions to the viscous Burgers equation, Osaka J. Math., 44 (2007), 99-119. [27] T. P. Witelski and A. J. Bernoff, Self-similar asymptotics for linear and nonlinear diffusion equations, Stud. Appl. Math., 100 (1998), 153-193. doi: 10.1111/1467-9590.00074.
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