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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 |
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. |
show all references
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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