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Abstract
The present paper is centered on a class of real quintic moment problem. We state some conditions for the existence of a representative measure and we provide it explicitly. We also state some cases where no representative measure exists. Some numerical examples are presented to illustrate construction of the representative measure as well as to highlight the conflicts behind the irresolvability.
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References
- N. I. Akhiezer et M. G. Krein, Some questions in the theory of moments , Translations of Mathematical Monographs, Amer. Math. Soc., Providence, 1962.
- C. Bayer and J. Teichmann, The proof of Tchakaloff’s theorem , Proc. Amer. Math. Soc. 134, no. 10, (2006), 3035–3040.
- R. E. Curto and L. A. Fialkow, Solution of the truncated complex moment problem for flat data , Mem. Amer. Math. Soc., 119, no. 568, (1996).
- R. E. Curto and L. A. Fialkow, Flat extensions of positive moment matrices: recursively generated relations , Mem. Amer. Math. Soc., 136, no. 648, (1998).
- R. E. Curto and L. A. Fialkow, Flat extensions of positive moment matrices: relations in analytic or conjugate terms , Nonselfadjoint operator algebras, operator theory, and related topics, (1998), 59–82.
- R. E. Curto and L. A. Fialkow, Solution of the singular quartic moment problem , J. Operator Theory, 48, no. 2, (2002), 315–354.
- R. E. Curto and L. A. Fialkow, Truncated K-moment problems in several variables, J. Operator Theory, 54, no. 1, (2005) 189–226.
- R. E. Curto, L. A. Fialkow and H. M. M¨oller, The extremal truncated moment problem, Integral Equations Operator Theory, 60, no. 2, (2008), 177–200.
- R. E. Curto and L. A. Fialkow, Recursively determined representing measures for bivariate truncated moment sequences , J. Operator Theory, 70, no. 2, (2013), 401–436.
- R. E. Curto and S. Yoo, Non-extremal sextic moment problems , J. Funct. Anal., 269, no. 3, (2015), 758–780.
- R. E. Curto and S. Yoo, Concrete solution to the nonsingular quartic binary moment problem, Proc. Amer. Math. Soc., 144, no. 1, (2016), 249–258.
- R. E. Curto and S. Yoo, The division algorithm in sextic truncated moment problems, Integral Equations Operator Theory, 87, no. 4, (2017), 515–528.
- R. E. Curto and S. Yoo, A new approach to the nonsingular cubic binary moment problem, Ann. Funct. Anal., 9, no. 4, (2018), 525–536.
- P. J. di Dio et K. Schmdgen, The multidimensional truncated moment problem: Atoms, determinacy, and core variety , J. Funct. Anal, 274, no. 11, (2018), 3124–3148.
- R. G. Douglas, On majorization and range inclusion of operators in Hilbert space , Proc. Amer. Math. Soc., 17, no. 2, (1966) 413–416.
- L. A. Fialkow, Truncated multivariable moment problems with finite variety , J. Operator Theory, 60, no. 2, (2008), 343–377.
- L. A. Fialkow, The core variety of a multisequence in the truncated moment problem , J. Math. Anal. Appl., 456, no. 2, (2017), 946–969.
- D. P. Kimsey, The cubic complex moment problem , Integral Equations Operator Theory, 80, no. 3, (2014), 353–378.
- M. Laurent, Sums of squares, moment matrices and optimization over polynomials , in Emerging applications of algebraic geometry, IMA Volumes in Mathematics and its Applications book series, 149, Springer, New York, 157–270.
- H. Richter, Parameterfreie Abschtzung und Realisierung von Erwartungswerten , Bltter der Deutsch. Ges. Versicherungsmath, 3, 147–161, (1957).
- Y. L. Shmuljan, An operator hellinger integral , Matematicheskii Sbornik, 91, no. 4, (1959), 381–430.
- K. Schm¨udgen, The moment problem, Graduate Texts in Mathematics, Springer, 277, 2017.
- J. Stochel, Solving the truncated moment problem solves the full moment problem , Glasg. Math. J., 43, no. 3, (2001), 335–341.
- S. Yoo, Extremal sextic truncated moment problems , PhD thesis, University of Iowa, 2011
- S. Yoo, Sextic moment problems on 3 parallel lines , Bull. Korean Math. Soc., 54, no. 1 (2017), 299–318
References
N. I. Akhiezer et M. G. Krein, Some questions in the theory of moments , Translations of Mathematical Monographs, Amer. Math. Soc., Providence, 1962.
C. Bayer and J. Teichmann, The proof of Tchakaloff’s theorem , Proc. Amer. Math. Soc. 134, no. 10, (2006), 3035–3040.
R. E. Curto and L. A. Fialkow, Solution of the truncated complex moment problem for flat data , Mem. Amer. Math. Soc., 119, no. 568, (1996).
R. E. Curto and L. A. Fialkow, Flat extensions of positive moment matrices: recursively generated relations , Mem. Amer. Math. Soc., 136, no. 648, (1998).
R. E. Curto and L. A. Fialkow, Flat extensions of positive moment matrices: relations in analytic or conjugate terms , Nonselfadjoint operator algebras, operator theory, and related topics, (1998), 59–82.
R. E. Curto and L. A. Fialkow, Solution of the singular quartic moment problem , J. Operator Theory, 48, no. 2, (2002), 315–354.
R. E. Curto and L. A. Fialkow, Truncated K-moment problems in several variables, J. Operator Theory, 54, no. 1, (2005) 189–226.
R. E. Curto, L. A. Fialkow and H. M. M¨oller, The extremal truncated moment problem, Integral Equations Operator Theory, 60, no. 2, (2008), 177–200.
R. E. Curto and L. A. Fialkow, Recursively determined representing measures for bivariate truncated moment sequences , J. Operator Theory, 70, no. 2, (2013), 401–436.
R. E. Curto and S. Yoo, Non-extremal sextic moment problems , J. Funct. Anal., 269, no. 3, (2015), 758–780.
R. E. Curto and S. Yoo, Concrete solution to the nonsingular quartic binary moment problem, Proc. Amer. Math. Soc., 144, no. 1, (2016), 249–258.
R. E. Curto and S. Yoo, The division algorithm in sextic truncated moment problems, Integral Equations Operator Theory, 87, no. 4, (2017), 515–528.
R. E. Curto and S. Yoo, A new approach to the nonsingular cubic binary moment problem, Ann. Funct. Anal., 9, no. 4, (2018), 525–536.
P. J. di Dio et K. Schmdgen, The multidimensional truncated moment problem: Atoms, determinacy, and core variety , J. Funct. Anal, 274, no. 11, (2018), 3124–3148.
R. G. Douglas, On majorization and range inclusion of operators in Hilbert space , Proc. Amer. Math. Soc., 17, no. 2, (1966) 413–416.
L. A. Fialkow, Truncated multivariable moment problems with finite variety , J. Operator Theory, 60, no. 2, (2008), 343–377.
L. A. Fialkow, The core variety of a multisequence in the truncated moment problem , J. Math. Anal. Appl., 456, no. 2, (2017), 946–969.
D. P. Kimsey, The cubic complex moment problem , Integral Equations Operator Theory, 80, no. 3, (2014), 353–378.
M. Laurent, Sums of squares, moment matrices and optimization over polynomials , in Emerging applications of algebraic geometry, IMA Volumes in Mathematics and its Applications book series, 149, Springer, New York, 157–270.
H. Richter, Parameterfreie Abschtzung und Realisierung von Erwartungswerten , Bltter der Deutsch. Ges. Versicherungsmath, 3, 147–161, (1957).
Y. L. Shmuljan, An operator hellinger integral , Matematicheskii Sbornik, 91, no. 4, (1959), 381–430.
K. Schm¨udgen, The moment problem, Graduate Texts in Mathematics, Springer, 277, 2017.
J. Stochel, Solving the truncated moment problem solves the full moment problem , Glasg. Math. J., 43, no. 3, (2001), 335–341.
S. Yoo, Extremal sextic truncated moment problems , PhD thesis, University of Iowa, 2011
S. Yoo, Sextic moment problems on 3 parallel lines , Bull. Korean Math. Soc., 54, no. 1 (2017), 299–318