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On Polynomial Maximum Entropy Method for Classical Moment Problem

Published online by Cambridge University Press:  21 December 2015

Jiu Ding ,
Noah H. Rhee  and
Chenhua Zhang
Jiu Ding*
Affiliation:
Department of Mathematics, The University of Southern Mississippi, Hattiesburg, MS 39406-5045, USA
Noah H. Rhee
Affiliation:
Department of Mathematics and Statistics, University of Missouri–Kansas City, Kansas City, MO 64110-2499, USA
Chenhua Zhang
Affiliation:
Department of Mathematics, The University of Southern Mississippi, Hattiesburg, MS 39406-5045, USA
*
*Corresponding author. Email:jiu.ding@usm.edu (J. Ding), rheen@umkc.edu (N. H. Rhee), chenhua.zhang@usm.edu (C. H. Zhang)
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Abstract

The maximum entropy method for the Hausdorff moment problem suffers from ill conditioning as it uses monomial basis {1,x,x2,...,xn}. The maximum entropy method for the Chebyshev moment probelm was studied to overcome this drawback in. In this paper we review and modify the maximum entropy method for the Hausdorff and Chebyshev moment problems studied in and present the maximum entropy method for the Legendre moment problem. We also give the algorithms of converting the Hausdorff moments into the Chebyshev and Lengendre moments, respectively, and utilizing the corresponding maximum entropy method.

Keywords

41A3565D0765J10Classical moment problemmonomialsChebyshev polynomialsLegendre polynomials
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 8 , Issue 1 , February 2016 , pp. 117 - 127
Copyright
Copyright © Global-Science Press 2016 

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References

[1]Ding, J., A maximum entropy method for solving Frobenius-Perron operator equations, Appl. Math. Comput., 93 (1998), pp. 155–168.Google Scholar
[2]Ding, J., Jin, C., Rhee, N. and Zhou, A., A maximum entropy method based on piecewise linear functions for the recovery of a stationary density of interval mappings, J. Stat. Phys., 145(6) (2011), pp. 1620–1639.CrossRefGoogle Scholar
[3]Ding, J. and Mead, L., Maximum entropy approximation for Lyaponov exponents of chaostic maps, J. Math. Phys., 43(5) (2002), pp. 2518–2522.Google Scholar
[4]Ding, J. and Rhee, N., A maximum entropy method based on orthogonal polynomials for Frobenius-Perron operators, Adv. Appl. Math. Mech., 3(2) (2011), pp. 204–218.CrossRefGoogle Scholar
[5]Ding, J. and Rhee, N., Birkhoff’s ergodic theorem and the piecewise-constant maximum entropy method for Frobenius-Perron operators, Int. J. Comput. Math., 89(8) (2012), pp. 1083–1091.Google Scholar
[6]Ding, J. and Rhee, N., A unified maximum entropy method via spline functions for Frobenius-Perron operators, Numer. Algb. Control Optim., 3(2) (2013), pp. 235–245.Google Scholar
[7]Ding, J. and Zhou, A., Statistical Properties of Deterministic Systems, Springer, 2009.Google Scholar
[8]Jayne, E. T., Information theory and statistical mechanics, Phys. Rev., 106 (1957), pp. 620–630.Google Scholar
[9]Lasota, A. and Mackey, M., Chaos, Fractals, and Noises, Second Edition, Springer-Verlag, New York, 1994.Google Scholar
[10]Mead, L. R. and Papanicolaou, N., Maximum entropy in the problem of moments, J. Math. Phys., 25 (1984), pp. 2404–2417.Google Scholar
[11]Natanson, I. P., Constructive Function Theory, Frederick Ungar, New York, 1965.Google Scholar