ANECESSARY AND REQUIRED CONDITION OF HAMBURGER MOMENT PROBLEM

  • SANAP GK Head, Department of Mathematics, Sunderrao Solanke Mahavidyalaya, Majalgaon. Dist.Beed (MS) India.,
  • RAUT NK Ex.Head,Dept.Of Physics,Sunderrao Solanke Mahavidyalaya Majalgaon,Dist.Beed (M.S.)India.

Abstract

Objective: This present paper deals with necessary condition of Hamburger moment problem and polynomial which is not identically and non-negative sequence and semi-definite nature of a moment sequence.


Materials and Methods: If we suppose that (sn)n>=0 is a sequence of real numbers, the moment problem on I consists of solving the following three problems:


 There exists a positive measure on I with moment(sn)n>=0.


 This positive measure uniquely determined by the moments(sn)n>=0.


 The moment problem on [0,1) is referred to as Hausdroff moment problem and the moment problem on R is called Hamburger moment problem and the [0, ∞) is called Stieltjes moment problem.


 


Results: For n be an arbitrary non-negative integer, and sub-interval tn in every sub-interval is not greater than such that . The function V(t) in terms of operator M is ifα(t) had infinitely many points of non-decrease, then for every positive polynomial P(t) not identically zero,      21 TT n n μ t d t         2 t d t t t t 1p 0 i TT n i 1 i n 1 i 21             


. .      2 μ μ μ t V M m 20 n                      n0 k k k 0t d t P μ t P M


Conclusion: For increasing function α(t) has a finite number of points of non-increase. Every non-negative sequence is either definite or semi-definite.

Keywords: Hamburger moment problem, Moment sequence, monotonic sequence, Rational number, Stieltjes integral

References

1. 1.Riesz M. Sur le probleme des. Moments. Arkiv. Mat Fys,1923, Vol. 16,1-52.
2. 2.Hamburger H. Uber eine Erweiterung des Stieltjesschen Moment problems. Math.Ann.I, 1920, Vol. 81, 235-319.
3. 3.Hamburger H. Uber eine Erweiterung des Stieltjesschen Moment problems. MathAnn.II,1921,Vol. 82,120-164.
4. 4.Hamburger,H. Uber eine Erweiterung des Stieltjesschen Moment problems. MathAnn.III,1921,Vol. 82, 168-187.
5. 5.Hardy GH, Littlewood JE, and Hardy GH. Little-Wood JE and Plya G. Lecture Notes in Mathematics, Springer, 2002.
6. 6.Harny GH, Titchmarsh EC.Solution of an integral equation. Journal of the London Mathematical Society,1929,Vol. 4,300- 304.
7. 7.Boas RP, JR.Stieltjes moment problem for functions of bounded variation.Bulletin of the American Mathematical Society,1939,Vol. 45, 399-404.
8. Boas, R. P., JR.Functions with positive derivatives. Duke Mathematical Journal, 1941, Vol. 8,163-172.
9. Boas RP, JR.Widder DV.The iterated Stieltjes transform. Transactions of the American Mathematical Society,1939,Vol. 45,1-72.
10. Boas RP, JR, Widder DV.An inversion formula for the Laplace integral.Duke Mathematical Journal,1940a,Vol. 6,1-26.
11. Boas RP, JR, Widder DV.Functions with positive differences,Duke Mathematical Journal,1940b,Vol. 7,496 503.
12. Gaverssss DP.Observing stochastic processes and approximate transform inversion. Operations Research,1966,14,444-459.
13. Landau.H. Classical background of the moment problem, Moments inMathematics,1987,1-15
14. Stochel J. Stochel JB.On the xth roots of a movement sequence, J. Math and l. App. 1396 ,2012,786 – 800.
15. Widder DV. Necessary and sufficient conditions for the representation of a function as a Laplace integral. Transacti6ns of the American Mathematical Society, 1931, Vol. 33, 851-892
16. BeningVE, Korolev VK.Statistical estimation of parameter of fractionally stable distribution. J. Math. Science (NY) ,2013,189 (6),899 – 902.
17. 17.Berg C.Moment problems and polynomial approximation,Annales de la Faculty des Sciences de Toulouse Mat.1994, Vol .5, 9-32.
18. Evans GA,Chung KC.Laplace transform inversion using optimal contours in the complex plane. Int. J. Comput. Math.2000,73, 531-543.
19. Feller W. Completely monotone functions and sequences. Duke Mathematical Journal, 1939, Vol. 5,662-663.
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GK, S., & NK, R. (2020). ANECESSARY AND REQUIRED CONDITION OF HAMBURGER MOMENT PROBLEM. Innovare Journal of Sciences, 8(7), 32-36. Retrieved from https://innovareacademics.in/journals/index.php/ijs/article/view/38523
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