ANECESSARY AND REQUIRED CONDITION OF HAMBURGER MOMENT PROBLEM

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.