Stieltjes moment problem

From Wikipedia, the free encyclopedia
Jump to navigation Jump to search

In mathematics, the Stieltjes moment problem, named after Thomas Joannes Stieltjes, seeks necessary and sufficient conditions for a sequence { mn, : n = 0, 1, 2, ... } to be of the form

for some measure μ. If such a function μ exists, one asks whether it is unique.

The essential difference between this and other well-known moment problems is that this is on a half-line [0, ∞), whereas in the Hausdorff moment problem one considers a bounded interval [0, 1], and in the Hamburger moment problem one considers the whole line (−∞, ∞).

Existence[edit]

Let

and

Then { mn : n = 1, 2, 3, ... } is a moment sequence of some measure on with infinite support if and only if for all n, both

mn : n = 1, 2, 3, ... } is a moment sequence of some measure on with finite support of size m if and only if for all , both

and for all larger

Uniqueness[edit]

There are several sufficient conditions for uniqueness, for example, Carleman's condition, which states that the solution is unique if

References[edit]

  • Reed, Michael; Simon, Barry (1975), Fourier Analysis, Self-Adjointness, Methods of modern mathematical physics, 2, Academic Press, p. 341 (exercise 25), ISBN 0-12-585002-6