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Stieltjes moment problem

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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

Contents

m n = ∫ 0 ∞ x n d μ ( x )

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

Let

Δ n = [ m 0 m 1 m 2 ⋯ m n m 1 m 2 m 3 ⋯ m n + 1 m 2 m 3 m 4 ⋯ m n + 2 ⋮ ⋮ ⋮ ⋱ ⋮ m n m n + 1 m n + 2 ⋯ m 2 n ]

and

Δ n ( 1 ) = [ m 1 m 2 m 3 ⋯ m n + 1 m 2 m 3 m 4 ⋯ m n + 2 m 3 m 4 m 5 ⋯ m n + 3 ⋮ ⋮ ⋮ ⋱ ⋮ m n + 1 m n + 2 m n + 3 ⋯ m 2 n + 1 ] .

Then { mn : n = 1, 2, 3, ... } is a moment sequence of some measure on [ 0 , ∞ ) with infinite support if and only if for all n, both

det ( Δ n ) > 0   a n d   det ( Δ n ( 1 ) ) > 0.

{ mn : n = 1, 2, 3, ... } is a moment sequence of some measure on [ 0 , ∞ ) with finite support of size m if and only if for all n ≤ m , both

det ( Δ n ) > 0   a n d   det ( Δ n ( 1 ) ) > 0

and for all larger n

det ( Δ n ) = 0   a n d   det ( Δ n ( 1 ) ) = 0.

Uniqueness

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

∑ n ≥ 1 m n − 1 / ( 2 n ) = ∞   .

References

Stieltjes moment problem Wikipedia
(Text) CC BY-SA


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