Krein's condition - Alchetron, The Free Social Encyclopedia

In mathematical analysis, Krein's condition provides a necessary and sufficient condition for exponential sums

Statement

Let μ be an absolutely continuous measure on the real line, dμ(x) = f(x) dx. The exponential sums

∑ k = 1 n a k exp ⁡ ( i λ k x ) , a k ∈ C , λ k ≥ 0

are dense in L₂(μ) if and only if

∫ − ∞ ∞ − ln ⁡ f ( x ) 1 + x 2 d x = ∞ .

Let μ be as above; assume that all the moments

m n = ∫ − ∞ ∞ x n d μ ( x ) , n = 0 , 1 , 2 , …

of μ are finite. If

∫ − ∞ ∞ − ln ⁡ f ( x ) 1 + x 2 d x < ∞

holds, then the Hamburger moment problem for μ is indeterminate; that is, there exists another measure ν ≠ μ on R such that

m n = ∫ − ∞ ∞ x n d ν ( x ) , n = 0 , 1 , 2 , …

This can be derived from the "only if" part of Krein's theorem above.

Let

f ( x ) = 1 π exp ⁡ { − ln 2 ⁡ x } ;

the measure dμ(x) = f(x) dx is called the Stieltjes–Wigert measure. Since

∫ − ∞ ∞ − ln ⁡ f ( x ) 1 + x 2 d x = ∫ − ∞ ∞ ln 2 ⁡ x + ln ⁡ π 1 + x 2 d x < ∞ ,

the Hamburger moment problem for μ is indeterminate.

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