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Krein's condition

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

{ ∑ k = 1 n a k exp ⁡ ( i λ k x ) , a k ∈ C , λ k ≥ 0 } , {\displaystyle \left\{\sum _{k=1}^{n}a_{k}\exp(i\lambda _{k}x),\quad a_{k}\in \mathbb {C} ,\,\lambda _{k}\geq 0\right\},}

to be dense in a weighted L2 space on the real line. It was discovered by Mark Krein in the 1940s. A corollary, also called Krein's condition, provides a sufficient condition for the indeterminacy of the moment problem.

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 {\displaystyle \sum _{k=1}^{n}a_{k}\exp(i\lambda _{k}x),\quad a_{k}\in \mathbb {C} ,\,\lambda _{k}\geq 0}

are dense in L2(μ) if and only if

∫ − ∞ ∞ − ln ⁡ f ( x ) 1 + x 2 d x = ∞ . {\displaystyle \int _{-\infty }^{\infty }{\frac {-\ln f(x)}{1+x^{2}}}\,dx=\infty .}

Indeterminacy of the moment problem

Let μ be as above; assume that all the moments

m n = ∫ − ∞ ∞ x n d μ ( x ) , n = 0 , 1 , 2 , … {\displaystyle m_{n}=\int _{-\infty }^{\infty }x^{n}d\mu (x),\quad n=0,1,2,\ldots }

of μ are finite. If

∫ − ∞ ∞ − ln ⁡ f ( x ) 1 + x 2 d x < ∞ {\displaystyle \int _{-\infty }^{\infty }{\frac {-\ln f(x)}{1+x^{2}}}\,dx<\infty }

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 , … {\displaystyle m_{n}=\int _{-\infty }^{\infty }x^{n}\,d\nu (x),\quad n=0,1,2,\ldots }

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

Example

Let

f ( x ) = 1 π exp ⁡ { − ln 2 ⁡ x } ; {\displaystyle f(x)={\frac {1}{\sqrt {\pi }}}\exp \left\{-\ln ^{2}x\right\};}

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 < ∞ , {\displaystyle \int _{-\infty }^{\infty }{\frac {-\ln f(x)}{1+x^{2}}}dx=\int _{-\infty }^{\infty }{\frac {\ln ^{2}x+\ln {\sqrt {\pi }}}{1+x^{2}}}\,dx<\infty ,}

the Hamburger moment problem for μ is indeterminate.

References

  1. Krein, M.G. (1945). "On an extrapolation problem due to Kolmogorov". Doklady Akademii Nauk SSSR. 46: 306–309.
  2. Stoyanov, J. (2001) , "Krein_condition", Encyclopedia of Mathematics, EMS Press
  3. Berg, Ch. (1995). "Indeterminate moment problems and the theory of entire functions". J. Comput. Appl. Math. 65 (1–3): 1–3, 27–55. doi:10.1016/0377-0427(95)00099-2. MR 1379118.
  4. Akhiezer, N. I. (1965). The Classical Moment Problem and Some Related Questions in Analysis. Oliver & Boyd.
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