Krein's condition

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

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

to be dense in a weighted L₂ space on the real line. It was discovered by Mark Krein in the 1940s.^[1] A corollary, also called Krein's condition, provides a sufficient condition for the indeterminacy of the moment problem.^[2][3]

Statement

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

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

are dense in L₂(μ) if and only if

${\displaystyle \int _{-\infty }^{\infty }{\frac {-\ln f(x)}{1+x^{2}}}\,dx=\infty .}$

Indeterminacy of the moment problem

Let μ be as above; assume that all the moments

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

of μ are finite. If

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

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

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

This can be derived from the "only if" part of Krein's theorem above.^[4]

Example

Let

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

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

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

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", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
3. Berg, Ch. (1995). "Indeterminate moment problems and the theory of entire functions". J. Comput. Appl. Math. 65: 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.