Krein's condition

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

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

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

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

are dense in L₂(μ) if and only if

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

Indeterminacy of the moment problem

Let μ be as above; assume that all the moments

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

of μ are finite. If

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

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

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

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

Example

Let

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

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

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

the Hamburger moment problem for μ is indeterminate.