Krein's condition

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

$\{\sum _{k=1}^n{a}_k\exp (i{\lambda }_{k}x),a_k\in ℂ,{\lambda }_k\ge 0\},$

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

$\sum _{k=1}^n{a}_k\exp (i{\lambda }_{k}x),a_k\in ℂ,{\lambda }_k\ge 0$

are dense in L₂(μ) if and only if

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

Indeterminacy of the moment problem

Let μ be as above; assume that all the moments

$m_n=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}{x}^{n}d\mu (x),n=0,1,2,\dots$

of μ are finite. If

$\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\frac{-\ln f(x)}{1+x^2}dx<\mathrm{\infty }$

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

$m_n=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}{x}^{n}d\nu (x),n=0,1,2,\dots$

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

Example

Let

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

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

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

the Hamburger moment problem for μ is indeterminate.