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 &mu; be an absolutely continuous measure on the real line, d&mu;(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(&mu;) if and only if


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

Indeterminacy of the moment problem
Let &mu; 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 &mu; are finite. If


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

holds, then the Hamburger moment problem for &mu; is indeterminate; that is, there exists another measure &nu; ≠ &mu; 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 d&mu;(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 &mu; is indeterminate.