Krein's condition


In mathematical analysis, Krein's condition provides a necessary and sufficient condition for exponential sums
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 = f dx. The exponential sums
are dense in L2 if and only if

Indeterminacy of the moment problem

Let μ be as above; assume that all the moments
of μ are finite. If
holds, then the Hamburger moment problem for μ is indeterminate; that is, there exists another measure ν ≠ μ on R such that
This can be derived from the "only if" part of Krein's theorem above.

Example

Let
the measure dμ = f dx is called the Stieltjes–Wigert measure. Since
the Hamburger moment problem for μ is indeterminate.