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.
[1] A corollary, also called Krein's condition, provides a sufficient condition for the indeterminacy of the moment problem.
[2][3] Let μ be an absolutely continuous measure on the real line, dμ(x) = f(x) dx.
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.
Since the Hamburger moment problem for μ is indeterminate.