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.^[1] A corollary, also called Krein's condition, provides a sufficient condition for the indeterminacy of the moment problem.^[2][3]

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.^[4]

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.

References

1. Krein, M.G. (1945). "On an extrapolation problem due to Kolmogorov". Doklady Akademii Nauk SSSR 46: 306–309.
2. Stoyanov, J. (2001), "Krein_condition", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
3. Berg, Ch. (1995). "Indeterminate moment problems and the theory of entire functions". J. Comput. Appl. Math. 65: 1–3, 27–55. doi:10.1016/0377-0427(95)00099-2. MR 1379118.
4. Akhiezer, N. I. (1965). The Classical Moment Problem and Some Related Questions in Analysis. Oliver & Boyd.