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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 \( L_2(\mu ) \) 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

^ Krein, M.G. (1945). "On an extrapolation problem due to Kolmogorov". Doklady Akademii Nauk SSSR 46: 306–309.
^ Stoyanov, J. (2001), "Krein_condition", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1556080104
^ Berg, Ch. (1995). "Indeterminate moment problems and the theory of entire functions". J. Comput. Appl. Math. 65: 1–3, 27–55

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