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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 \( L_2(\mu ) \) 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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