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Krein's condition

From Wikipedia, the free encyclopedia

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

Statement

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Let μ be an absolutely continuous measure on the real line, dμ(x) = f(x) dx. The exponential sums

are dense in L2(μ) if and only if

Indeterminacy of the moment problem

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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.[4]

Example

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Let

the measure dμ(x) = f(x) dx is called the Stieltjes–Wigert measure. Since

the Hamburger moment problem for μ is indeterminate.

References

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  1. ^ Krein, M.G. (1945). "On an extrapolation problem due to Kolmogorov". Doklady Akademii Nauk SSSR. 46: 306–309.
  2. ^ Stoyanov, J. (2001) [1994], "Krein_condition", Encyclopedia of Mathematics, EMS Press
  3. ^ Berg, Ch. (1995). "Indeterminate moment problems and the theory of entire functions". J. Comput. Appl. Math. 65 (1–3): 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.