In mathematical analysis, Krein's condition provides a necessary and sufficient condition for exponential sums
![{\displaystyle \left\{\sum _{k=1}^{n}a_{k}\exp(i\lambda _{k}x),\quad a_{k}\in \mathbb {C} ,\,\lambda _{k}\geq 0\right\},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/56af84da606169b844db2021cfae83a5b0c7c3f0)
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
![{\displaystyle \sum _{k=1}^{n}a_{k}\exp(i\lambda _{k}x),\quad a_{k}\in \mathbb {C} ,\,\lambda _{k}\geq 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/417391beb5baf7bca222d9a988bfa83a798fcff8)
are dense in L2(μ) if and only if
![{\displaystyle \int _{-\infty }^{\infty }{\frac {-\ln f(x)}{1+x^{2}}}\,dx=\infty .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e44b1e672834f0dfd1a1289c31c778be2a0fa065)
Indeterminacy of the moment problem
Let μ be as above; assume that all the moments
![{\displaystyle m_{n}=\int _{-\infty }^{\infty }x^{n}d\mu (x),\quad n=0,1,2,\ldots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/123e9097fdee35e562088f70e142ec7ca2df0525)
of μ are finite. If
![{\displaystyle \int _{-\infty }^{\infty }{\frac {-\ln f(x)}{1+x^{2}}}\,dx<\infty }](https://wikimedia.org/api/rest_v1/media/math/render/svg/0076959fa36692e46d7c11a9f9deff0e0b22c08b)
holds, then the Hamburger moment problem for μ is indeterminate; that is, there exists another measure ν ≠ μ on R such that
![{\displaystyle m_{n}=\int _{-\infty }^{\infty }x^{n}\,d\nu (x),\quad n=0,1,2,\ldots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/13900f7a526f8fe571cc48a4128bff36dab8cb3a)
This can be derived from the "only if" part of Krein's theorem above.[4]
Example
Let
![{\displaystyle f(x)={\frac {1}{\sqrt {\pi }}}\exp \left\{-\ln ^{2}x\right\};}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e82cc4cdd20fc3e94af1cb81db285d04e565a9ae)
the measure dμ(x) = f(x) dx is called the Stieltjes–Wigert measure. Since
![{\displaystyle \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 ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d9a7d7203b5c0e4b290160e78144c9642175ceb7)
the Hamburger moment problem for μ is indeterminate.
References