Carleman's condition

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In mathematics, particularly, in analysis, Carleman's condition gives a sufficient condition for the determinacy of the moment problem. That is, if a measure μ satisfies Carleman's condition, there is no other measure ν having the same moments as μ. The condition was discovered by Torsten Carleman in 1922.[1]

Hamburger moment problem[edit]

For the Hamburger moment problem (the moment problem on the whole real line), the theorem states the following:

Let μ be a measure on R such that all the moments

m_n = \int_{-\infty}^{+\infty} x^n \, d\mu(x)~, \quad n = 0,1,2,\cdots

are finite. If

\sum_{n=1}^\infty m_{2n}^{-\frac{1}{2n}} = + \infty,

then the moment problem for mn is determinate; that is, μ is the only measure on R with (mn) as its sequence of moments.

Stieltjes moment problem[edit]

For the Stieltjes moment problem, the sufficient condition for determinacy is [clarification needed]

\sum_{n=1}^\infty m_{n}^{-\frac{1}{2n}} = + \infty. \,

Notes[edit]

References[edit]

  • Akhiezer, N. I. (1965). The Classical Moment Problem and Some Related Questions in Analysis. Oliver & Boyd.