Carleman's condition

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

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

are finite. If

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

For the Stieltjes moment problem, the sufficient condition for determinacy is

Notes

gollark: That was fictional! It was also really stupid!
gollark: Also, that would effectively just turn over control to whoever writes the objective function/manages the computing stuff involved.
gollark: Doing all governance tasks basically requires AGI. We do not *have* AGI, and if we get it there will be bigger problems.
gollark: And, as someone who knows more about machine learning/AI than you (41025 kilooffense), we cannot actually just sidestep the issue by turning over governance to AI.
gollark: This global government would obviously be quite powerful. People would want it to do their preferred thing.

References

  • Akhiezer, N. I. (1965). The Classical Moment Problem and Some Related Questions in Analysis. Oliver & Boyd.CS1 maint: ref=harv (link)
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