Моменттер мәселесі: математикалық талдау және қолданыстары

Trying to map moments to a measure that generates them

In mathematics, a moment problem arises as the result of trying to invert the mapping that takes a measure to the sequence of moments

More generally, one may consider

for an arbitrary sequence of functions .

In the classical setting, is a measure on the real line, and is the sequence In this form the question appears in probability theory, asking whether there is a probability measure having specified mean, variance and so on, and whether it is unique. There are three named classical moment problems: the Hamburger moment problem in which the support of is allowed to be the whole real line; the Stieltjes moment problem, for ; and the Hausdorff moment problem for a bounded interval, which without loss of generality may be taken as
The moment problem also extends to complex analysis as the trigonometric moment problem in which the Hankel matrices are replaced by Toeplitz matrices and the support of μ is the complex unit circle instead of the real line.

The uniqueness of in the Hausdorff moment problem follows from the Weierstrass approximation theorem, which states that polynomials are dense under the uniform norm in the space of continuous functions on For the problem on an infinite interval, uniqueness is a more delicate question. There are distributions, such as log normal distributions, which have finite moments for all the positive integers but where other distributions have the same moments.

An important variation is the truncated moment problem, which studies the properties of measures with fixed first k moments (for a finite k). Results on the truncated moment problem have numerous applications to extremal problems, optimisation and limit theorems in probability theory.