On the application of maximum entropy to the moments problem

Abstract

The maximum-entropy approach to the solution of classical inverse problem of moments, in which one seeks to reconstruct a function p(x) [where x∈(0,+∞)] from the values of a finite set N+1 of its moments, is studied. It is shown that for N≥4 such a function always exists, while for N=2 and N=3 the acceptable values of the moments are singled out analytically. The paper extends to the general case where the results were previously bounded to the case N=2.

Maximum Entropy for Reduced Moment Problems

“…We remark that there are restrictions on the solvability for any N ≥ 2 (see for example 7 ) in contrast to the result presented in articles by Tagliani and Frontini 8,19 which negates restrictions for N ≥ 4.…”

“…In particular, moment problems on bounded intervals in R have been considered. 16 In a series of papers 8,19,20 the Hamburger and Stieltjes moment problems are considered (the preprint --preprint --preprint --preprint --preprint latter differs from the Hamburger case presented above in the domain Ω = [0, ∞) and possibly in the highest moment function which need not be of even order). The authors use a technique which is different from the one presented here and which relies on special properties of the moment problems under consideration.…”

“…6 The case N = 3 has also been treated in the literature. 13,19 The resulting conditions require knowledge of the moments of…”

Transition-event durations in one-dimensional activated processes

“…Reconstructing a function approximately from a finite number of moments has been studied, e.g., by maximum entropy method [66,67,68], continued fraction approach [69,70], and Talenti method [71,72] and perhaps other techniques. Here we follow Hon and Wei's work [72] to reconstruct the density ρ b (t) in a similar way.…”

Lyapunov exponents and the natural invariant density determination of chaotic maps: an iterative maximum entropy ansatz

“…The classical moment problem (CMP) is an archetypal example of an inverse problem that involves reconstruction of a non-negative density distribution from a knowledge of (usually finite) moments [1,2,3,4,5,6]. The CMP is an important inverse problem that has attracted researchers from many diverse fields of science and engineering ranging from geological prospecting, computer tomography, medical imaging to transport in complex inhomogeneous media [7].…”