Abstract
Given the first n moments of an unknown function x̄ on the unit interval, a common estimate of x̄ is ψ(π n ), where π n is a polynomial of degree n taking values in a prescribed interval, ψ is a given monotone function, and π n is chosen so that the moments of ψ(π n ) equal those of x̄. This moment-matching procedure is closely related to best entropy estimation of x̄: two classical cases arise when ψ is the exponential function (corresponding to the Boltzmann-Shannon entropy) and the reciprocal function (corresponding to the Burg entropy). General conditions ensuring the existence and uniqueness of π n are given using convex programming duality techniques, and it is shown that the estimate ψ(π n ) converges uniformly to x̄ providing x̄ is sufficiently smooth.
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