Hausdorff moment problem and maximum entropy: A unified approach

Hausdor€ moment problem in the approach of maximum entropy is reconsidered. Some aspects, as existence condition, entropy convergence, entropy decreasing, stability are studied through a uni®ed approach. Results known in literature and some new ones are illustrated Ó 1999 Elsevier Science Inc. All rights reserved. Keywords: Convex hull; Entropy; Hankel determinant; Moment problem 1. Introduction The recovering of a probability density f …x† of which a ®nite number of moments are known represents a classical indetermined problem as there are, in general, an in®nite variety of functions with the ®rst M ‡ 1 moments l0; . . . ; lM . To stabilize the problem it is necessary to impose some further condition which will act as a constraint within the function space and lead to a unique solution. The maximum entropy (ME) approach [1] o€ers a de®nite procedure for the reconstruction of an approximant fM…x† under the condition that the ®rst M ‡ 1 moments be equal to the true moments lj; j ˆ 0; . . . ;M . In particular we study Hausdor€ ®nite moment problem in which the goal is to recover a probability density f …x† : ‰0; 1Š ! R‡ from the knowledge of ®nitely many moments lj, j ˆ 0; . . . ;M . Applied Mathematics and Computation 105 (1999) 291±305 www.elsevier.nl/locate/amc 1 E-mail: [email protected]. 0096-3003/99/$ ± see front matter Ó 1999 Elsevier Science Inc. All rights reserved. PII: S0 0 96 -3 0 03 (9 8 )1 00 8 4X Several interesting analytical and computational aspects of the Hausdor€ moment problem have been tackled in literature in the framework of the ME, as (1) existence of the solution [2], (2) convergence [3±5], (3) entropy decreasing, (4) stability [6]. All the quoted aspects have been tackled by resorting to spread of techniques. In this paper we reconsider some analytical and computational aspects of the problem by a uni®ed approach allowing us to obtain both results previously found and new ones concerning the stability.