The Hausdorff moments in statistical mechanics

Abstract

A new method for solving the Hausdorff moment problem is presented which makes use of Pollaczek polynomials. This problem is severely ill posed; a regularized solution is obtained without any use of prior knowledge. When the problem is treated in the L2 space and the moments are finite in number and affected by noise or round-off errors, the approximation converges asymptotically in the L2 norm. The method is applied to various questions of statistical mechanics and in particular to the determination of the density of states. Concerning this latter problem the method is extended to include distribution valued densities. Computing the Laplace transform of the expansion a new series representation of the partition function Z(β) (β=1/kBT) is obtained which coincides with a Watson resummation of the high-temperature series for Z(β).