Moment-theory approximations for nonnegative spectral densities

Abstract

Moment‐theory approximations constructed from finite numbers of spectral power moments are described for continuous, nonnegative spectral densities and associated Stieltjes integrals. Derivatives of the mean (Stieltjes) values of the nth‐order Tchebycheff bounds on nondecreasing distributions provide the appropriate approximations to the associated spectral densities. The nth‐order Tchebycheff density so defined is shown to be real, nonnegative, and continuous on the real axis, to have 2n−4 continuous derivatives there, and to support 2n‐2 positive‐integer power moments. Related approximations to the associated Stieltjes integral are obtained from corresponding principal‐value quadratures. The Tchebycheff densities are convergent in the limit of large numbers of spectral moments for determined moment problems, but they are not solutions of reduced moment problems of appropriate finite order. An illustrative application in the case of normal‐mode lattice vibrations in a diatomic chain indicates that the Tchebycheff densities are suitably convergent, and provide faithful images of the forbidden band gap and Van Hove singularities present.