Strict positive definiteness on spheres via disk polynomials

Strict Positive Definiteness on Spheres via Disk Polynomials

Positive Definiteness of Multivariate Densities Based on Hermite Polynomials

This paper develops both univariate and multivariate distributions based on Gram-Charlier and Edgeworth expansions, attempting to ensure non negativity by exploiting the orthogonal properties of the Hermite polynomials. The article motivates the problems underlying some specifications (in particular those involving other conditional moments beyond the variance) and provides empirical examples c...

Extending Positive Definiteness

The main result of this paper gives criteria for extendibility of mappings defined on symmetric subsets of ∗-semigroups to positive definite ones. By specifying the mappings in question we obtain new solutions of relevant issues in harmonic analysis concerning truncations of some important multivariate moment problems, like complex, two-sided complex and multidimensional trigonometric moment pr...

Positive definiteness of tridiagonal matrices via the numerical range

Differential Criteria for Positive Definiteness

We show how the Mellin transform can be used to derive differential criteria for integral representations of functions as scale mixtures, and apply this idea to derive differential criteria for positive definiteness of functions in a manner similar to that of Gneiting (1999,2001). We also give a simple derivation of Williamson’s result on multiply monotone functions (Williamson, 1956), with a l...