Positive definiteness on products via generalized Stieltjes and other functions

Abstract

. Let ( X , ρ ) and ( Y , σ ) be quasi-metric spaces and λ a fi xed positive real number. This paper establishes the positive de fi niteness of functions of the form on X × Y , where r (cid:2) λ , f belongs to the convex cone of all generalized Stieltjes functions of order λ , and g and h are positive valued conditionally negative de fi nite functions on ( X , ρ ) and ( Y , σ ) , respectively. As a bypass, it establishes the positive de fi niteness of functions of the form H for a generalized complete Bernstein function f of order λ , under the same assumptions on r , g and h . The paper also provides necessary and suf fi cient conditions for the strict positive de fi niteness of the two models when the spaces involved are metric. The two results yield addi- tional methods to construct positive de fi nite and strictly positive de fi nite functions on a product of metric spaces by integral transforms.