On a certain class of positive definite functions and measures on locally compact Abelian groups and inner-product spaces

Abstract

Let L be a lattice in Rn and φ the L-periodic Gaussian function on Rn given by φ(x)=∑y∈Le−‖x−y‖2. The paper was motivated by the following observation: φ(x+y)φ(x−y) is a positive definite function of two variables x,y. We say that a positive definite function φ on an Abelian group G is cross positive definite (c.p.d.) if, for each z∈G, the function φ(x+y+z)φ(x−y) of two variables x,y is positive definite. We say that a finite Radon measure on a locally compact Abelian group is c.p.d. if its Fourier transform is a c.p.d. function on the dual group. We investigate properties of c.p.d. functions and measures and give some integral characterizations. It is proved that the classes of c.p.d. functions and measures are closed with respect to certain natural operations. In particular, products, convolutions and Fourier transforms of c.p.d. functions and measures are c.p.d., whenever defined.