Positive definite temperature functions and a correspondence to positive temperature functions

Abstract

SynopsisPositive definite temperature functionsu(x, t) in ℝn+1= {(x, t)|x∈ ℝn,t> 0} are characterised bywhere μ is a positive measure satisfying that for every ℰ > 0,is finite. A transformis introduced to give an isomorphism between the class ofall positive definite temperature functions and the class of all possible temperature functions inThen correspondence given bygeneralises the Bochner–Schwartz Theorem for the Schwartz distributions and extends Widder's correspondence characterising some subclass of the positive temperature functions by the Fourier-Stieltjes transform.