Some generalizations of the problem of positive definiteness of a piecewise linear function

Abstract

In this paper, we consider the following problem concerning functions that are positive definite on a real linear space E (the set Φ(E)).Letf,g∈Φ(E)be such thatf(0)=g(0)andf≢g. We are looking for the setPf,g:={λ∈C:fλ∈Φ(E)}, wherefλ(x):=λf(x)+(1−λ)g(x),λ∈C. A particular case of this problem is g(x)≡f(ax), with a>0, a≠1, and we want to find the set Pf,a:=Pf,g. When f(x)=(1−|x|)+ we come up with the positive definiteness of a piecewise liner function. It is well-known that the set Pf,g is always an interval of the real axis. In case E=Rn, the expressions for endpoints of that interval were found. We also provide examples of functions f∈Φ(Rn), for which the set Pf,a was found, including those functions that depend on Euclidean norm (radial positive definite functions).