SOME GENERALIZATIONS OF THE PROBLEM OF POSITIVE DEFINITENESS OF A PIECEWISE-LINEAR FUNCTION
In this paper we consider the following problem for positive definite functions on a real linear space (the class Φ(𝐸)). Let 𝑓, ∈ Φ(𝐸) be such that 𝑓(0) = 𝑔(0) and 𝑓 ̸≡ 𝑔. The problem is to find the set 𝑃𝑓,𝑔 := {𝜆 ∈ C : ∈ Φ(𝐸)}, where 𝑓𝜆(𝑥) := 𝜆𝑓(𝑥) + (1 - 𝜆)𝑔(𝑥), ∈ C. We also study this problem for 𝑔(𝑥) ≡ 𝑓(𝑎𝑥), where > 0, 𝑎 ̸= 1 and it is required to find the set 𝑃𝑓,𝑎 := 𝑃𝑓,𝑔. In case of 𝑓(𝑥) = (1 - |𝑥|)+ this problem is reduced to the problem of positive definiteness of piecewise-linear function.We prove that 𝑃𝑓,𝑔 is a closed interval of R. In the case = R𝑛 we obtain formulas for endpoints of this interval. We also give examples of functions ∈ Φ(R𝑛) for which the set 𝑃𝑓,𝑎 is found, including examples of radial positive definite functions.