On the positive definiteness of some functions related to the Schoenberg problem

Abstract

For a broad class of functions f: [0,+∞) → ℝ, we prove that the function f(ρ^λ(x)) is positive definite on a nontrivial real linear space E if and only if 0 ≤ λ ≤ α(E, ρ). Here ρ is a nonnegative homogeneous function on E such that ρ(x) ≢ 0 and α(E, ρ) is the Schoenberg constant.