Radial positive definite functions and Schoenberg matrices with negative eigenvalues

Abstract

The main object under consideration is a class Φ n ∖ Φ n + 1 \Phi _n\backslash \Phi _{n+1} of radial positive definite functions on R n \mathbb {R}^n which do not admit radial positive definite continuation on R n + 1 \mathbb {R}^{n+1} . We find certain necessary and sufficient conditions on the Schoenberg representation measure ν n u _n of f ∈ Φ n f\in \Phi _n for f ∈ Φ n + k f\in \Phi _{n+k} , k ∈ N k\in \mathbb {N} . We show that the class Φ n ∖ Φ n + k \Phi _n\backslash \Phi _{n+k} is rich enough by giving a number of examples. In particular, we give a direct proof of Ω n ∈ Φ n ∖ Φ n + 1 \Omega _n\in \Phi _n\backslash \Phi _{n+1} , which avoids Schoenberg’s theorem; Ω n \Omega _n is the Schoenberg kernel. We show that Ω n ( a ⋅ ) Ω n ( b ⋅ ) ∈ Φ n ∖ Φ n + 1 \Omega _n(a\cdot )\Omega _n(b\cdot )\in \Phi _n\backslash \Phi _{n+1} for a ≠ b aot =b . Moreover, for the square of this function we prove the surprisingly much stronger result Ω n 2 ( a ⋅ ) ∈ Φ 2 n − 1 ∖ Φ 2 n \Omega _n^2(a\cdot )\in \Phi _{2n-1}\backslash \Phi _{2n} . We also show that any f ∈ Φ n ∖ Φ n + 1 f\in \Phi _n\backslash \Phi _{n+1} , n ≥ 2 n\ge 2 , has infinitely many negative squares. The latter means that for an arbitrary positive integer N N there is a finite Schoenberg matrix S X ( f ) := ‖ f ( | x i − x j | n + 1 ) ‖ i , j = 1 m \mathcal {S}_X(f) := \|f(|x_i-x_j|_{n+1})\|_{i,j=1}^{m} , X := { x j } j = 1 m ⊂ R n + 1 X := \{x_j\}_{j=1}^m \subset \mathbb {R}^{n+1} , which has at least N N negative eigenvalues.