2015/02/25 by Leonid Golinskiĭ, M. M. Malamud, Golinskii, L. +3
Computer Science · Mathematics · #42A82 #42B10 #47B37 #Advanced Mathematical Modeling in Engineering #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Analysis (math.FA) #Matrix Theory and Algorithms #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.1502.07179
openalex publication_date 2015/02/25 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
The main object under consideration is a class Φn\backslashΦn+1 of radial positive definite functions on \Rn which do not admit radial positive definite continuation on \Rn+1. We find certain necessary and sufficient conditions for the Schoenberg representation measure νn of f∈ Φn in order that the inclusion f∈ Φn+k, k∈\N, holds. We show that the class Φn\backslashΦn+k is rich enough by giving a number of examples. In particular, we give a direct proof of Ωn∈Φn\backslashΦn+1, which avoids Schoenberg's theorem, Ωn is the Schoenberg kernel. We show that Ωn(a⋅)Ωn(b⋅)∈Φn\backslashΦn+1, for a\not=b. Moreover, for the square of this function we prove surprisingly much stronger result: Ωn2(a⋅)∈Φ2n-1\backslashΦ2n. We also show that any f∈Φn\backslashΦn+1, n≥2, has infinitely many negative squares. The latter means that for an arbitrary positive integer N there is a finite Schoenberg matrix \kSX(f) := ‖f(|xi-xj|n+1)‖i,j=1m, X := \xj\j=1m ⊂\Rn+1, which has at least N negative eigenvalues.