2014/03/10 by Leonid Golinskiĭ, M. M. Malamud, Golinskii, L. +3
Computer Science · Mathematics · #33C10 #42A82 #42B10 #47B37 #Advanced Mathematical Modeling in Engineering #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical Analysis and Transform Methods #Numerical methods in inverse problems #Spectral Theory (math.SP)
paper · pdf · doi:10.48550/arxiv.1403.2234
openalex publication_date 2014/03/10 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
Given a function f on the positive half-line \R+ and a sequence (finite or infinite) of points X=\xk\k=1ω in \Rn, we define and study matrices \kSX(f)=‖f(|xi-xj|)‖i,j=1ω called Schoenberg's matrices. We are primarily interested in those matrices which generate bounded and invertible linear operators SX(f) on ℓ2(\N). We provide conditions on X and f for the latter to hold. If f is an ℓ2-positive definite function, such conditions are given in terms of the Schoenberg measure σ(f). We also approach Schoenberg's matrices from the viewpoint of harmonic analysis on \Rn, wherein the notion of the strong X-positive definiteness plays a key role. In particular, we prove that each radial ℓ2-positive definite function is strongly X-positive definite whenever X is separated. We also implement a "grammization" procedure for certain positive definite Schoenberg's matrices. This leads to Riesz--Fischer and Riesz sequences (Riesz bases in their linear span) of the form \kFX(f)=\f(x-xj)\xj∈ X for certain radial functions f∈ L2(\Rn). Examples of Schoenberg's operators with various spectral properties are presented.