Schoenberg Matrices of Radial Positive Definite Functions and Riesz Sequences of Translates in $$L^2({\mathbb R}^n)$$ L 2 ( R n )

Abstract

Given a function f on the positive half-line R + and a sequence (finite or infinite) of points X = {x k } ω k=1 in R n , we define and study matrices S X (f ) = f (|x i − x j |) ω i,j=1 called Schoenberg's matrices. We are primarily interested in those matrices which generate bounded and invertible linear operators S X (f ) on ℓ 2 (N). We provide conditions on X and f for the latter to hold. If f is an ℓ 2 -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 R n , wherein the notion of the strong X-positive definiteness plays a key role. In particular, we prove that each radial ℓ 2 -positive definite function is strongly Xpositive 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 F X (f ) = {f (x − x j )} x j ∈X for certain radial functions f ∈ L 2 (R n ). Examples of Schoenberg's operators with various spectral properties are presented.Mathematics Subject Classification (2010). 42A82, 42B10, 33C10, 47B37

Representations of hypergeometric functions for arbitrary parameter values and their use

“…Asymptotic formulas [3, 16.11.6,16.11.7] show that the condition t d−1 f ∈ L 1 (R + ) is satisfied for dimensions d < min(a). Finally, [14,Theorem 1.7] shows that under the conditions of Theorem 8, the functions p+1 F p (a; b; −r 2 ) and p F p (a; b; −r 2 ) are strongly X positive definite for any separated set X. The definition of strongly positive definite functions is found in [14, Definition 1.5], their importance is also explained in [14] and references therein.…”

“…Hence, to compute the last integral we can drop the absolute value, substitute t = u 2 and use (42). The inequality is in fact strict for all real z = 0 as can be seen from (14) by the mean value theorem.…”

“…The class of n-RPD functions is denoted by Φ n . The above definition and many further details can be found, for instance, in the two recent papers [14,15]. The class Φ n has been characterized by Schoenberg in 1938: Classes Φ n are known to be nested: Φ n+1 ⊂ Φ n , and the inclusion is proper.…”

“…Remark. We mention further connections of hypergeometric functions with two more functional classes considered in [14]. For nonnegative, monotone decreasing functions f , normalized by f (0) = 1 the authors proved that the Schoenberg operator associated with the matrix [f ( t i − t j n )] m i,j=1 , m ≤ ∞, is bounded on l 2 under the additional condition t d−1 f ∈ L 1 (R + ), where d is the dimension of the linear span of t 1 , .…”

Radial positive definite functions and Schoenberg matrices with negative eigenvalues

“…The second one is direct and has nothing to do with Schoenberg's theorem. We just manufacture a set X = {x j } n+3 j=1 ⊂ R n+1 such that the corresponding matrix S X (Ω n ) = Ω n (|x i − x j |) n+3 i,j=1 called the Schoenberg matrix (see [5] for a detailed account of this object) has at least one negative eigenvalue (Proposition 2.2).…”