Conditionally Negative Definite Functions

A Gneiting-Like Method for Constructing Positive Definite Functions on Metric Spaces

“…Let X be the unit sphere S d in R d+1 endowed with is usual geodesic distance ρ d and let Y = [0, π/2] and Z = R n both endowed with their usual Euclidean distances σ and τ respectively. The function g given by the formula g(t) = 3 − cos t, t ∈ [0, π], belongs to CND S d , ρ d while results proved in [17] show that, if s ∈…”

“…In fact, we can let (Z, τ ) be a quasi-metric space which is isometrically embedded in an infinite-dimensional Hilbert space. [20] and the many concrete examples of functions in CND(R, ρ) listed in [17].…”

A Gneiting-Like Method for Constructing Positive Definite Functions on Metric Spaces

“…belongs to CND(S d , ρ d ) while results proved in [17] points that, if s ∈ (0, 2], then the function h given by h(u, v) = 1 + sin u + v s , (u, v) ∈ [0, π/2] × [0, ∞), belongs to CND(Y × Z, σ, τ ). It is also easily seen that g(t) > g(0) for all t ∈ (0, π] and h(u, v) > h(0, 0) for (u, v) ∈ [0, π/2] × [0, ∞) with u + v > 0.…”

“…defines a function G r in SP D(X, Y, Z, ρ, σ d , τ d ′ ), as long as f comes from S λ . The interested reader can implement quite more complicated examples along the same lines by using the characterization of functions in CND(S d , σ d ) obtained in [20] and the many concrete examples of functions in CND(R, ρ) listed in [17].…”

Matrix valued positive definite kernels related to the generalized Aitken's integral for Gaussians

“…These two classes of kernels will be denoted by CN D q (X) and SCN D q (X), respectively. Examples of kernels in CN D 1 (X) and SCN D 1 (X) can be found in [4] while connections between the classes P D 1 (X) and CN D 1 (X) are described in [3,4,10].…”