2018/07/06 by Ahmed Arafat, Arafat, Ahmed, Pablo Gregori +3
Computer Science · Mathematics · #33C45 #42A82 #62M30 #Advanced Mathematical Modeling in Engineering #FOS: Mathematics #Functional Analysis (math.FA) #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations #Statistics Theory (math.ST) #math.FA #math.ST #msc:33C45 #msc:42A82 #msc:62M30 #stat.TH
paper · pdf · doi:10.48550/arxiv.1807.02363
18 pages, 3 figures
arxiv created 2018/07/06 · openalex publication_date 2018/07/06 · arxiv updated 2018/07/09 · openalex created_date 2022/08/04 · openalex updated_date 2026/07/28
We consider the class Ψd of continuous functions ψ\colon [0,π] → ℝ, with ψ(0)=1 such that the associated isotropic kernel C(ξ,η)= ψ(θ(ξ,η)) ---with ξ,η∈ \mathbbSd and θ the geodesic distance--- is positive definite on the product of two d-dimensional spheres \mathbbSd. We face Problems 1 and 3 proposed in the essay Gneiting (2013b). We have considered an extension that encompasses the solution of Problem 1 solved in Fiedler (2013), regarding the expression of the d-Schoenberg coefficients of members of Ψd as combinations of 1-Schoenberg coefficients. We also give expressions for the computation of Schoenberg coefficients of the exponential and Askey families for all even dimensions through recurrence formula. Problem 3 regards the curvature at the origin of members of Ψd of local support. We have improved the current bounds for determining this curvature, which is of applied interest at least for d=2.