Positive definite functions on spheres

Schoenberg, I. J.

doi:10.1215/s0012-7094-42-00908-6

Cited by 544 publications

(433 citation statements)

References 6 publications

Supporting

Mentioning

423

Contrasting

Unclassified

Order By: Relevance

“…If the matrix A is positive definite then is called a strictly positive definite kernel; see [15,25]. We define the kernel from a univariate function φ :…”

Section: Finite Dimensional Subspacesmentioning

confidence: 99%

Preconditioners for pseudodifferential equations on the sphere with radial basis functions

et al. 2009

View full text Add to dashboard Cite

In a previous paper a preconditioning strategy based on overlapping domain decomposition was applied to the Galerkin approximation of elliptic partial differential equations on the sphere. In this paper the methods are extended to more general pseudodifferential equations on the sphere, using as before spherical radial basis functions for the approximation space, and again preconditioning the illconditioned linear systems of the Galerkin approximation by the additive Schwarz method. Numerical results are presented for the case of hypersingular and weakly singular integral operators on the sphere S 2 .

show abstract

“…If the matrix A is positive definite then is called a strictly positive definite kernel; see [15,25]. We define the kernel from a univariate function φ :…”

Section: Finite Dimensional Subspacesmentioning

confidence: 99%

Preconditioners for pseudodifferential equations on the sphere with radial basis functions

et al. 2009

View full text Add to dashboard Cite

show abstract

“…Schoenberg [11] (see also Bingham [4]) has shown that every positive definite function on the N-sphere has the form Theorem I for N=I and 2=2 is the case H=3 of the above Corollary. Also, it is now easy to find examples of pl .... , pn, for various H> 3, such that equality in (3.8) does not imply that the -+Ph can be made into the vertices of a regular H-gon by an appropriate choice of signs.…”

Section: P R O O F Of Theorem I For N = 1 and 2 =mentioning

confidence: 99%

An extremal characterization of regular simplices via the addition theorem for Gegenbauer polynomials

Stolarsky

1976

Geom Dedicata

View full text Add to dashboard Cite

Let p~, ..., Pn+2 be unit vectors in E n+~. Then the sum of the squares of all the (N+2, 2) inner products which they determine is minimal if and only if + signs can be chosen so that the _+p~ are the vertices of a regular simplex. The proof involves a special case of the addition theorem for Gegenbauer polynomials.* AMS(MOS) subject classifications (1970). Primary 52A25, 52A40.Geometriae Dedicata 5 (1976) 229-238. All Rights Reserved

show abstract

“…From Proposition 1.1.2, we see that the set of all positive definite kernels is closed under point-wise convergence and convex cone stable under multiplication (Schoenberg, 1942).…”

Section: If the Limitmentioning

confidence: 92%

“…The coefficients b n,d in (1.35) have interesting properties as they can used to determine whether a given function defined on S d is positive definite function (see Schoenberg (1942); Gneiting (2013a)). Now, we can define the concept of positive definite function on the sphere.…”

Section: Zonal Spherical Harmonics Functionsmentioning

confidence: 99%

“…As an application of the spectral representation of the isotropic positive definite functions on spheres provided by Schoenberg (1942), we studied the equivalence of two Gaussian measures on the sphere. Equivalence and orthogonality of probability measures are useful tools when assessing asymptotic properties of both prediction and estimation for Gaussian fields.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Mathematical Developments on Isotropic Positive Definite Functions on Spheres

Moahmmed¹

View full text Add to dashboard Cite

Abstract:In this thesis, we investigate some problems related to positive definite functions on the sphere. As known, positive definite functions play an important role in representation theory, probability theory, stochastic processes, harmonic analysis and machine learning. This role is emphasized a long the thesis.First, equivalence of Gaussian measures represents a fundamental tool to establish the properties of maximum likelihood estimators as well as kriging predictions under fixed domain asymptotics. Classical results for the equivalence and orthogonality of measures associated to Gaussian fields on bounded sets of R d are available in the literature. The present work considers Gaussian fields defined over spheres of R d+1 , with covariance functions depending on the great circle distance. We provide necessary and sufficient conditions for the equivalence of two Gaussian measures with two different covariance models with associated d-Schoenberg sequences. As an example we study equivalence of Gaussian measures for some parametric families of covariance functions valid on spheres. A simulation study explores the consistency of the maximum likelihood estimator associated to the covariance parameters of some covariance models on the sphere. We face Problems 1 and 3 proposed in the essay by Gneiting (2013b) and related to the d-Schoenberg coefficients in the series expansion of members of Ψ d for a given d. Such problems have precise implications for the simulation of Gaussian fields on spheres as well as for applications to geostatistical data, with special emphasis to atmospheric sciences. We also show how to deduce the 2-Schoenberg coefficients of given parametric families of members of the class Ψ d , called respectively exponential and Askey families, for which the 1-coefficients were available but the 2-dimensional case was still elusive.Third, we propose and define a family of marked point processes in a noncompact semisimple Lie groups. We first generate Lévy processes via marked point processes by using jump-diffusion processes. We then build a family of Markov processes in a maximal compact subgroup of a given semisimple Lie group. With all my heart, thank to you all.

show abstract

Positive definite functions on spheres

Cited by 544 publications

References 6 publications

Preconditioners for pseudodifferential equations on the sphere with radial basis functions

Preconditioners for pseudodifferential equations on the sphere with radial basis functions

An extremal characterization of regular simplices via the addition theorem for Gegenbauer polynomials

Mathematical Developments on Isotropic Positive Definite Functions on Spheres

Contact Info

Product

Resources

About