Isotropic Positive Definite Functions on Spheres

Han Feng; Yan Ge

doi:10.1016/j.jfa.2021.109287

Isotropic Positive Definite Functions on Spheres

Han Feng
, Yan Ge^*

^*Corresponding author for this work

Department of Mathematics

Research output: Journal Publications and Reviews › RGC 21 - Publication in refereed journal › peer-review

3 Citations (Scopus)

Abstract

In this paper, we investigate the relationship between positive definite functions on the unit sphere S^d and on the Euclidean space R^d. For the dimension d to be odd, a new technique is developed to establish the inheritance of positive (semi-)definite property from R^dto S^dand the converse. For d = 2, it is proved that a function defined by f_θ,_δ(t) = (θ − t)^δ₊, δ ≥ d + 1/2 is positive definite on the unit sphere S²by restricting θ in an absolute range. Our results can verify a conjecture proposed by R.K. Beatson, W. zu Castell, Y. Xu and a sharp Pólya type criterion for positive definite functions on spheres.

Original language	English
Article number	109287
Journal	Journal of Functional Analysis
Volume	282
Issue number	1
Online published	19 Oct 2021
DOIs	https://doi.org/10.1016/j.jfa.2021.109287
Publication status	Published - 1 Jan 2022

Bibliographical note

Research Unit(s) information for this publication is provided by the author(s) concerned.

Research Keywords

Positive definite functions
Sphere
Positive integrals
Jacobi polynomials

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title = "Isotropic Positive Definite Functions on Spheres",

abstract = "In this paper, we investigate the relationship between positive definite functions on the unit sphere Sd and on the Euclidean space Rd. For the dimension d to be odd, a new technique is developed to establish the inheritance of positive (semi-)definite property from Rd to Sd and the converse. For d = 2, it is proved that a function defined by fθ,δ(t) = (θ − t)δ+, δ ≥ d + 1/2 is positive definite on the unit sphere S2 by restricting θ in an absolute range. Our results can verify a conjecture proposed by R.K. Beatson, W. zu Castell, Y. Xu and a sharp P{\'o}lya type criterion for positive definite functions on spheres.",

keywords = "Positive definite functions, Sphere, Positive integrals, Jacobi polynomials",

author = "Han Feng and Yan Ge",

note = "Research Unit(s) information for this publication is provided by the author(s) concerned.",

year = "2022",

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doi = "10.1016/j.jfa.2021.109287",

language = "English",

volume = "282",

journal = "Journal of Functional Analysis",

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TY - JOUR

T1 - Isotropic Positive Definite Functions on Spheres

AU - Feng, Han

AU - Ge, Yan

N1 - Research Unit(s) information for this publication is provided by the author(s) concerned.

PY - 2022/1/1

Y1 - 2022/1/1

N2 - In this paper, we investigate the relationship between positive definite functions on the unit sphere Sd and on the Euclidean space Rd. For the dimension d to be odd, a new technique is developed to establish the inheritance of positive (semi-)definite property from Rd to Sd and the converse. For d = 2, it is proved that a function defined by fθ,δ(t) = (θ − t)δ+, δ ≥ d + 1/2 is positive definite on the unit sphere S2 by restricting θ in an absolute range. Our results can verify a conjecture proposed by R.K. Beatson, W. zu Castell, Y. Xu and a sharp Pólya type criterion for positive definite functions on spheres.

AB - In this paper, we investigate the relationship between positive definite functions on the unit sphere Sd and on the Euclidean space Rd. For the dimension d to be odd, a new technique is developed to establish the inheritance of positive (semi-)definite property from Rd to Sd and the converse. For d = 2, it is proved that a function defined by fθ,δ(t) = (θ − t)δ+, δ ≥ d + 1/2 is positive definite on the unit sphere S2 by restricting θ in an absolute range. Our results can verify a conjecture proposed by R.K. Beatson, W. zu Castell, Y. Xu and a sharp Pólya type criterion for positive definite functions on spheres.

KW - Positive definite functions

KW - Sphere

KW - Positive integrals

KW - Jacobi polynomials

UR - https://www.scopus.com/pages/publications/85117572720

UR - https://www.scopus.com/pages/publications/85117572720#tab=impact

U2 - 10.1016/j.jfa.2021.109287

DO - 10.1016/j.jfa.2021.109287

M3 - RGC 21 - Publication in refereed journal

SN - 0022-1236

VL - 282

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

IS - 1

M1 - 109287

ER -

Isotropic Positive Definite Functions on Spheres

Abstract

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Research Keywords

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ECS: Approximation in weighted spaces on regular domains

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Isotropic Positive Definite Functions on Spheres

Abstract

Bibliographical note

Research Keywords

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ECS: Approximation in weighted spaces on regular domains

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