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One-step recurrences for stationary random fields on the sphere.
Symmetry Integr. Geom. Methods Appl. 12:043 (2016)
Recurrences for positive definite functions in terms of the space dimension have been used in several fields of applications. Such recurrences typically relate to properties of the system of special functions characterizing the geometry of the underlying space. In the case of the sphere Sd−1 ⊂ ℝd the (strict) positive definiteness of the zonal function f(cos θ) is determined by the signs of the coefficients in the expansion of f in terms of the Gegenbauer polynomials {Cn λ}, with λ = (d − 2)/2. Recent results show that classical differentiation and integration applied to f have positive definiteness preserving properties in this context. However, in these results the space dimension changes in steps of two. This paper develops operators for zonal functions on the sphere which preserve (strict) positive definiteness while moving up and down in the ladder of dimensions by steps of one. These fractional operators are constructed to act appropriately on the Gegenbauer polynomials {Cn λ}.
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Publication type
Article: Journal article
Document type
Scientific Article
Keywords
Fractional Integration ; Gegenbauer Polynomials ; Positive Definite Zonal Functions ; Ultraspherical Expansions; Positive-definite Kernels; Homogeneous Spaces
e-ISSN
1815-0659
Quellenangaben
Volume: 12,
Article Number: 043
Publisher
SIGMA
Publishing Place
Kyiv 4
Non-patent literature
Publications
Reviewing status
Peer reviewed
Institute(s)
Institute of Computational Biology (ICB)