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Beatson, R.K.* ; zu Castell, W.

One-step recurrences for stationary random fields on the sphere.

Symmetry Integr. Geom. Methods Appl. 12:043 (2016)
Verlagsversion DOI
Open Access Gold
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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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Publikationstyp Artikel: Journalartikel
Dokumenttyp Wissenschaftlicher Artikel
Schlagwörter Fractional Integration ; Gegenbauer Polynomials ; Positive Definite Zonal Functions ; Ultraspherical Expansions; Positive-definite Kernels; Homogeneous Spaces
e-ISSN 1815-0659
Quellenangaben Band: 12, Heft: , Seiten: , Artikelnummer: 043 Supplement: ,
Verlag SIGMA
Verlagsort Kyiv 4