5,091 research outputs found
One-Step Recurrences for Stationary Random Fields on the Sphere
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 the (strict) positive definiteness of the zonal function
is determined by the signs of the coefficients in the
expansion of in terms of the Gegenbauer polynomials , with
. Recent results show that classical differentiation and
integration applied to 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
Thinplate Splines on the Sphere
In this paper we give explicit closed forms for the semi-reproducing kernels
associated with thinplate spline interpolation on the sphere. Polyharmonic or
thinplate splines for were introduced by Duchon and have become
a widely used tool in myriad applications. The analogues for are the thin plate splines for the sphere. The topic was first
discussed by Wahba in the early 1980's, for the case. Wahba
presented the associated semi-reproducing kernels as infinite series. These
semi-reproducing kernels play a central role in expressions for the solution of
the associated spline interpolation and smoothing problems. The main aims of
the current paper are to give a recurrence for the semi-reproducing kernels,
and also to use the recurrence to obtain explicit closed form expressions for
many of these kernels. The closed form expressions will in many cases be
significantly faster to evaluate than the series expansions. This will enhance
the practicality of using these thinplate splines for the sphere in
computations
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