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2 Journal of Applied Analysis

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Differentiable positive definite kernels on spheres

Menegatto V. A., Oliveira C. P., Peron A. P.

Journal of Applied Analysis

2009

Vol. 15, nr 1

101-117

We analyze term-by-term differentiability of uniformly convergent series of the form [wzór], where Sm-1 is the unit sphere in Rm, pk ≥ 0, k = 0,1,..., [wzór] Pk > 0 , and {Yk} is a sequence of spherical harmonics or even more general functions. Since this class of kernels includes the continuous positive definite kernels on ,Sm-1, the results in this paper will show that, under certain conditions, the action of convenient differential operators on positive definite (strictly positive definite) kernels on Sm-1 generate positive definite kernels.

Orthogonal bases for spaces of complex spherical harmonics

Menegatto V. A., Oliveira C. P.

Journal of Applied Analysis

2005

Vol. 11, nr 1

113-132

This paper proposes an inductive method to construct bases for spaces of spherical harmonics over the unit sphere Ω 2q of Cq. The bases are shown to have many interesting properties, among them orthogonality with respect to the inner product of L²(Ω 2q). As a bypass, we study the inner product [f,g] = f(D)(g(z))(0) over the space P(Cq) of polynomials in the variables [wzór], in which f(D) is the differential operator with symbol f(z). On the spaces of spherical harmonics, it is shown that the inner product [. , .] reduces to a multiple of the L²(Ω 2q) inner product. Bi-orthogonality in (F(Cq), [. , .] ) is fully investigated.