C. P. Oliveira

Strictly positive definite kernels on subsets of the complex plane

Let (z, w) @? @? x @? (zw) be a positive definite kernel and B a subset of @?. In this paper, we seek conditions in order that the restriction (z, w) @? B x B(zw) be strictly positive definite. Since this problem has been solved recently in the cases in which B is either @? or the unit circle in @?, our purpose here is twofold: to present some results we obtained when attempting to solve the problem for the above and other choices of B and to acquaint the audience with some other questions that remain. For two different classes of subsets, we completely characterize the strict positive definiteness of the kernel. We include a complete discussion of the case in which B is the unit circle of @?, making a comparison with the classical problem of strict positive definiteness on the real circle.

Differentiable positive definite kernels on spheres

Abstract We analyze term-by-term differentiability of uniformly convergent series of the form , where Sm –1 is the unit sphere in , 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

Abstract This paper proposes an inductive method to construct bases for spaces of spherical harmonics over the unit sphere Ω2 q of . The bases are shown to have many interesting properties, among them orthogonality with respect to the inner product of L 2(Ω2 q ). As a bypass, we study the inner product over the space of polynomials in the variables z, , in which ƒ(D̅) is the differential operator with symbol ƒ(z̄). On the spaces of spherical harmonics, it is shown that the inner product [·, ·] reduces to a multiple of the L 2(Ω2 q ) inner product. Bi-orthogonality in (, [·, ·]) is fully investigated.

Exact point-distributions over the complex sphere

We study point-distributions over the surface of the unit sphere in unitary space that generate quadrature rules which are exact for spherical polynomials up to a certain bi-degree. In this first stage, we explore several different characterizations for this type of point sets using standard tools such as, positive definiteness, reproducing kernel techniques, linearization formulas, etc. We find bounds on the cardinality of a point-distribution, without discussing the deeper question regarding best bounds. We include examples, construction methods and explain, via isometric embeddings from real to complex spheres, the proper connections with the so-called spherical designs.

Operators associated with conditionally positive definite kernels

Abstract Conditionally positive definite kernels are frequently used in multi-dimensional data fitting. Starting witha single conditionally positive definite kernel on a domain Ω and a set of interpolation points in Ω, the interpolant consists of a linear combination of translates by the interpolation points of the kernel plus a low degree polynomial. A major problem is to identify which conditionally positive definite kernels can be used to carry out the interpolation. In this paper, we study linear and nonlinear operators that leave the class of conditionally positive definite kernels invariant. The operators are useful in the description of the class of strictly conditionally positive definite kernels with respect to a polynomial space P on Ω, when a description of the class of strictly conditionally positive definite kernels with respect to a subspace of P is available.

Mixed integral identities involving unit spheres and balls in complex context

In this paper, we derive integral identities relating both unit spheres and unit balls of several dimensions in the complex setting. More specifically, we find a chain of equations involving either balls or balls and spheres of different dimensions. In addition, as a result almost independent we prove a prototype of the Funk–Hecke formula for embedded subspheres within the unit sphere of ℂq, allowing for a closed-form expression for the computation of the eigenvalues.