Related papers: Differentiable positive definite functions on two-…

Highly localized kernels based on orthogonal polynomials have been studied and utilized over several regular domains. Much of the results deduced via these kernels can be treated uniformly in the framework of localizable spaces of…

The seminal theorem of I.J. Schoenberg characterizes positive definite (p.d.) kernels on the unit sphere $S^{n-1}$ invariant under the automorphisms of the sphere. We obtain two generalizations of this theorem for p.d. kernels on fiber…

This survey is an introduction to positive definite kernels and the set of methods they have inspired in the machine learning literature, namely kernel methods. We first discuss some properties of positive definite kernels as well as…

We show that isotropic positive definite functions on the $d$-dimensional sphere which are $2k$ times differentiable at zero have $2k+[(d-1)/2]$ continuous derivatives on $(0,\pi)$. This result is analogous to the result for radial positive…

We tackle the problem of optimizing over all possible positive definite radial kernels on Riemannian manifolds for classification. Kernel methods on Riemannian manifolds have recently become increasingly popular in computer vision. However,…

Positive definite functions are very important in both theory and applications of approximation theory, probability and statistics. In particular, identifying strictly positive definite kernels is of great interest as interpolation problems…

Convolution is an important tool in the construction of positive definite kernels on a manifold. This contribution provides conditions on an $L^2$-positive definite and zonal kernel on the unit sphere of $\mathbb{C}^q$ in order that the…

We supply a Fourier characterization for the real, continuous, isotropic and strictly positive definite kernels on a product of circles.

In this article, we use the knowledge of positive definite tensors to develop a concept of positive definite multi-kernels to construct the kernel-based interpolants of scattered data. By the techniques of reproducing kernel Banach spaces,…

This work concerns the construction and characterization of product kernels for multivariate approximation from a finite set of discrete samples. To this end, we consider composing different component kernels, each acting on a…

We give two new global and algorithmic constructions of the reproducing kernel Hilbert space associated to a positive definite kernel. We further present ageneral positive definite kernel setting using bilinear forms, and we provide new…

We consider kernels of discrete convolution operators or, equivalently, homogeneous solutions of partial difference operators and show that these solutions always have to be exponential polynomials. The respective polynomial space in…

We present a characterization for a positive definite operator valued kernel to be universal or $C_{0}$-universal, and apply these characterizations to a family of operator valued kernels that are shown to be well behaved. Later, we obtain…

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 theory of positive kernels and associated reproducing kernel Hilbert spaces, especially in the setting of holomorphic functions, has been an important tool for the last several decades in a number of areas of complex analysis and…

We consider the class $\Psi_d$ of continuous functions $\psi \colon [0,\pi] \to \mathbb{R}$, with $\psi(0)=1$ such that the associated isotropic kernel $C(\xi,\eta)= \psi(\theta(\xi,\eta))$ ---with $\xi,\eta \in \mathbb{S}^d$ and $\theta$…

We study classes of reproducing kernels $K$ on general domains; these are kernels which arise commonly in machine learning models; models based on certain families of reproducing kernel Hilbert spaces. They are the positive definite kernels…

We propose a framework for 2D shape analysis using positive definite kernels defined on Kendall's shape manifold. Different representations of 2D shapes are known to generate different nonlinear spaces. Due to the nonlinearity of these…

In this paper, we investigate the relationship between positive definite functions on the unit sphere $\sph$ and on the Euclidean space $\RR^d$. For the dimension $d$ to be odd, a new technique is developed to establish the inheritance of…

We prove that any isotropic positive definite function on the sphere can be written as the spherical self-convolution of an isotropic real-valued function. It is known that isotropic positive definite functions on d-dimensional Euclidean…