Related papers: Generalizations of Schoenberg's theorem on positiv…

We present a characterization for the continuous, isotropic and positive definite kernels on a product of spheres along the lines of a classical result of I. J. Schoenberg on positive definiteness on a single sphere. We also discuss a few…

We provide a characterization for the continuous positive definite kernels on $\mathbb R^d$ that are invariant to linear isometries, i.e. invariant under the orthogonal group $O(d)$. Furthermore, we provide necessary and sufficient…

The paper introduces new sufficient conditions of strict positive definiteness for kernels on d-dimensional spheres which are not radially symmetric but possess specific coefficient structures. The results use the series expansion of the…

We study positive definiteness of kernels $K(x,y)$ on two-point homogeneous spaces. As opposed to the classical case, which has been developed and studied in the existing literature, we allow the kernel to have an (integrable) singularity…

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 $\theta$…

In 1942 I. J. Schoenberg proved that a function is positive definite in the unit sphere if and only if this function is a positive linear combination of the Gegenbauer polynomials. In this paper we extend Schoenberg's theorem for…

Schoenberg's theorem for the complex Hilbert sphere proved by Christensen and Ressel in 1982 by Choquet theory is extended to the following result: Let L denote a locally compact group and let \overline{\D} denote the closed unit disc in…

The paper introduces a new characterisation of strictly positive definiteness for kernels on the 2-sphere without assuming the kernel to be radially (isotropic) or axially symmetric. The results use the series expansion of the kernel in…

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…

The present work develops certain analytical tools required to construct and compute invariant kernels on the space of complex covariance matrices. The main result is the $\mathrm{L}^1$--Godement theorem, which states that any invariant…

We present a proof of positivity of an invariant kernel, which is of basic importance for the Staruszkiewicz theory of the quantum Coulomb field. Presented proof of positivity is independent of the Staruszkiewicz theory and is based on the…

In this paper we study reproducing kernel Hilbert spaces of arbitrary smoothness on the sphere $\mathbb{S}^d \subset \mathbb{R}^{d+1}$. The reproducing kernel is given by an integral representation using the truncated power function…

This work aims to prove that the classical Gaussian kernel, when defined on a non-Euclidean symmetric space, is never positive-definite for any choice of parameter. To achieve this goal, the paper develops new geometric and analytical…

In this paper we study continuous kernels on compact two point homogeneous spaces which are positive definite and zonal (isotropic). Such kernels were characterized by R. Gangolli some forty years ago and are very useful for solving…

We present a necessary and sufficient condition for the strict positive definiteness of a real, continuous, isotropic and positive definite kernel on a two-point compact homogeneous space. The characterization adds to others previously…

Let $E$ be a smooth bundle with fiber an $n$-dimensional real projective space $\mathbb{R}P^n$. We show that, if every fiber carries a positively curved pointwise strongly $1/4$-pinched Riemannian metric that varies continuously with…

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 present, among other things, a necessary and sufficient condition for the strict positive definiteness of an isotropic and positive definite kernel on the cartesian product of a circle and a higher dimensional sphere. The result…

In this paper we obtain a description of the Grothendieck group of complex vector bundles over the classifying space of a p-local finite group in terms of representation rings of subgroups of its Sylow. We also prove a stable elements…

For the real, continuous, isotropic and positive definite kernels on a product of spheres, one may consider not only its usual strict positive definiteness but also strict positive definiteness restrict to the points of the product that…