Related papers: The Mercer-Young Theorem for Matrix-Valued Kernels…

The classical Mercer's theorem claims that a continuous positive definite kernel $K({\mathbf x}, {\mathbf y})$ on a compact set can be represented as $\sum_{i=1}^\infty \lambda_i\phi_i({\mathbf x})\phi_i({\mathbf y})$ where…

Mercer's expansion and Mercer's theorem are cornerstone results in kernel theory. While the classical Mercer's theorem only considers continuous symmetric positive definite kernels, analogous expansions are effective in practice for…

The study presents a vector-valued extension of the classical Mercer theorem within the framework of reproducing kernel Hilbert spaces defined over Kaplansky-Hilbert modules associated with the algebra of essentially bounded measurable…

We extend the classical Mercer theorem to reproducing kernel Hilbert spaces whose elements are functions from a measurable space $X$into $\mathbb C^n$. Given a finite measure $\mu$ on $X$, we represent the reproducing kernel $K$ as…

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…

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…

In this paper, a spectral theorem is proved for self-adjoint cyclically compact partial integral operators in the space of functions with mixed norm, which is a Kaplansky--Hilbert module. The decomposition through eigenfunctions, integral…

Kernel theorems, in general, provide a convenient representation of bounded linear operators. For the operator acting on a concrete function space, this means that its action on any element of the space can be expressed as a generalised…

We consider a kernel based harmonic analysis of "boundary," and boundary representations. Our setting is general: certain classes of positive definite kernels. Our theorems extend (and are motivated by) results and notions from classical…

Strictly proper kernel scores are well-known tool in probabilistic forecasting, while characteristic kernels have been extensively investigated in the machine learning literature. We first show that both notions coincide, so that insights…

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 present new classes of positive definite kernels on non-standard spaces that are integrally strictly positive definite or characteristic. In particular, we discuss radial kernels on separable Hilbert spaces, and introduce broad classes…

In the paper we investigate a method of quantization based on the concept of positive definite kernel on a principal $G$-bundle with compact structural group G. For G=U(1) our approach leads to Kostant-Souriau geometric quantization as well…

This work establishes a characterization theorem for (generalized) Young measures generated by symmetric derivatives of functions of bounded deformation (BD) in the spirit of the classical Kinderlehrer-Pedregal theorem. Our result places…

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…

This is a survey article on Mercer's Theorem in its most general form and its relations with the theory of reproducing kernel Hilbert spaces and the spectral theory of compact operators. We provide a modern introduction to the basics of the…

Based on direct integrals, a framework allowing to integrate a parametrised family of reproducing kernels with respect to some measure on the parameter space is developed. By pointwise integration, one obtains again a reproducing kernel…

Let $k$ be an uncountable algebraically closed field of positive characteristic and let $S$ be a smooth projective connected surface over $k$. We extend the theorem on the Gysin kernel from [20, Theorem 5.1] to also be true over $k$, where…

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 generalize a fundamental theorem on positive matrix semigroups stating that each component is either strictly positive for all times or identically zero ("L\'evy's Theorem"). Our proof of this fact that does not require the matrices to…