Characterizations of linear Volterra integral equations with nonnegative kernels

We first introduce the notion of positive linear Volterra integral equations. Then, we offer a criterion for positive equations in terms of the resolvent. In particular, equations with nonnegative kernels are positive. Next, we obtain a variant of the Paley-Wiener theorem for equations of this class and its extension to perturbed equations. Furthermore, we get a Perron-Frobenius type theorem for linear Volterra integral equations with nonnegative kernels. Finally, we give a criterion for positivity of the initial function semigroup of linear Volterra integral equations and provide a necessary and sufficient condition for exponential stability of the semigroups.     

On the Discrete Galerkin Method for Fredholm Integral Equations of the Second Kind

In the present paper, we refine some previous results on thediscrete Galerlcin method and the discrete iterated Galerkinmethod for Fredholm integral equations of the second kind. Byconsidering discrete inner products and discrete projectionson the same node points but with different quadrature rules,we are able to treat more appropriately kernels with discontinuousderivatives. In particular, for Green's function kernels weobtain a Nystr?m-type method which has the same order of convergenceas the corresponding Nystr?m method for infinitely smooth kernels.     

RECURSIVE REPRODUCING KERNELS HILBERT SPACES USING THE THEORY OF POWER KERNELS

The main objective of this work is to decompose orthogonally the reproducing kernels Hilbert space using any conditionally positive definite kernels into smaller ones by introducing the theory of power kernels, and to show how to do this decomposition recursively. It may be used to split large interpolation problems into smaller ones with different kernels which are related to the original kernels. To reach this objective, we will reconstruct the reproducing kernels Hilbert space for the normalized and the extended kernels and give the recursive algorithm of this decomposition.     

Recursive Kernels

This paper is an extension of earlier papers [8, 9] on the “native” Hilbert spaces of functions on some domain Ω ⊂ R ^d in which conditionally positive definite kernels are reproducing kernels. Here, the focus is on subspaces of native spaces which are induced via subsets of Ω, and we shall derive a recursive subspace structure of these, leading to recursively defined reproducing kernels. As an application, we get a recursive Neville-Aitken-type interpolation process and a recursively defined orthogonal basis for interpolation by translates of kernels.     

Markov chains on $${{\mathbb {Z}}^+}$$ Z + : analysis of stationary measure via harmonic functions approach

Queueing Systems - We suggest a method for constructing a positive harmonic function for a wide class of transition kernels on $${{\mathbb {Z}}^+}$$ . We also find natural conditions under which...     

A solution to Schrödinger's problem of non-linear integral equations

A solution of Schrödinger's system of non-linear integral equations determines the rate function of a large deviation principle for kernels with prescribed marginal distributions. This kind of large deviation principle has some meaning in quantum mechanics.Diffusion equations associated with Schrödinger equations have typically transition functions with singular creation and killing. Hence they provide measurable non-negative generally unbounded kernels which may vanish on sets with positive measure and which can possess infinite mass.For Schrödinger systems with such kernels, a solution is proved to exist uniquely in terms of a product measure. It is obtained from a variational principle for the local adjoint of a product measure endomorphism. The generally unbounded factors of the solution are characterized by integrability properties.     

Tight frame expansions of multiscale reproducing kernels in Sobolev spaces

Multiscale kernels are a new type of positive definite reproducing kernels in Hilbert spaces. They are constructed by a superposition of shifts and scales of a single refinable function and were introduced in the paper of R. Opfer [Multiscale kernels, Adv. Comput. Math. (2004), in press]. By applying standard reconstruction techniques occurring in radial basis function- or machine learning theory, multiscale kernels can be used to reconstruct multivariate functions from scattered data. The multiscale structure of the kernel allows to represent the approximant on several levels of detail or accuracy. In this paper we prove that multiscale kernels are often reproducing kernels in Sobolev spaces. We use this fact to derive error bounds. The set of functions used for the construction of the multiscale kernel will turn out to be a frame in a Sobolev space of certain smoothness. We will establish that the frame coefficients of approximants can be computed explicitly. In our case there is neither a need to compute the inverse of the frame operator nor is there a need to compute inner products in the Sobolev space. Moreover we will prove that a recursion formula between the frame coefficients of different levels holds. We present a bivariate numerical example illustrating the mutiresolution and data compression effect.     

Conditionally positive definite dot product kernels

This paper deals with conditionally positive definite kernels on Euclidean spaces. The focus here is on dot product kernels, that is, those depending on the inner product of the variables. Among the results, we include some properties relating conditional positive definiteness and standard convolution in the line and also results related to the characterization of the conditionally positive definite dot product kernels with respect to finite-dimensional polynomial spaces. We also introduce and characterize two large classes of strictly conditionally positive definite dot product kernels.     

Locally stationary stochastic processes and Weyl symbols of positive operators

The paper treats locally stationary stochastic processes. A connection with the Weyl symbols of positive operators is observed and explored. We derive necessary conditions on the two functions that constitute the covariance function of a locally stationary stochastic process, some of which use this connection to time-frequency analysis and pseudodifferential operators. Finally, we discuss briefly the subclass of Cohen’s class of time–frequency representations having separable kernels, which is related to locally stationary stochastic processes.     

Eigenvalue estimates of positive integral operators with analytic kernels

In this paper, we exhibit canonical positive definite integral kernels associated with simply connected domains. We give lower bounds for the eigenvalues of the sums of such kernels.     

Power Series Kernels

We introduce a class of analytic positive definite multivariate kernels which includes infinite dot product kernels as sometimes used in machine learning, certain new nonlinearly factorizable kernels, and a kernel which is closely related to the Gaussian. Each such kernel reproduces in a certain “native” Hilbert space of multivariate analytic functions. If functions from this space are interpolated in scattered locations by translates of the kernel, we prove spectral convergence rates of the interpolants and all derivatives. By truncation of the power series of the kernel-based interpolants, we constructively generalize the classical Bernstein theorem concerning polynomial approximation of analytic functions to the multivariate case. An application to machine learning algorithms is presented.      

Eigenvalue Distribution of Positive Definite Kernels on Unbounded Domains

We study eigenvalues of positive definite kernels of L² integral operators on unbounded real intervals. Under the assumptions of integrability and uniform continuity of the kernel on the diagonal the operator is compact and trace class. We establish sharp results which determine the eigenvalue distribution as a function of the smoothness of the kernel and its decay rate at infinity along the diagonal. The main result deals at once with all possible orders of differentiability and all possible rates of decay of the kernel. The known optimal results for eigenvalue distribution of positive definite kernels in compact intervals are particular cases. These results depend critically on a 2-parameter differential family of inequalities for the kernel which is a consequence of positivity and is a differential generalization of diagonal dominance.     

Positive periodic solutions and optimal control for a distributed biological model of two interacting species

The paper deals with the existence of positive periodic solutions to a system of degenerate parabolic equations with delayed nonlocal terms and Dirichlet boundary conditions. Taking in each equation a meaningful function as a control parameter, we show that for a suitable choice of a class of such controls we have, for each of them, a time-periodic response of the system under different assumptions on the kernels of the nonlocal terms. Finally, we consider the problem of the minimization of a cost functional on the set of pairs: control-periodic response. The considered system may be regarded as a possible model for the coexistence problem of two biological populations, which dislike crowding and live in a common territory, under different kind of intra- and inter-specific interferences.     

On differentiability and analyticity of positive definite functions

We derive a set of differential inequalities for positive definite functions based on previous results derived for positive definite kernels by purely algebraic methods. Our main results show that the global behavior of a smooth positive definite function is, to a large extent, determined solely by the sequence of even-order derivatives at the origin: if a single one of these vanishes then the function is constant; if they are all non-zero and satisfy a natural growth condition, the function is real-analytic and consequently extends holomorphically to a maximal horizontal strip of the complex plane.     

The S′-convolution with singular kernels in the Euclidean case and the product domain case

We characterize those tempered distributions which are S′-convolvable with a given class of singular convolution kernels. We study both, the Euclidean case and the product domain case. In the Euclidean case, we consider a class of kernels that includes Riesz kernels, Calderón–Zygmund singular convolution kernels, finite part distributions defined by hypersingular convolution kernels, and Hörmander multipliers. In the product domain case, we consider a class of singular kernels introduced by Fefferman and Stein as a generalization of the n-dimensional Hilbert kernel.     

第Ⅲ类非自共轭锥上的Gamma函数

本文将Ｇａｍｍａ函数及Ｓｉｅｇｅｌ积分推广到一般的第Ⅲ类非自共轭锥上,作为其应用,显式给出了以这些锥为底的管状域（也称第一类Ｓｉｅｇｅｌ域）的Ｃａｕｃｈｙ－Ｓｚｅｇｏ核和形式Ｐｏｉｓｓｏｎ核。关键词     

Tame kernels of pure cubic fields

In this paper, we study the p-rank of the tame kernels of pure cubic fields. In particular, we prove that for a fixed positive integer m, there exist infinitely many pure cubic fields whose 3-rank of the tame kernel equal to m. As an application, we determine the 3-rank of their tame kernels for some special pure cubic fields.     

On the properties of Sard kernels and multiple error estimates for bounded linear functionals of bivariate functions with application to non-product cubature

Each Sard kernels theorem supplies multiple error estimates for a bounded linear functional, provided the underlying multivariate function is sufficiently smooth. The error estimation due to Sard generally concerns the $L_1$ -norms of Sard kernels and the supremum norms of different partial derivatives of a given function. This article presents a verified method for computing Sard error constants. To assist the verified computation, we examine relevant properties of Sard kernels for general bounded linear functionals of bivariate functions. The derived $L_1$ -norms, together with interval Taylor arithmetic, make the multiple error estimates possible. We demonstrate the flexibility and superiority of Sard kernels method by different numerical examples that concern non-product cubature for bivariate functions.     

A Definiteness Theory for Cubature Formulae of Order Two

Let Ω ⊂ ℝ^d be a compact convex set of positive measure. A cubature formula will be called positive definite (or a pd-formula, for short) if it approximates the integral ∫_Ω f(x) dx of every convex function f from below. The pd-formulae yield a simple sharp error bound for twice continuously differentiable functions. In the univariate case (d = 1), they are the quadrature formulae with a positive semidefinite Peano kernel of order two. As one of the main results, we show that there is a correspondence between pd-formulae and partitions of unity on Ω. This is a key for an investigation of pd-formulae without employing the complicated multivariate analogue of Peano kernels. After introducing a preorder, we establish criteria for maximal pd-formulae. We also find a lower bound for the error constant of an optimal pd-formula. Finally, we describe a phenomenon which resembles a property of Gaussian formulae.