Radial kernels and their reproducing kernel Hilbert spaces

We describe how to use Schoenberg’s theorem for a radial kernel combined with existing bounds on the approximation error functions for Gaussian kernels to obtain a bound on the approximation error function for the radial kernel. The result is applied to the exponential kernel and Student’s kernel. To establish these results we develop a general theory regarding mixtures of kernels. We analyze the reproducing kernel Hilbert space (RKHS) of the mixture in terms of the RKHS’s of the mixture components and prove a type of Jensen inequality between the approximation error function for the mixture and the approximation error functions of the mixture components.     

Local properties of Hilbert spaces of Dirichlet series

We show that the asymptotic behavior of the partial sums of a sequence of positive numbers determine the local behavior of the Hilbert space of Dirichlet series defined using these as weights. This extends results recently obtained describing the local behavior of Dirichlet series with square summable coefficients in terms of local integrability, boundary behavior, Carleson measures and interpolating sequences. As these spaces can be identified with functions spaces on the infinite-dimensional polydisk, this gives new results on the Dirichlet and Bergman spaces on the infinite-dimensional polydisk, as well as the scale of Besov-Sobolev spaces containing the Drury-Arveson space on the infinite-dimensional unit ball. We use both techniques from the theory of sampling in Paley-Wiener spaces, and classical results from analytic number theory.     

Approximate solution of the Fredholm integral equation of the first kind in a reproducing kernel Hilbert space

An approach for solving Fredholm integral equations of the first kind is proposed for in a reproducing kernel Hilbert space (RKHS). The interest in this problem is strongly motivated by applications to actual prospecting. In many applications one is puzzled by an ill-posed problem in space C[a,b] or L²[a,b], namely, measurements of the experimental data can result in unbounded errors of solutions of the equation. In this work, the representation of solutions for Fredholm integral equations of the first kind is obtained if there are solutions and the stability of solutions is discussed in RKHS. At the same time, a conclusion is obtained that approximate solutions are also stable with respect to _∞ or _L² in RKHS. A numerical experiment shows that the method given in the work is valid.     

Adjoints of composition operators on Hilbert spaces of analytic functions

We observe that a formula for the adjoint of a composition operator, known only for special symbols in some spaces of analytic functions, actually holds for every admissible symbol and in any Hilbert space of analytic functions with reproducing kernels. Along with some new results, all known formulas for the adjoint obtained so far follow easily as a consequence, some in an improved form.     

Berezin symbol and invertibility of operators on the functional Hilbert spaces

We give in terms of reproducing kernel and Berezin symbol the sufficient conditions ensuring the invertibility of some linear bounded operators on some functional Hilbert spaces.     

Continuum versus discrete networks,graph Laplacians,and reproducing kernel Hilbert spaces

Motivated by applications to machine learning, we construct a reversible and irreducible Markov chain whose state space is a certain collection of measurable sets of a chosen l.c.h. space $\mathfrak{X}$. We study the resulting network (connected undirected graph), including transience, Royden and Riesz decompositions, and kernel factorization. We describe a construction for Hilbert spaces of signed measures which comes equipped with a new notion of reproducing kernels and there is a unique solution to a regularized optimization problem involving the approximation of $L^2$ functions by functions of finite energy. The latter has applications to machine learning (for Markov random fields, for example).     

Norm estimates for functions of two operators on tensor products of Hilbert spaces

Analytic operator valued functions of two operators on tensor products of Hilbert spaces are considered. A precise norm estimate is established. Applications to operator differential equations are also discussed. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Weak consistency of the Support Vector Machine Quantile Regression approach when covariates are functions

This paper deals with a nonparametric estimation of conditional quantile regression when the explanatory variable X takes its values in a bounded subspace of a functional space X and the response Y takes its values in a compact of the space Y?R. The functional observations, X₁,…,X_n, are projected onto a finite dimensional subspace having a suitable orthonormal system. The X_i’s will be characterized by their coordinates in this basis. We perform the Support Vector Machine Quantile Regression approach in finite dimension with the selected coefficients. Then we establish weak consistency of this estimator. The various parameters needed for the construction of this estimator are automatically selected by data-splitting and by penalized empirical risk minimization.     

On lifting of operators to Hilbert spaces induced by positive selfadjoint operators

We introduce the notion of induced Hilbert spaces for positive unbounded operators and show that the energy spaces associated to several classical boundary value problems for partial differential operators are relevant examples of this type. The main result is a generalization of the Krein-Reid lifting theorem to this unbounded case and we indicate how it provides estimates of the spectra of operators with respect to energy spaces.     

The density condition and distinguishedness for weighted Fréchet spaces of holomorphic functions

We consider weighted Fréchet spaces of holomorphic functions which are defined as countable intersections of weighted Banach spaces of type H^∞. We study when these spaces have Stefan Heinrich's density condition and when they are distinguished.     

Approximation by polynomials and rational functions in weighted rearrangement invariant spaces

Let Γ be a Dini-smooth curve in the complex plane, and let G:=IntΓ. We prove some direct and inverse theorems of approximation theory by algebraic polynomials and rational functions in the weighted rearrangement invariant Smirnov spaces EX(G,ω) defined on G.     

On the Riemann‐Hilbert boundary value problem for generalized analytic functions in the framework of variable exponent spaces

Let Γ be a simple closed curve that bounds the finite domain D , z =z (ζ )=z (r e ^{i ?} ) be the conformal mapping of the circle {ζ :|ζ |<1} onto the domain D . Furthermore, let the functions A (z ), B (z ) be given on D and U ^{s ,2}(A ;B ;D ) be the set of regular solutions of the equation . We call the Smirnov class E ^{p (t )}(A ;B ;D ) the set of those generalized functions W in D for which where p (t ) is a positive measurable function on Γ. We consider the Riemann‐Hilbert problem: Define a function W (z ) from the class E ^{p (t )}(A ;B ;D ) for which the equality, is fulfilled almost everywhere on Γ. It is assumed that Γ is a piecewise‐smooth curve without external peaks; , p is Log Hölder continuous and the function belongs to the class A (p (t );Γ), which is the generalization of the well‐known Simonenko class A (p ;Γ), where p is constant. The solvability conditions are established, and solutions are constructed.     

Representation of certain infinitely divisible probability measures on Banach spaces

Subclasses L₀ ? L₁ ? … ? L_∞ of the class L₀ of self-decomposable probability measures on a Banach space are defined by means of certain stability conditions. Each of these classes is closed under translation, convolution and passage to weak limits. These subclasses are analogous to those defined earlier by K. Urbanik on the real line and studied in that context by him and by the authors. A representation is given for the characteristic functionals of the measures in each of these classes on conjugate Banach spaces. On a Hilbert space it is shown that L_∞ is the smallest subclass of L₀ with the closure properties above containing all the stable measures.