Kernel Hilbert Space Of Functions Research Articles

Hilbert–Schmidt regularity of symmetric integral operators on bounded domains with applications to SPDE approximations

Regularity estimates for an integral operator with a symmetric continuous kernel on a convex bounded domain are derived. The covariance of a mean-square continuous random field on the domain is an example of such an operator. The estimates are of the form of Hilbert–Schmidt norms of the integral operator and its square root, composed with fractional powers of an elliptic operator equipped with homogeneous boundary conditions of either Dirichlet or Neumann type. These types of estimates, which couple the regularity of the driving noise with the properties of the differential operator, have important implications for stochastic partial differential equations on bounded domains as well as their numerical approximations. The main tools used to derive the estimates are properties of reproducing kernel Hilbert spaces of functions on bounded domains along with Hilbert–Schmidt embeddings of Sobolev spaces. Both non-homogeneous and homogeneous kernels are considered. In the latter case, results in a general Schatten class norm are also provided. Important examples of homogeneous kernels covered by the results of the paper include the class of Matérn kernels.

Grassmann geometry of zero sets in reproducing kernel Hilbert spaces

Let H be a reproducing kernel Hilbert space of functions on a set X. We study the problem of finding a minimal geodesic of the Grassmann manifold of H that joins two subspaces consisting of functions which vanish on given finite subsets of X. We establish a necessary and sufficient condition for existence and uniqueness of geodesics, and we then analyze it in examples. We discuss the relation of the geodesic distance with other known metrics when the mentioned finite subsets are singletons. We find estimates on the upper and lower eigenvalues of the unique self-adjoint operators which define the minimal geodesics, which can be made more precise when the underlying space is the Hardy space. Also for the Hardy space we discuss the existence of geodesics joining subspaces of functions vanishing on infinite subsets of the disk, and we investigate when the product of projections onto this type of subspaces is compact.

Complex hyperbolic geometry and Hilbert spaces with complete Pick kernels

Suppose H is a finite dimensional reproducing kernel Hilbert space of functions on X. If H has the complete Pick property then there is an isometric map, Φ, from X, with the metric induced by H, into complex hyperbolic space, CHn, with its pseudohyperbolic metric. We investigate the relationships between the geometry of Φ(X) and the function theory of H and its multiplier algebra.

Reproducing Kernel Hilbert Space vs. Frame Estimates

We consider conditions on a given system F of vectors in Hilbert space H, forming a frame, which turn H into a reproducing kernel Hilbert space. It is assumed that the vectors in F are functions on some set Ω . We then identify conditions on these functions which automatically give H the structure of a reproducing kernel Hilbert space of functions on Ω. We further give an explicit formula for the kernel, and for the corresponding isometric isomorphism. Applications are given to Hilbert spaces associated to families of Gaussian processes.

Reproducing kernel Hilbert spaces generated by the binomial coefficients

We study a reproducing kernel Hilbert space of functions defined on the positive integers and associated to the binomial coefficients. We introduce two transforms, which allow us to develop a related harmonic analysis in this Hilbert space. Finally, we mention connections with the theory of discrete analytic functions, statistics, and with the quantum case.

Imbeddings and bounds for functions with [formula omitted]-Laplacians

Spaces of locally integrable functions on R n that vanish at ∞ and whose gradient and Laplacian are in L p ( R n ; R n ) , L q ( R n ) respectively are defined. A representation theorem for such functions is described and properties of the fundamental solution of the modified Laplacian operator are used to prove L r and supremum norm inequalities when n = 3 . Imbedding results for these spaces into L r ( R 3 ) and C 0 ( R 3 ) when 1 ⩽ p < 3 are described. The case p = q = 2 yields a reproducing kernel Hilbert space of functions on R 3 . Some different estimates for solutions of the finite mass and energy solutions of Poissonʼs equation on R 3 are found using these results.

Randomly shifted lattice rules with the optimal rate of convergence for unbounded integrands

We study the multivariate integration problem ∫ R d f ( x ) ρ ( x ) d x , with ρ being a product of univariate probability density functions. We assume that f belongs to a weighted tensor-product reproducing kernel Hilbert space of functions whose mixed first derivatives, when multiplied by a weight function ψ , have bounded L 2 -norms. After mapping into the unit cube [ 0 , 1 ] d , the transformed integrands are typically unbounded or have huge derivatives near the boundary, and thus fail to lie in the usual function space setting where many good results have been established. In our previous work, we have shown that randomly shifted lattice rules can be constructed component-by-component to achieve a worst case error bound of order O ( n − 1 / 2 ) in this new function space setting. Using a more clever proof technique together with more restrictive assumptions, in this article we improve the results by proving that a rate of convergence close to the optimal order O ( n − 1 ) can be achieved with an appropriate choice of parameters for the function space. The implied constants in the big- O bounds can be independent of d under appropriate conditions on the weights of the function space.

Representation of Simple Symmetric Operators with Deficiency Indices (1, 1) in de Branges Space

Recently it has been shown that any regular simple symmetric operator with deficiency indices (1, 1) is unitarily equivalent to the operator of multiplication in a reproducing kernel Hilbert space of functions on the real line with the Kramer sampling property. This work has been motivated, in part, by potential applications to signal processing and mathematical physics. In this paper we exploit well-known results about de Branges–Rovnyak spaces and characteristic functions of symmetric operators to prove that any such a symmetric operator is in fact unitarily equivalent to multiplication by the independent variable in a de Branges space of entire functions. This leads to simple new results on the spectra of such symmetric operators, on when multiplication by z is densely defined in de Branges–Rovnyak spaces in the upper half plane, and to sufficient conditions for there to be an isometry from a given subspace of \({L^2 (\mathbb{R}, d\nu)}\) onto a de Branges space of entire functions which acts as multiplication by a measurable function.

Symmetric Operators and Reproducing Kernel Hilbert Spaces

We establish the following sufficient operator-theoretic condition for a subspace $${S \subset L^2 (\mathbb{R}, d\nu)}$$ to be a reproducing kernel Hilbert space with the Kramer sampling property. If the compression of the unitary group U(t) := e itM generated by the self-adjoint operator M, of multiplication by the independent variable, to S is a semigroup for t ≥ 0, if M has a densely defined, symmetric, simple and regular restriction to S, with deficiency indices (1, 1), and if ν belongs to a suitable large class of Borel measures, then S must be a reproducing kernel Hilbert space with the Kramer sampling property. Furthermore, there is an isometry which acts as multiplication by a measurable function which takes S onto a reproducing kernel Hilbert space of functions which are analytic in a region containing $${\mathbb{R}}$$ , and are meromorphic in $${\mathbb{C}}$$ . In the process of establishing this result, several new results on the spectra and spectral representations of symmetric operators are proven. It is further observed that there is a large class of de Branges functions E, for which the de Branges spaces $${\mathcal{H}(E) \subset L^{2}(\mathbb{R}, |E(x)|^{-2}dx)}$$ are examples of subspaces satisfying the conditions of this result.

On the power of standard information for multivariate approximation in the worst case setting

We study multivariate approximation with the error measured in L ∞ and weighted L 2 norms. We consider the worst case setting for a general reproducing kernel Hilbert space of functions of d variables with a bounded or integrable kernel. Here d can be arbitrarily large. We analyze algorithms that use standard information consisting of n function values, and we are especially interested in the optimal order of convergence, i.e., in the maximal exponent b for which the worst case error of such an algorithm is of order n − b . We prove that b ∈ [ 2 p 2 / ( 2 p + 1 ) , p ] for weighted L 2 approximation and b ∈ [ 2 p ( p − 1 / 2 ) / ( 2 p + 1 ) , p − 1 / 2 ] for L ∞ approximation, where p is the optimal order of convergence for weighted L 2 approximation among all algorithms that may use arbitrary linear functionals, as opposed to function values only. Under a mild assumption on the reproducing kernels we have p > 1 / 2 . It was shown in our previous paper that the optimal order for L ∞ approximation and linear information is p − 1 / 2 . We do not know if our bounds are sharp for standard information. We also study tractability of multivariate approximation, i.e., we analyze when the worst case error bounds depend at most polynomially on d and n − 1 . We present necessary and sufficient conditions on tractability and illustrate our results for the weighted Korobov spaces with arbitrary smoothness and for the weighted Sobolev spaces with the Wiener sheet kernel. Tractability conditions for these spaces are given in terms of the weights defining these spaces.

Linear information versus function evaluations for [formula omitted]-approximation

We study algorithms for the approximation of functions, the error is measured in an L 2 norm. We consider the worst case setting for a general reproducing kernel Hilbert space of functions. We analyze algorithms that use standard information consisting in n function values and we are interested in the optimal order of convergence. This is the maximal exponent b for which the worst case error of such an algorithm is of order n - b . Let p be the optimal order of convergence of all algorithms that may use arbitrary linear functionals, in contrast to function values only. So far it was not known whether p > b is possible, i.e., whether the approximation numbers or linear widths can be essentially smaller than the sampling numbers. This is (implicitly) posed as an open problem in the recent paper [F.Y. Kuo, G.W. Wasilowski, H. Woźniakowski, On the power of standard information for multivariate approximation in the worst case setting, J. Approx. Theory, to appear] where the authors prove that p > 1 2 implies b ⩾ 2 p 2 / ( 2 p + 1 ) > p - 1 2 . Here we prove that the case p = 1 2 and b = 0 is possible, hence general linear information can be exponentially better than function evaluation. Since the case p > 1 2 is quite different, it is still open whether b = p always holds in that case.

On Learning Vector-Valued Functions

In this letter, we provide a study of learning in a Hilbert space of vectorvalued functions. We motivate the need for extending learning theory of scalar-valued functions by practical considerations and establish some basic results for learning vector-valued functions that should prove useful in applications. Specifically, we allow an output space Y to be a Hilbert space, and we consider a reproducing kernel Hilbert space of functions whose values lie in Y. In this setting, we derive the form of the minimal norm interpolant to a finite set of data and apply it to study some regularization functionals that are important in learning theory. We consider specific examples of such functionals corresponding to multiple-output regularization networks and support vector machines, for both regression and classification. Finally, we provide classes of operator-valued kernels of the dot product and translation-invariant type.

Uniform Distribution, Discrepancy, and Reproducing Kernel Hilbert Spaces

In this paper we define a notion of uniform distribution and discrepancy of sequences in an abstract set E through reproducing kernel Hilbert spaces of functions on E. In the case of the finite-dimensional unit cube these discrepancies are very closely related to the worst case error obtained for numerical integration of functions in a reproducing kernel Hilbert space. In the compact case we show that the discrepancy tends to zero if and only if the sequence is uniformly distributed in our sense. Next we prove an existence theorem for such uniformly distributed sequences and investigate the relation to the classical notion of uniform distribution. Some examples conclude this paper.

Uncertainty relations for wavefunctions

It is proved that when the Hilbert space of quantum mechanics is a reproducing kernel Hilbert space of functions defined on the phase space, the generating function of the entropy can be used as a measure of the concentration of the wavefunctions in the phase space. The sharp upper bound obtained for the generating function of the entropy proves that the wavefunctions cannot be arbitrarily peaked in the phase space. The most peaked wavefunctions are those defined by the reproducing kernels.