Sampling inequalities play an important role in deriving error estimates for various reconstruction processes. They provide quantitative estimates on a Sobolev norm of a function, defined on a bounded domain, in terms of ...

We provide an interpretation for the covariance of the predictive process of Bayesian Gaussian process regression as reproducing kernel of a subset of the Cameron Martin space of the prior. We demonstrate that this ......

In this paper we provide a finite-sample and an infinite-sample representer theorem for the concatenation of (linear combinations of) kernel functions of reproducing kernel Hilbert spaces. These results serve as mathematical ...

In this article, we present a cost-benefit analysis of the approximation in tensor products of Hilbert spaces of Sobolev-analytic type. The Sobolev part is defined on a finite dimensional domain, whereas the analytical ......

In this article, we present kernel methods for the approximation of quantities of interest which are derived from solutions of parametric partial differential equations. We explicitly construct a reproducing kernel Hilbert ...

We consider the reconstruction of a function stemming from a reproducing kernel Hilbert space using data which is perturbed by a deterministic error of maximal size ε_∞. The accuracy ε_∞ ...

Many physical and mathematical models involve random fields in their input data. Examples are ordinary differential equations, partial differential equations and integro–differential equations with uncertainties in the ...

We present a theoretical framework for reproducing kernel based reconstruction methods in certain generalized Besov spaces based on positive, essentially self-adjoint operators. An explicit representation of the reproducing ...

We consider reproducing kernels K : Ω x Ω → ℝ in multiscale series expansion form, i.e., kernels of the form K (x, y) = ∑_ℓ∈ℕλ_ℓ∑_j∈Iℓ

We derive sampling inequalities for discrete point sets which are of anisotropic tensor product form. Such sampling inequalities can be used to prove convergence for arbitrary stable reconstruction processes. As usual in ...