Approximation by Translates of a Positive Definite Function

Approximation by Translates of a Radial Function

Multivariate Interpolation and Approximation by Translates of a Basis Function

This contribution will touch the following topics: Short introduction into the theory of multivariate interpolation and approximation by nitely many (irregular) translates of a (not necessarily radial) basis function, motivated by optimal recovery of functions from discrete samples. Native spaces of functions associated to conditionally positive definite functions, and relations between such sp...

Student Seminar Talk APPROXIMATION BY TRANSLATES OF A FUNCTION

We discuss the following problem: given φ ∈ L2(R) and a sequence Λ ⊂ R, is it possible that the system of translates {φ(x − λ)}λ∈Λ spans the whole space, and how sparse the spectrum Λ might be? 1. Systems of translates 1.1. Let a function φ ∈ L(R) and a discrete sequence Λ ⊂ R be given. We are interested to know if the system of translates {φ(x− λ)}λ∈Λ (1.1) is complete in L(R), that is, if the...

On Multivariate Approximation by Integer Translates of a Basis Function

Approximation properties of the dilations of the integer translates of a smooth function, with some derivatives vanishing at infinity, are studied. The results apply to fundamental solutions of homogeneous elliptic operators and to “shifted” fundamental solutions of the iterated Laplacian. Following the approach from spline theory, the question of polynomial reproduction by quasiinterpolation i...

Armin Iske * Scattered Data Approximation by Positive Definite Kernel Functions

Kernel functions are suitable tools for scattered data interpolation and approximation. We first review basic features of kernel-based multivariate interpolation, before we turn to the construction and the characterization of positive definite kernels and their associated reproducing kernel Hilbert spaces. The optimality of the resulting kernel-based interpolation scheme is shown. Moreover, we ...