Extension Theorems for Spaces Arising from Approximation by Translates of a Basic Function

Approximation by Translates of a Radial Function

APPROXIMATION IN Lp(Rd) FROM SPACES SPANNED BY THE PERTURBED INTEGER TRANSLATES OF A RADIAL FUNCTION

The problem of approximating smooth L p-functions from spaces spanned by the integer translates of a radially symmetric function is very well understood. In case the points of translation, , are scattered throughout R d , the approximation problem is only well understood in the \stationary" setting. In this work, we provide lower bounds on the obtainable approximation orders in the \non-station...

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...