Completeness of Integer Translates in Function Spaces on R

compactifications and function spaces on weighted semigruops

chapter one is devoted to a moderate discussion on preliminaries, according to our requirements. chapter two which is based on our work in (24) is devoted introducting weighted semigroups (s, w), and studying some famous function spaces on them, especially the relations between go (s, w) and other function speces are invesigated. in fact this chapter is a complement to (32). one of the main fea...

On the Properties of the Integer Translates of a Square Integrable Function in L(R)

(1.1) 〈ψ〉 = span{ψ(· − k) ≡ ψk : k ∈ Z} That is, 〈ψ〉 is the L2(R) closure of the space generated by the finite linear combinations of the integer translates, (Tkψ)(x) ≡ ψ(x − k) ≡ ψk(x), of ψ ∈ L2(R). Such a space is often called the principal shift invariant space generated by ψ. In general, a closed subspace V ⊂ L2(R) is shift invariant if and only if TkV ⊂ V for all translates Tk, k ∈ Z. 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...

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...

A new class of function spaces on domains of R^d and its relations to classical function spaces