Controlled approximation and a characterization of the local approximation order

The local approximation order from a scale (Sh) of approximating functions on IR m is characterized in terms of the linear spac (and its Fourier transform) of the finitely many compactly supported functions φ whose integer translates φ(· − j), j ∈ ZZ, span the space S = S1 from which the scale is derived. This provides a correction of similar results stated and proved, in part, by Strang and Fix. The term “controlled approximation” was introduced in 1970 by Strang [St]; see also [FS]. It concerns approximations of the form ∑ φ∈Φ φ ∗ cφ, with Φ a finite collection of functions on IR of compact support and φ ∗ c the function obtained from φ by convolution with some “sequence” c : ZZ → IR; i.e., φ ∗ c := ∑ j∈ZZm φ(· − j)c(j). If, more generally, c is some function on IR, we will still just write φ ∗ c instead of the correct but more complicated φ ∗ (c ZZm). The function u to be approximated lies in the Sobolev space W k p (IR ) with norm ‖u‖k,p := ∑ j≤k |u|j,p, where |u|j,p := ∑ |α|=j ‖Du‖p and ‖u‖p := ‖u‖Lp(IRm). We denote by W k p,c(IR ) the subspace of W k p (IR ) of compactly supported functions. The approximations are, more explicitly, of the form σh ∑ φ∈Φ φ ∗ cφ  /h with σhf := f(·/h). Concerning the degree of approximation to u ∈ W k p (IR) achievable by proper choice of the weights cφ, Strang and Fix [SF, Theorem II] state the following result. In its statement and subsequent analysis, the normalized multivariate monomials appear often enough to deserve an abbreviation of their own. We will use [[ ]] to stand for the function IR → IR : x 7→ x/α! (and will use standard multi-index notation throughout). In particular, D[[ ]] = [[ ]]α−β , and this holds even when β 6≤ α, since then [[ ]]α−β = 0, by convention. Further, Πj will denote the collection of polynomials on IR of total degree ≤ j. Finally, f̂ will denote the Fourier transform of f — i.e., f̂(ξ) := ∫ IRme −iξx f(x) dx, with ξx the scalar product. 1980 Mathematics Subject Classification. Primary 41A25, 41A15, 41A63, 65N30.