Wavelets and Differentiation

The approximation conditions of Strang and Fix are first recalled. An elementary result on the differentiation of a finite elements approximation is proved, followed by a result by Lemarié on the differentiation of a wavelet decomposition. Daubechies’ spline example is detailed. 1 The Strang and Fix conditions 1.1 Main result The following theorem relates the approximating properties of a discrete shift invariant operator to its ability to reproduce polynomials. Theorem 1 (Fix-Strang [1]) Let K ∈ LLoc(R× R) such that K(t + 1, s + 1) = K(t, s) a.e. (1) ∃M s.t. K(t, s) = 0 if |t − s| ≥ M (2) and, for δ > 0, define Pδ as Pδf(t) = 1 δ ∫ R K ( t δ , s δ ) f(s)ds (3) Then the two following statements are equivalent: ∗Centre Automatique et Systèmes, École Nationale Supérieure des Mines de Paris, 35 rue Saint Honoré, 77305 Fontainebleau Cedex FRANCE, e-mail: [email protected], http://cas.ensmp.fr/ ̃chaplais/ †Université Paris-Sud Orsay