The Strang and Fix Conditions

Then Strang and Fix conditions are recalled and proved. The Strang and Fix conditions caracterize the approximating properties of a shift invariant localized operator by its ability to reconstruct polynomials. In particular, it is used to relate the number of vanishing moments of a wavelet to the order of approximation provided by the corresponding multiresolution analysis. Article [1] is generally given as a reference for this result. This note is intented to be an alternative to it, since the original paper has probably disappeared from most bookshelves. 1 Notations • L2(R) is the space of integrable functions with a finite energy. H (R) is defined recursively by H(R) = L2(R) and H (R) is the space of functions in L2(R) with derivatives in HN−1(R). If f ∈ H (R), f (N) denotes the n derivative of f . • In what follows, K ∈ LLoc(R× R) is a kernel such that K(x + 1, y + 1) = K(x, y) a.e. (1) ∃M s.t. K(x, y) = 0 if |x − y| ≥ M, (2) that is, K is a localized kernel and it commutes with integer shifts. Because of the localization property (2), K is of finite energy with respect to the separate variables x and y. Because of the shift invariance property (1), it cannot be of finite energy with respect to the joint variables (x, y) (unless it is zero). ∗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/ †The author wishes to thank P. G. Lemarié for introducing him to the mysteries of the Stang and Fix conditions and providing him the original proof on which this paper is based.