Generalization of the Strang and Fix Conditions to the Product Operator

We first recall the linear approximation of Strang and Fix [1], in the context of multiresolution analysis and wavelets. The original theorem is concerned with general finite element approximations. Theorem 1 (Fix-Strang) Let p ∈ N, φ a (bi)orthogonal scaling function, and ψ * its conjugate wavelet. The following three conditions are equivalent • for any 0 ≤ k < p, there exists a polynomial θ k of degree k such that +∞ n=−∞ n k φ(t − n) = θ k (t) • ψ * has p vanishing moments • ∃C (∀f ∈ H p+1 (R)) (∀j ≤ 0) P j f − f L 2 ≤ C2 j(p+1) f (p+1) L 2 where H p+1 is the Sobolev space of functions with p + 1 derivatives in L 2 (R). The following theorem generalizes the Strang and Fix condition to the approximation of the product operator. Theorem 2 Assume that * is a product operator such that (V 0 , +, *) be a commutative ring, that * commutes with the shift of length L ∈ N, and that the product of two compactly supported functions is zero beyond a certain fixed distance. We define * j on V j by rescaling. Let N ≤ p, with p as in theorem 1. The two following conditions are equivalent: