Multi-Resolution Analysis of Multiplicity d: Applications to Dyadic Interpolation

AbstractThis paper studies the Multi-Resolution Analyses of multiplicity d (d ∈ N*), that is, the families (Vn)n∈Z of closed subspaces in L2(R) such that Vn ⊂ Vn + 1, Vn + 1 = DVn, where Dƒ(x) = ƒ(2x), and such that there exists a Riesz basis for V0 of the form {φi(· − k), i = 1, . . . , d,k ∈ Z}, with φ1, . . . , φd ∈ V0. Using the Fourier transform, we prove that Φ̂(λ) = t[φ̂1(λ), . . . , φ̂d(λ)] = H(λ/2)Φ̂(λ/2), where H is in the set Md of continuous 1-periodic functions taking values in M(d, C). If d = 1, the definition corresponds to the standard Multi-Resolution Analyses, and one can characterize the regular 1-periodic complex-valued functions H (called, then, scaling filters) which yield a Multi-Resolution Analysis. In this paper, we generalize this study to d ≥ 2 by giving conditions on H ∈ Md so that there exists Φ̂ = t[φ̂1, . . . , φ̂d] in L2(R, Cd) solution of Φ̂(λ) = H(λ/2)Φ̂(λ/2), and so that the integer translates of φ1, . . . , φd form a Riesz family. Then, the latter span the space V0 of a Multi-Resolution Analysis of multiplicity d. We show that the conditions on H focus on the zeros of det H(·) and on simple spectral hypotheses for the operator PH defined on Md by PHF(λ) = H(λ/2)F(λ/2)H(λ/2)* + H(λ/2 + 1/2)F(λ/2 + 1/2)H(λ/2 + 1/2)*. Finally, we explore connections with the order r dyadic interpolation schemes, where r ∈ N*