Projective Multi-Resolution Analyses for L2(?2)

Abstract

We define the notion of "projective" multiresolution analyses, for which, by definition, the initial space corresponds to a finitely generated projective module over the algebra C(T n ) of continuous complex-valued functions on an n-torus. The case of ordinary multi-wavelets is that in which the projective module is actually free. We discuss the properties of projective multiresolution analyses, including the frames which they provide for L 2 (R n ). Then we show how to construct examples for the case of any diagonal 2 × 2 dilation matrix with integer entries, with initial module specified to be any fixed finitely generated projective C(T 2 )-module. We compute the isomorphism classes of the corresponding wavelet modules.In classical wavelet theory one uses multi-resolution analyses to construct (multi-) wavelets and their corresponding orthonormal bases or frames for L 2 (R n ). In almost all applications the scaling functions and wavelets have continuous Fourier transforms. This continuity is a significant and interesting condition. If one requires just a bit more, then one finds that in the frequency domain one is dealing with what are called projective modules over C(T n ) (or equivalently, with the spaces of continuous cross-sections of complex vector bundles over T n ). The case of a single scaling function or wavelet corresponds to the free module of rank 1, whereas the case of several orthogonal scaling functions or wavelets corresponds to free modules of higher rank. This leads one to ask whether wavelet theory carries over to the case of general projective modules over C(T n ). It is the purpose of this article to show that the answer is affirmative.For a given dilation matrix A we define a projective multiresolution analysis for L 2 (R n ) to be an increasing sequence {V j } of subspaces having the usual properties, with the one exception that instead of V 0 being the linear span of the integer translates of one or Math Subject Classifications. Primary: 46L99; secondary: 42C15, 46H25, 47A05.

Representations, Wavelets, and Frames

“…If λ is rational, say λ = a b , a, b ∈ Z in lowest terms, the authors determine completely to what extent X λ supports a family of mutually orthogonal exponential functions. The article by Packer (a colleague and frequent collaborator of Larry) gives an overview of projective multiresolution analyses, as first defined by M. Rieffel and as formalized by Packer and Rieffel in [50]. After presenting a survey of Rieffel's original approach, she concentrates on two 2 × 2 dilation matrices that are not similar to diagonal matrices with integer entries.…”

Projective multi-resolution analyses arising from direct limits of Hilbert modules

“…is a Hilbert C(T n )-module [19,Proposition 7]. We freely use key properties of established by Packer and Rieffel, and especially Propositions 13 and 14 of [19].…”

“…To see this, we recall from [19,Proposition 3] that is complete, and apply Proposition 1.3 with R k the inclusion of C 0 ((−k, k)) in . The induced map R ∞ has dense range by [19,Proposition 4], and hence is surjective.…”

“…Part (b) then says that the sequence {V k } is a projective multi-resolution analysis for the (necessarily closed) submodule Q(Y ∞ ) of X. We will explain in Example 3.7 how these projective resolution analyses give the ones considered by Packer and Rieffel in [19].…”

“…and is the closure of the set {f (e 2πix )φ(x) : f ∈ C(T n )}; in other words, V 0 is the closure of the free module over C(T n ) generated by φ. Packer and Rieffel have shown that one can also obtain projective multi-resolution analyses in which the initial module V 0 is the closure of a finitely generated projective module over C(T n ) [19]. Over the past few years, various authors have realised that Hilbert modules over C * -algebras provide a fertile environment for studying multi-resolution analyses, wavelets and frames [11], [21], [23], [18], [7], and Packer and Rieffel worked in that context throughout.…”