Projective Multi-resolution Analyses for L 2 (r 2 )

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 paper 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 more scaling functions, we only require that V 0 be the (projective) module of inverse Fourier transforms of continuous cross-sections of a complex vector bundle over T n. The precise definition is given in Definition 3.1. We show in Section 3 that projective multi-resolution analyses give in a natural way wavelets which determine normalized tight frames for L 2 (R n). This seems to be a somewhat new way of constructing tight frames for L 2 (R n). We note …