Construction of Parseval wavelets from redundant filter systems

Abstract

We consider wavelets in L^2(R^d) which have generalized multiresolutions. This means that the initial resolution subspace V_0 in L^2(R^d) is not singly generated. As a result, the representation of the integer lattice Z^d restricted to V_0 has a nontrivial multiplicity function. We show how the corresponding analysis and synthesis for these wavelets can be understood in terms of unitary-matrix-valued functions on a torus acting on a certain vector bundle. Specifically, we show how the wavelet functions on R^d can be constructed directly from the generalized wavelet filters.Comment: 34 pages, AMS-LaTeX ("amsproc" document class) v2 changes minor typos in Sections 1 and 4, v3 adds a number of references on GMRA theory and wavelet multiplicity analysis; v4 adds material on pages 2, 3, 5 and 10, and two more reference

Generalized Multiresolution Analyses with Given Multiplicity Functions

“…Our proof follows the general strategy suggested in [3,Lemma 3.3], but here we have been able to replace some of the grittier estimates with exact calculations, and those which remain are much sharper. The crux of the argument is the almost everywhere pointwise convergence of a sequence of averages, which we achieve by applying the reverse martingale convergence theorem.…”

“…Our new results include a general construction of filters for multiplicity functions (Proposition 7), and a criterion for the purity of an associated isometry which improves a key result in [3] (Theorem 8).…”

“…GMRAs were introduced in [4], and have since been studied in [5] and [3]; other authors had previously observed that much of the theory still works when we use Γ in place of the classical translation group Z n (see [1] and [9], for example).…”

“…Consider the multiplicity function for dilation by 2 in R corresponding to the Journé wavelet, previously studied in [8], [2] and [3]. The multiplicity function for this minimally supported frequency wavelet is given by…”

“…Then in §3 we prove our main theorem. We first show that our multiplicity function m admits a low-pass filter, which is a matrix H of functions on Γ satisfying relations, introduced in [3], which generalize those of quadrature mirror filters. From H we build an isometry S H on a Hilbert space K, following an idea which goes back at least to [7], and Theorem 8 says that when the filter is low-pass, S H is a pure isometry.…”