Frames, Modular Functions for Shift-invariant Subspaces and Fmra Wavelet Frames

We introduce the concept of the modular function for a shiftinvariant subspace that can be represented by normalized tight frame generators for the shift-invariant subspace and prove that it is independent of the selections of the frame generators for the subspace. We shall apply it to study the connections between the dimension functions of wavelet frames for any expansive integer matrix A and the multiplicity functions for general multiresolution analysis (GMRA). Given a frame mutiresolution analysis (FMRA), we show that the standard construction formula for orthonormal multiresolution analysis wavelets does not yield wavelet frames unless the underlying FMRA is an MRA. A modified explicit construction formula for FMRA wavelet frames is given in terms of the frame scaling functions and the low-pass filters. 1. Preliminaries A frame for a separable Hilbert space H is a sequence of vectors {fj} in H such that there exist constants C1, C2 > 0 such that C1‖f‖2 ≤ ∑ j |〈f, fj〉| ≤ C2‖f‖2 holds for all f ∈ H . If C1 = C2 = 1, we say that {fj} is a normalized tight frame. For a d× d real expansive matrix A (i.e., all the eigenvalues of A are required to have absolute values greater than 1), an A-dilation (orthogonal) wavelet is a single function ψ ∈ L(R) with the property that ψ m, (t) := {|detA| m 2 ψ(At− ) : m ∈ Z, ∈ Z} is an orthonormal basis for L(R). More generally, ψ will be called a normalized tight A-dilation wavelet frame if {ψ m, } forms a normalized tight frame for L(R). In what follows we will use the term “wavelet frame” to denote the normalized tight ones. The dilation operator δA and the translation operator T are defined by: (δAf)(t) = |detA|f(At), (T f)(t) = f(t− ) Received by the editors February 25, 2002 and, in revised form, November 11, 2003. 2000 Mathematics Subject Classification. Primary 42C15, 47B38.