Wavelet frames and shift-invariant subspaces of periodic functions

Abstract

We introduce the concept of the modular function for a shift-invariant 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 multires-olution 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 {f j } in H such that there exist constants C 1 , C 2 > 0 such that C 1 f 2 2 ≤ j ||f, f j | 2 ≤ C 2 f 2 2 holds for all f ∈ H. If C 1 = C 2 = 1, we say that {f j } 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 2 (R d) with the property that ψ A m,, (t) := {|detA| m 2 ψ(A m t −) : m ∈ Z, , ∈ Z d } is an orthonormal basis for L 2 (R d). More generally, ψ will be called a normalized tight A-dilation wavelet frame if {ψ A m,, } forms a normalized tight frame for L 2 (R d). In what follows we will use the term "wavelet frame" to denote the normalized tight ones. The dilation operator δ A and the translation operator T are defined by: (δ A f)(t) = |detA| 1/2 f (At), (T f)(t) = f (t −)

Pairs of oblique duals in spaces of periodic functions

“…Most statements are easy consequences of the definition and were also observed in earlier papers (see, e.g., [5,7,8]), so we only prove (1.5). Given f ∈ L 2 (0, 2π), we have by (ii) that…”

“…The theory of periodic wavelets via polyphase splines in [5,6,7] was developed for the more general multidimensional setting of L 2 ((0, 2π) s ), where s ∈ N, as minimal additional efforts compared to the one-dimensional case were needed. Likewise, all the results here on oblique duals for finite-dimensional spaces in L 2 (0, 2π) could be easily extended to the multidimensional setting.…”

“…, r; ℓ ∈ R}; (1.1) the underlying multiresolution analysis. The multiple generator setup of (1.1) reflects the situations encountered with multiple refinable functions, multiwavelets and wavelet frames; see, e.g., [5,6,7]. Our purpose here is to obtain expansions of the functions in V of the form…”

“…Polyphase splines. It turns out to be convenient to express the functions φ m generating the vector space V in (1.1) in terms of the so-called polyphase splines introduced in [5] and further studied in [7]. Polyphase splines are generalizations of the orthogonal splines formulated in [8] for the situation of V having a single generating function, i.e., r = 1.…”

Characterizations of dual multiwavelet frames of periodic functions

“…More general constructions of periodic orthogonal wavelets or biorthogonal multidimensional multiwavelets emerged from non-stationary PMRA's in the sense that different scaling functions and wavelets are involved at different scales [22,27]. Constructions of tight periodic (multi)wavelet frames were obtained in [20,21,23,25,33]. In [32], a dual pair of periodic wavelet frames for L 2 [0, 2π] was derived from 2πperiodization of a pair of functions on R satisfying certain conditions and in [33] these conditions were relaxed.…”