Projections and dyadic Parseval frame MRA wavelets

Abstract

Abstract. A classical theorem attributed to Naimark states that, given a Parseval frame B in a Hilbert space H, one can embed H in a larger Hilbert space K so that the image of B is the projection of an orthonormal basis for K. In the present work, we revisit the notion of Parseval frame MRA wavelets from [11] and [12] and produce an analog of Naimark's theorem for these wavelets at the level of their scaling functions. We aim to make this discussion as self-contained as possible and provide a different point of view on Parseval frame MRA wavelets than that of [11] and [12].1. Notation and Preliminary Remarks 1.1. Notation. The Fourier transform in this paper will be denoted byWe will identify subsets of the torus, T = R/Z, with subsets of R which are invariant under translations by integers -in particular, functions on T can be considered to be 1-periodic function on R and the Lebesgue measure we associate to T has total mass 1. We will frequently use the operatorsIn particular, D j and T k are unitary maps corresponding to the dyadic dilations and integer translations. We will let ψ jk = D j T k ψ = 2 j/2 ψ(2 j ·−k). Definition 1.1. We will use the notation m • g to denote the function which satisfies m • g = m · g, whenever this function is well-defined.Remark. The above definition makes sense, for instance, if g ∈ L 2 (R) and m ∈ L ∞ (R): then m · g is an L 2 (R) function, and so m • g is well-defined since the Fourier transform is an isometry on L 2 (R). Though this notation is perhaps new, the above operation is actually quite common in a variety of areas of analysis. It comes up repeatedly in the study of shift-invariant spaces, e.g. in [7]. We should take a moment to reference a few of the authors who have made significant contributions to the study of shift invariant spaces upon which much of the present work rests: Helson, [6], de Boor, DeVore, and Ron, [4] and [5], Bownik, [1], as well as many of the references contained in those papers. The operation m • g operation also comes up in the study of singular integrals -for example, the Hilbert transform corresponds to such a map when, say, g ∈ L 2 (R) and m is (a multiple of) the function which is 1 for positive reals and −1 for negative reals.In everything we discuss below -unless otherwise noted -equalities between functions are taken to be equalities in the almost everywhere sense; equalities between sets are taken to be equalities up to sets of measure zero. All functions we will discuss below are measurable.1.2. Dyadic Parseval Frame MRA Wavelets. The term wavelet can mean a variety of things in different contexts. For our purposes, a wavelet will be a function ψ ∈ L 2 (R) so that {ψ jk : j, k ∈ Z} linearly generates L 2 (R) in some way -these are more accurately described as dyadic wavelets since the dilations are dyadic dilations. Classically, wavelets were always taken to be orthonormal wavelets; that is, functions ψ so that {ψ jk : j, k ∈ Z} corresponded to orthonormal bases of L 2 (R) (e.g. the Haar wavelet, Shannon wavelet, Daubechies wavelet...

A High-Resolution Dyadic Transform for Non-Stationary Signal Analysis

“…On the other hand, if A = 1 then the frame is an orthonormal basis and is represented by dyadic wavelets [25][26][27]. If we rewrite Equation (22) for orthonormal bases, we can reach the conclusion that the use of dyadic wavelet does not modify the energy of the transformed signal (see Equation ( 23)).…”