Generalized Multiresolution Analyses with Given Multiplicity Functions

Abstract

Abstract. Generalized multiresolution analyses are increasing sequences of subspaces of a Hilbert space H that fail to be multiresolution analyses in the sense of wavelet theory because the core subspace does not have an orthonormal basis generated by a fixed scaling function. Previous authors have studied a multiplicity function m which, loosely speaking, measures the failure of the GMRA to be an MRA. When the Hilbert space H is L 2 (R n ), the possible multiplicity functions have been characterized by Baggett and Merrill. Here we start with a function m satisfying a consistency condition which is known to be necessary, and build a GMRA in an abstract Hilbert space with multiplicity function m.

Classification of generalized multiresolution analyses

“…Using α * to relate the representations π| V 1 and π| V 0 , it is shown in [6] and more generally in [4] that multiplicity functions for a GMRA must satisfy the following consistency equation:…”

Direct limits, multiresolution analyses, and wavelets

“…3) 3 When K = T and β(z) = z N , this is same as the measure constructed by Dutkay in [8, Proposition 4.2(i)]. In our notation, his defining property is…”

“…Now suppose that μ : Γ → U(H ) is a unitary representation such that Sμ γ = μ α(γ ) S for γ ∈ Γ . Then we proved in [3,Theorem 5(d)] that there is a representation μ ∞ of Γ on H ∞ characterized by μ ∞ (γ )U n = U n μ α n (γ ) ; we then have S ∞ μ ∞ (γ ) = μ ∞ (α(γ ))S ∞ , and the triple ({V n }, μ ∞ , S −1 ∞ ) is a generalized multiresolution analysis (GMRA) for H ∞ if and only if S is a pure isometry.…”

“…We proved in [3,Theorem 5] that there is a unitary operator S ∞ on H ∞ characterized by S ∞ U n = U n S = U n−1 , and that the subspaces V n of H ∞ defined by…”

Markov measures and extended zeta functions

“…To make our paper more accessible, we offer below a few pointers to the relevant literature. Readers familiar with one of these areas, but perhaps not the others, may wish to check the following references covering aspects of these areas used below: operator algebras, Cuntz algebras and their representations ( [13], [4], [10], [8], [16], [31], [22], [24], [25], [26], [40]) ; multiresolutions and their diverse uses ( [1], [12], [3], [23], [2], [14] , [27] ) ; Zeta functions ( [6], [39], [20], [38], [37], [36]); and Markov measures ( [32], [33], [21], [5], [51], [52], [7], [9]). We further use results from harmonic analysis, such as ( [19], [29], [15], [17], [18], [49]).…”