Direct limits, multiresolution analyses, and wavelets

Abstract

A multiresolution analysis for a Hilbert space realizes the Hilbert space as the direct limit of an increasing sequence of closed subspaces. In a previous paper, we showed how, conversely, direct limits could be used to construct Hilbert spaces which have multiresolution analyses with desired properties. In this paper, we use direct limits, and in particular the universal property which characterizes them, to construct wavelet bases in a variety of concrete Hilbert spaces of functions. Our results apply to the classical situation involving dilation matrices on L 2 (R n ), the wavelets on fractals studied by Dutkay and Jorgensen, and Hilbert spaces of functions on solenoids.

KMS states on C*-algebras associated to local homeomorphisms

“…We can construct measures on E ∞ by viewing it as an inverse limit lim ← − (E n , r n ), where r n : E n+1 → E n takes ν = ν 1 ν 2 · · · ν n ν n+1 to ν 1 ν 2 · · · ν n . Then any family of measures δ n on E n such that δ n+1 (Z(ν) ∩ E n+1 ) = δ n (Z(ν)) for |ν| = n gives a measure δ on E ∞ such that δ(Z(ν)) = δ n (Z(ν)) for |ν| = n (see, for example, Lemma 6.1 of [1]). We can construct such a sequence by taking δ 0 = ε, inductively choosing weights w e such that r(e)=v w e = ε v , recursively choosing {w νe ∈ [0, ∞) : νe ∈ E n+1 } such that r(e)=s(ν) w νe = w ν , and setting δ n+1 (νe) = w νe .…”

Phase transition on Exel crossed products associated to dilation matrices

“…However, it does not seem to be so easy to find natural Hilbert space representations of O(M L ) in which the KMS log | det A| state is a vector state. [4], or more precisely, one of the more general sort studied in [2].…”

“…As for the system in [23], the circular symmetry at β = log N which disappears for β > log N is not apparently realised by an action of T on T (N ⋊ N N). In [23], though, this circular symmetry persists for β ∈ [1,2], as a result of the more complicated convergence issues for the series representations of the normalising factors.…”

Classification of generalized multiresolution analyses

“…Dutkay and Jorgensen [16] pioneered the study of wavelets in function spaces on fractals, with later work by D'Andrea et al [14]. Larsen, Raeburn and coworkers then showed that these and other interesting examples can be constructed via direct limits [19,4,5]. Dutkay et al [9,17] constructed MRAs and super-wavelets in Hilbert spaces formed by direct sums of L 2 (R d ) to orthonormalize examples such as the Cohen wavelet.…”