Wavelet filters and infinite-dimensional unitary groups (original) (raw)
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A ug 2 00 0 Wavelet filters and infinite-dimensional unitary groups
2000
Abstract. In this paper, we study wavelet filters and their dependence on two numbers, the scale N and the genus g. We show that the wavelet filters, in the quadrature mirror case, have a harmonic analysis which is based on representations of the C∗-algebra ON . A main tool in our analysis is the infinite-dimensional group of all maps T → U (N) (where U (N) is the group of all unitary N-by-N matrices), and we study the extension problem from low-pass filter to multiresolution filter using this group.
Fe b 20 00 Wavelet filters and infinite-dimensional unitary groups
2000
Abstract. In this paper, we study wavelet filters and their dependence on two numbers, the scale N and the genus g. We show that the wavelet filters, in the quadrature mirror case, have a harmonic analysis which is based on representations of the C∗-algebra ON . A main tool in our analysis is the infinite-dimensional group of all maps T → U (N) (where U (N) is the group of all unitary N-by-N matrices), and we study the extension problem from low-pass filter to multiresolution filter using this group.
Unitary matrix functions, wavelet algorithms, and structural properties of wavelets
2004
Some connections between operator theory and wavelet analysis: Since the mid eighties, it has become clear that key tools in wavelet analysis rely crucially on operator theory. While isolated variations of wavelets, and wavelet constructions had previously been known, since Haar in 1910, it was the advent of multiresolutions, and sub-band filtering techniques which provided the tools for our ability to now easily create efficient algorithms, ready for a rich variety of applications to practical tasks. Part of the underpinning for this development in wavelet analysis is operator theory. This will be presented in the lectures, and we will also point to a number of developments in operator theory which in turn derive from wavelet problems, but which are of independent interest in mathematics. Some of the material will build on chapters in a new wavelet book, co-authored by the speaker and Ola Bratteli, see http://www.math.uiowa.edu/˜jorgen/. Contents Abstract 1 1. Introduction 3 1.1. Index of terminology in math and in engineering 4 1.1.1. Some background on Hilbert space 1.1.2. Connections to group theory 1.1.3. Some background on matrix functions in mathematics and in engineering 1.
In this paper we are discussing various aspects of wavelet filters. While there are earlier studies of these filters as matrix valued functions in wavelets, in signal processing, and in systems, we here expand the framework. Motivated by applications, and by bringing to bear tools from reproducing kernel theory, we point out the role of non-positive definite Hermitian inner products (negative squares), for example Krein spaces, in the study of stability questions. We focus on the non-rational case, and establish new connections with the theory of generalized Schur functions and their associated reproducing kernel Pontryagin spaces, and the Cuntz relations.
Isometries, shifts, Cuntz algebras and multiresolution wavelet analysis of scale N
1996
In this paper we show how wavelets originating from multiresolution analysis of scale N give rise to certain representations of the Cuntz algebras O_N, and conversely how the wavelets can be recovered from these representations. The representations are given on the Hilbert space L^2(T) by (S_i\xi)(z)=m_i(z)\xi(z^N). We characterize the Wold decomposition of such operators. If the operators come from wavelets they are shifts, and this can be used to realize the representation on a certain Hardy space over L^2(T). This is used to compare the usual scale-2 theory of wavelets with the scale-N theory. Also some other representations of O_N of the above form called diagonal representations are characterized and classified up to unitary equivalence by a homological invariant.
Electronic Research Announcements of the American Mathematical Society, 2003
In this note, we announce a general method for the construction of nonseparable orthogonal wavelet bases of L 2 ( R n ) , L^2(\mathbb {R}^n), where n ≥ 2. n\geq 2. Hence, we prove the existence of such type of wavelet bases for any integer n ≥ 2. n\geq 2. Moreover, we show that this construction method can be extended to the construction of n n -D multiwavelet matrix filters.
The dilation property for abstract Parseval wavelet systems
In this work we introduce a class of discrete groups called wavelet groups that are generated by a discrete group Γ0 (translations) and a cyclic group Γ1 (dilations), and whose unitary representations naturally give rise to a wide variety of wavelet systems generated by the pseudo-lattice Γ = Γ1Γ0. We prove a condition in order that a Parseval frame wavelet system generated by Γ can be dilated to an orthonormal basis that is also generated by Γ via a super-representation. For a subclass of groups where Γ0 is Heisenberg, we show that this condition always holds, and we cite a number of familiar examples as applications.
Decomposition of wavelet representations and Martin boundaries
Journal of Functional Analysis, 2012
We study a decomposition problem for a class of unitary representations associated with wavelet analysis, wavelet representations, but our framework is wider and has applications to multi-scale expansions arising in dynamical systems theory for non-invertible endomorphisms.
Using the notions and tools from realization in the sense of systems theory, we establish an explicit and new realization formula for families of infinite products of rational matrix-functions of a single complex variable. Our realizations of these resulting infinite products have the following four features: 1) Our infinite product realizations are functions defined in an infinite-dimensional complex domain. 2) Starting with a realization of a single rational matrix-function MMM, we show that a resulting infinite product realization obtained from MMM takes the form of an (infinite-dimensional) Toeplitz operator with a symbol that is a reflection of the initial realization for MMM. 3) Starting with a subclass of rational matrix functions, including scalar-valued corresponding to low-pass wavelet filters, we obtain the corresponding infinite products that realize the Fourier transforms of generators of mathbfL_2(mathbbR)\mathbf L_2(\mathbb R)mathbfL_2(mathbbR) wavelets. 4) We use both the realizations for MMM and the...