Copy of 52572517-An-Introduction-to-Wavelets-Through-Linear-Algebra-2001-Michael-W-Frazier (original) (raw)
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The mathematical theory of wavelets
… analysis—a celebration (Il Ciocco, 2000), 2001
We present an overview of some aspects of the mathematical theory of wavelets. These notes are addressed to an audience of mathematicians familiar with only the most basic elements of Fourier Analysis. The material discussed is quite broad and covers several topics involving wavelets. Though most of the larger and more involved proofs are not included, complete references to them are provided. We do, however, present complete proofs for results that are new (in particular, this applies to a recently obtained characterization of "all" wavelets in section 4).
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.
1993
The past ten years have seen an explosion of research in the theory of wavelets and their applications. Theoretical accomplishments include development of new bases for many different function spaces and the characterization of orthonormal wavelets with compact support. Applications span the fields of signal processing, image processing and compression, data compression, and quantum mechanics. At the present time however, much of the literature remains highly mathematical, and consequently, a large investment of time is often necessary to develop a general understanding of wavelets and their potential uses. This paper thus seeks to provide an overview of the wavelet transform from an intuitive standpoint. Throughout the paper a signal processing frame of reference is adopted.
Ju n 20 19 System theory and orthogonal multi-wavelets
2019
In this paper we provide a complete and unifying characterization of compactly supported univariate scalar orthogonal wavelets and vector-valued or matrix-valued orthogonal multi-wavelets. This characterization is based on classical results from system theory and basic linear algebra. In particular, we show that the corresponding wavelet and multi-wavelet masks are identified with a transfer function F (z) = A+Bz(I −Dz) C, z ∈ D = {z ∈ C : |z| < 1}, of a conservative linear system. The complex matrices A, B, C, D define a block circulant unitary matrix. Our results show that there are no intrinsic differences between the elegant wavelet construction by Daubechies or any other construction of vector-valued or matrix-valued multiwavelets. The structure of the unitary matrix defined by A, B, C, D allows us to parametrize in a systematic way all classes of possible wavelet and multi-wavelet masks together with the masks of the corresponding refinable functions.
C A ] 2 4 Fe b 20 04 Measures in wavelet decompositions
2003
In applications, choices of orthonormal bases in Hilbert space H may come about from the simultaneous diagonalization of some specific abelian algebra of operators. This is the approach of quantum theory as suggested by John von Neumann; but as it turns out, much more recent constructions of bases in wavelet theory, and in dynamical systems, also fit into this scheme. However, in these modern applications, the basis typically comes first, and the abelian algebra might not even be made explicit. It was noticed recently that there is a certain finite set of non-commuting operators Fi, first introduced by engineers in signal processing, which helps to clarify this connection, and at the same time throws light on decomposition possibilities for wavelet packets used in pyramid algorithms. There are three interrelated components to this: an orthonormal basis, an abelian algebra, and a projection-valued measure. While the operators Fi were originally intended for quadrature mirror filters ...
On the Computation of Battle-Lemarie's Wavelets
Mathematics of Computation, 1994
We propose a matrix approach to the computation of Battle-Lemarié's wavelets. The Fourier transform of the scaling function is the product of the inverse F(x) of a square root of a positive trigonometric polynomial and the Fourier transform of a B-spline of order m. The polynomial is the symbol of a bi-infinite matrix B associated with a B-spline of order 2m. We approximate this bi-infinite matrix B2m by its finite section As , a square matrix of finite order. We use As to compute an approximation \s of x whose discrete Fourier transform is F(x). We show that xs converges pointwise to x exponentially fast. This gives a feasible method to compute the scaling function for any given tolerance. Similarly, this method can be used to compute the wavelets.