by Dmitri Betaneli.; Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1998.; Includes bibliographical references (p. 119-125).

Point sources in complex spacetime, which generate acoustic and electromagnetic pulsed-beam wavelets, are rigorously defined and computed with a view toward their realization.; Comment: 20 pages. Invited talk, "Multiscale Geometric Analysis" workshop, < http://www.ipam.ucla.edu/programs/mga2003/>

We give asymptotic approximations of the zeros of certain high degree polynomials. The zeros can be used to compute the filter coefficients in the dilation equations which define the compactly supported orthogonal Daubechies wavelets. Computational schemes are presented to obtain the numerical values of the zeros within high precision.

We give a complete characterization of the classes of weight functions for which the Haar wavelet system for $m$-dilations, $m= 2,3,\ldots$ is an unconditional basis in $L^p(\mathbb{R},w)$. Particulary it follows that higher rank Haar wavelets are unconditional bases in the weighted norm spaces $L^p(\mathbb{R},w)$, where $w(x) = |x|^{r}, r>p-1$. These weights can have very strong zeros at the origin. Which shows that the class of weight functions for which higher rank Haar wavelets are unconditional bases is much richer than it was supposed. One of main purposes of our study is to show that weights with strong zeros should be considered if somebody is studying basis properties of a given wavelet system in a weighted norm space.

Huygens' principle has a well-known problem with back-propagation due to the spherical nature of the secondary wavelets. We solve this by analytically continuing the surface of integration. If the surface is a sphere of radius $R$, this is done by complexifying $R$ to $R+ia$. The resulting complex sphere is shown to be a real bundle of disks with radius $a$ tangent to the sphere. Huygens' "secondary source points" are thus replaced by disks, and his spherical wavelets by well-focused pulsed beams propagating outward. This solves the back-propagation problem. The extended Huygens principle is a completeness relation for pulsed beams, giving a representation of a general radiation field as a superposition of such beams. Furthermore, it naturally yields a very efficient way to compute radiation fields because all pulsed beams missing a given observer can be ignored. Increasing $a$ sharpens the focus of the pulsed beams, which in turn raises the compression of the representation.; Comment: 49 pages, 14 figures

We introduce an equivalence relation on the set of single wavelets of L^2(R^n) associated with an arbitrary dilation matrix. The corresponding equivalence classes are characterized in terms of the support of the Fourier transform of wavelets and it is shown that each of these classes is non-empty.; Comment: 9 pages

We study the harmonic analysis of the quadrature mirror filters coming from multiresolution wavelet analysis of compactly supported wavelets. It is known that those of these wavelets that come from third order polynomials are parametrized by the circle, and we compute that the corresponding filters generate irreducible mutually disjoint representations of of the Cuntz algebra $ O_{2} $ except at two points on the circle. One of the two exceptional points corresponds to the Haar wavelet and the other is the unique point on the circle where the father function defines a tight frame which is not an orthonormal basis. At these two points the representation decomposes into two and three mutually disjoint irreducible representations, respectively, and the two representations at the Haar point are each unitarily equivalent to one of the three representations at the other singular point.; Comment: AMS-LaTeX; 29 pages, 6 figures incorporating 23 EPS diagrams and 1 LaTeX picture

In this paper we deal with the regression problem in a random design setting. We investigate asymptotic optimality under minimax point of view of various Bayesian rules based on warped wavelets and show that they nearly attain optimal minimax rates of convergence over the Besov smoothness class considered. Warped wavelets have been introduced recently, they offer very good computable and easy-to-implement properties while being well adapted to the statistical problem at hand. We particularly put emphasis on Bayesian rules leaning on small and large variance Gaussian priors and discuss their simulation performances comparing them with a hard thresholding procedure.

All wavelets can be associated to a multiresolution like structure, i.e. an incr easing sequence of subspaces of L^2(R). We consider the interaction of a wavel et and the translation operator in terms of which of the subspaces in this multi resolution like structure are invariant under the translation operator. This ac tion defines the notion of the translation invariance property of order n. In this paper we show that wavelets of all levels of translation invariance exist, first for the classic case of dilation by 2, and then for arbitrary integral di lation factors.; Comment: 17 pages; AMS-Latex

Perfect reconstruction filter banks can be used to generate a variety of wavelet bases. Using IIR linear phase filters one can obtain symmetry properties for the wavelet and scaling functions. In this paper we describe all possible IIR linear phase filters generating symmetric wavelets with any prescribed number of vanishing moments. In analogy with the well known FIR case, we construct and study a new family of wavelets obtained by considering maximal number of vanishing moments for each fixed order of the IIR filter. Explicit expressions for the coefficients of numerator, denominator, zeroes, and poles are presented. This new parameterization allows one to design linear phase quadrature mirror filters with many other properties of interest such as filters that have any preassigned set of zeroes in the stopband or that satisfy an almost interpolating property. Using Beylkin's approach, it is indicated how to implement these IIR filters not as recursive filters but as FIR filters.

We show how fundamental ideas from signal processing, multiscale theory and wavelets may be applied to non-linear dynamics. The problems from dynamics include iterated function systems (IFS), dynamical systems based on substitution such as the discrete systems built on rational functions of one complex variable and the corresponding Julia sets, and state spaces of subshifts in symbolic dynamics. Our paper serves to motivate and survey our recent results in this general area. Hence we leave out some proofs, but instead add a number of intuitive ideas which we hope will make the subject more accessible to researchers in operator theory and systems theory.; Comment: survey. v2: We have polished the writing and corrected some of the cross references and citations; and v2 has an acknowledgment paragraph added. Moreover, the ms has been converted to Birkhauser/OT style files. v3: added discussion of general themes of operator theory and how they complement the particular structure arising from multiscale theory, p. 22; regularized subsection numbering throughout; corrected some misspellings

We define a set of operators that localise a radial image in radial space and radial frequency simultaneously. We find the eigenfunctions of this operator and thus define a non-separable orthogonal set of radial wavelet functions that may be considered optimally concentrated over a region of radial space and radial scale space, defined via a doublet of parameters. We give analytic forms for their energy concentration over this region. We show how the radial function localisation operator can be generalised to an operator, localising any square integrable function in two dimensional Euclidean space. We show that the latter operator, with an appropriate choice of localisation region, approximately has the same eigenfunctions as the radial operator. Based on the radial wavelets we define a set of quaternionic valued wavelet functions that can extract local orientation for discontinuous signals and both orientation and phase structure for oscillatory signals. The full set of quaternionic wavelet functions are component wise orthogonal; hence their statistical properties are tractable, and we give forms for the variability of the estimates of the local phase and orientation, as well as the local energy of the image. By averaging estimates across wavelets...

A wavelet is a special case of a vector in a separable Hilbert space that generates a basis under the action of a collection, or system, of unitary operators. We will describe the operator-interpolation approach to wavelet theory using the local commutant of a system. This is really an abstract application of the theory of operator algebras to wavelet theory. The concrete applications of this method include results obtained using specially constructed families of wavelet sets. A frame is a sequence of vectors in a Hilbert space which is a compression of a basis for a larger space. This is not the usual definition in the frame literature, but it is easily equivalent to the usual definition. Because of this compression relationship between frames and bases, the unitary system approach to wavelets (and more generally: wandering vectors) is perfectly adaptable to frame theory. The use of the local commutant is along the same lines as in the wavelet theory. Finally, we discuss constructions of frames with special properties using targeted decompositions of positive operators, and related problems.; Comment: This is a semi-expository article based on a series of tutorial talks given by the author as part of the "Workshop on Functional and Harmonic Analyses of Wavelets and Frames" held Aug 4-7...

In this paper a simple method of construction of scaling function $\phi (x)$ and orthogonal wavelets with the compact support for any natural coefficient of scaling $N\ge 2$ is given. Examples of construction of wavelets for coefficients of scaling N=2 and N=3 are produced.; Comment: LaTeX2e, 15 pages

A generalisation of the Shannon complex wavelet is introduced, which is related to raised cosine filters. This approach is used to derive a new family of orthogonal complex wavelets based on the Nyquist criterion for Intersymbolic Interference (ISI) elimination. An orthogonal Multiresolution Analysis (MRA) is presented, showing that the roll-off parameter should be kept below 1/3. The pass-band behaviour of the Wavelet Fourier spectrum is examined. The left and right roll-off regions are asymmetric; nevertheless the Q-constant analysis philosophy is maintained. Finally, a generalisation of the (square root) raised cosine wavelets is proposed.; Comment: 8 pages, 14 figures

Let $\mathbb{H}$ be the three-dimensional Heisenberg group. We introduce a structure on the Heisenberg group which consists of the biregular representation of $\mathbb{H\times H}$ restricted to some discrete subset of $\mathbb{H\times H}$ and a free group of automorphisms $H$ singly generated and acting semi-simply on $\mathbb{H}$. Using well-known theorems borrowed from Gabor theory, we are able to construct simple and computable bandlimited discrete wavelets on the Heisenberg group. Moreover, we provide necessary and sufficient conditions for the existence of these wavelets.

In this paper, we describe some recent results obtained in the context of vector subdivision schemes which possess the so-called full rank property. Such kind of schemes, in particular those which have an interpolatory nature, are connected to matrix refinable functions generating orthogonal multiresolution analyses for the space of vector-valued signals. Corresponding multichannel (matrix) wavelets can be defined and their construction in terms of a very efficient scheme is given. Some examples illustrate the nature of these matrix scaling functions/wavelets.

Wavelets, known to be useful in non-linear multi-scale processes and in multi-resolution analysis, are shown to have a q-deformed algebraic structure. The translation and dilation operators of the theory associate with any scaling equation a non-linear, two parameter algebra. This structure can be mapped onto the quantum group $su_{q}(2)$ in one limit, and approaches a Fourier series generating algebra, in another limit. A duality between any scaling function and its corresponding non-linear algebra is obtained. Examples for the Haar and B-wavelets are worked out in detail.; Comment: 27 pages Latex, 3 figure ps

The objective of this work is to develop a systematic method of implementing the Wavelet-Galerkin method for approximating solutions of differential equations. The beginning of this project included understanding what a wavelet is, and then becoming familiar with some of the applications. The Wavelet-Galerkin method, as applied in this paper, does not use a wavelet at all. In actuality, it uses the wavelet's scaling function. The distinction between the two will be given in the following sections of this paper. The sections of this thesis will include defining wavelets and their scaling functions. This will give the reader valued insight to wavelets and Discrete Wavelet Transforms (DWT). Following this will be a section defining the Galerkin method. The purpose of this section will be to give the reader an understanding of how weighted residual methods work, in particular, the Galerkin Method. Next will be a section on how Scaling functions will be implemented in the Galerkin method, forming the Wavelet-Galerkin Method. The focus of this investigation will deal with solutions to a basic homogeneous differential equation. The solution of this basic equation will be analyzed using three separate, distinct methods, and then the results will be compared. These methods include the Wavelet-Galerkin Method...