Guido Weiss

Bio: Guido Weiss is an academic researcher from Washington University in St. Louis. The author has contributed to research in topics: Wavelet & Orthonormal basis. The author has an hindex of 39, co-authored 86 publications receiving 14777 citations. Previous affiliations of Guido Weiss include Michigan State University & Autonomous University of Madrid.

Introduction to Fourier Analysis on Euclidean Spaces.

Abstract: The authors present a unified treatment of basic topics that arise in Fourier analysis. Their intention is to illustrate the role played by the structure of Euclidean spaces, particularly the action of translations, dilatations, and rotations, and to motivate the study of harmonic analysis on more general spaces having an analogous structure, e.g., symmetric spaces.

A First Course on Wavelets

Abstract: Wavelet theory had its origin in quantum field theory, signal analysis, and function space theory In these areas wavelet-like algorithms replace the classical Fourier-type expansion of a function This unique new book is an excellent introduction to the basic properties of wavelets, from background math to powerful applications The authors provide elementary methods for constructing wavelets, and illustrate several new classes of wavelets The text begins with a description of local sine and cosine bases that have been shown to be very effective in applications Very little mathematical background is needed to follow this material A complete treatment of band-limited wavelets follows These are characterized by some elementary equations, allowing the authors to introduce many new wavelets Next, the idea of multiresolution analysis (MRA) is developed, and the authors include simplified presentations of previous studies, particularly for compactly supported waveletsSome of the topics treated include:Several bases generated by a single function via translations and dilationsMultiresolution analysis, compactly supported wavelets, and spline waveletsBand-limited waveletsUnconditionality of wavelet basesCharacterizations of many of the principal objects in the theory of wavelets, such as low-pass filters and scaling functionsThe authors also present the basic philosophy that all orthonormal wavelets are completely characterized by two simple equations, and that most properties and constructions of wavelets can be developed using these two equations Material related to applications is provided, and constructions of splines wavelets are presented Mathematicians, engineers, physicists, and anyone with a mathematical background will find this to be an important text for furthering their studies on wavelets

Littlewood-Paley Theory and the Study of Function Spaces

Abstract: Calderon's formula and a decomposition of $L^2(\mathbb R^n)$ Decomposition of Lipschitz spaces Minimality of $\dot B^0,1_1$ Littlewood-Paley theory The Besov and Triebel-Lizorkin spaces The $\varphi$ -transform Wavelets Calderon-Zygmund operators Potential theory and a result of Muckenhoupt-Wheeden Further applications.

The Fractal Geometry of Nature

Abstract: "...a blend of erudition (fascinating and sometimes obscure historical minutiae abound), popularization (mathematical rigor is relegated to appendices) and exposition (the reader need have little knowledge of the fields involved) ...and the illustrations include many superb examples of computer graphics that are works of art in their own right." Nature

A wavelet tour of signal processing

Abstract: Introduction to a Transient World. Fourier Kingdom. Discrete Revolution. Time Meets Frequency. Frames. Wavelet Zoom. Wavelet Bases. Wavelet Packet and Local Cosine Bases. An Approximation Tour. Estimations are Approximations. Transform Coding. Appendix A: Mathematical Complements. Appendix B: Software Toolboxes.

De-noising by soft-thresholding

Abstract: Donoho and Johnstone (1994) proposed a method for reconstructing an unknown function f on [0,1] from noisy data d/sub i/=f(t/sub i/)+/spl sigma/z/sub i/, i=0, ..., n-1,t/sub i/=i/n, where the z/sub i/ are independent and identically distributed standard Gaussian random variables. The reconstruction f/spl circ/*/sub n/ is defined in the wavelet domain by translating all the empirical wavelet coefficients of d toward 0 by an amount /spl sigma//spl middot//spl radic/(2log (n)/n). The authors prove two results about this type of estimator. [Smooth]: with high probability f/spl circ/*/sub n/ is at least as smooth as f, in any of a wide variety of smoothness measures. [Adapt]: the estimator comes nearly as close in mean square to f as any measurable estimator can come, uniformly over balls in each of two broad scales of smoothness classes. These two properties are unprecedented in several ways. The present proof of these results develops new facts about abstract statistical inference and its connection with an optimal recovery model. >

Ideal spatial adaptation by wavelet shrinkage

Abstract: SUMMARY With ideal spatial adaptation, an oracle furnishes information about how best to adapt a spatially variable estimator, whether piecewise constant, piecewise polynomial, variable knot spline, or variable bandwidth kernel, to the unknown function. Estimation with the aid of an oracle offers dramatic advantages over traditional linear estimation by nonadaptive kernels; however, it is a priori unclear whether such performance can be obtained by a procedure relying on the data alone. We describe a new principle for spatially-adaptive estimation: selective wavelet reconstruction. We show that variable-knot spline fits and piecewise-polynomial fits, when equipped with an oracle to select the knots, are not dramatically more powerful than selective wavelet reconstruction with an oracle. We develop a practical spatially adaptive method, RiskShrink, which works by shrinkage of empirical wavelet coefficients. RiskShrink mimics the performance of an oracle for selective wavelet reconstruction as well as it is possible to do so. A new inequality in multivariate normal decision theory which we call the oracle inequality shows that attained performance differs from ideal performance by at most a factor of approximately 2 log n, where n is the sample size. Moreover no estimator can give a better guarantee than this. Within the class of spatially adaptive procedures, RiskShrink is essentially optimal. Relying only on the data, it comes within a factor log 2 n of the performance of piecewise polynomial and variableknot spline methods equipped with an oracle. In contrast, it is unknown how or if piecewise polynomial methods could be made to function this well when denied access to an oracle and forced to rely on data alone.

The wavelet transform, time-frequency localization and signal analysis

Abstract: Two different procedures for effecting a frequency analysis of a time-dependent signal locally in time are studied. The first procedure is the short-time or windowed Fourier transform; the second is the wavelet transform, in which high-frequency components are studied with sharper time resolution than low-frequency components. The similarities and the differences between these two methods are discussed. For both schemes a detailed study is made of the reconstruction method and its stability as a function of the chosen time-frequency density. Finally, the notion of time-frequency localization is made precise, within this framework, by two localization theorems. >