**Author**: Yves Meyer

**Publisher:** Cambridge University Press

**ISBN:**

**Category:** Mathematics

**Page:** 244

**View:** 558

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## Wavelets and Operators:

Over the last two years, wavelet methods have shown themselves to be of considerable use to harmonic analysts and, in particular, advances have been made concerning their applications. The strength of wavelet methods lies in their ability to describe local phenomena more accurately than a traditional expansion in sines and cosines can. Thus, wavelets are ideal in many fields where an approach to transient behaviour is needed, for example, in considering acoustic or seismic signals, or in image processing. Yves Meyer stands the theory of wavelets firmly upon solid ground by basing his book on the fundamental work of Calderón, Zygmund and their collaborators. For anyone who would like an introduction to wavelets, this book will prove to be a necessary purchase.
## Wavelets in Geophysics

Applications of wavelet analysis to the geophysical sciences grew from Jean Morlet's work on seismic signals in the 1980s. Used to detect signals against noise, wavelet analysis excels for transients or for spatiallylocalized phenomena. In this fourth volume in the renown WAVELET ANALYSIS AND ITS APPLICATIONS Series, Efi Foufoula-Georgiou and Praveen Kumar begin with a self-contained overview of the nature, power, and scope of wavelet transforms. The eleven originalpapers that follow in this edited treatise show how geophysical researchers are using wavelets to analyze such diverse phenomena as intermittent atmospheric turbulence, seafloor bathymetry, marine and other seismic data, and flow in aquifiers. Wavelets in Geophysics will make informative reading for geophysicists seeking an up-to-date account of how these tools are being used as well as for wavelet researchers searching for ideas for applications, or even new points of departure. Key Features * Includes twelve original papers written by experts in the geophysical sciences * Provides a self-contained overview of the nature, power, and scope of wavelet transforms * Presents applications of wavelets to geophysical phenomena such as: * The sharp events of seismic data *Long memory processes, such as fluctuation in the level of the Nile * A structure preserving decomposition of turbulence signals
## Operator Methods in Wavelets, Tilings, and Frames

This volume contains the proceedings of the AMS Special Session on Harmonic Analysis of Frames, Wavelets, and Tilings, held April 13-14, 2013, in Boulder, Colorado. Frames were first introduced by Duffin and Schaeffer in 1952 in the context of nonharmonic Fourier series but have enjoyed widespread interest in recent years, particularly as a unifying concept. Indeed, mathematicians with backgrounds as diverse as classical and modern harmonic analysis, Banach space theory, operator algebras, and complex analysis have recently worked in frame theory. Frame theory appears in the context of wavelets, spectra and tilings, sampling theory, and more. The papers in this volume touch on a wide variety of topics, including: convex geometry, direct integral decompositions, Beurling density, operator-valued measures, and splines. These varied topics arise naturally in the study of frames in finite and infinite dimensions. In nearly all of the papers, techniques from operator theory serve as crucial tools to solving problems in frame theory. This volume will be of interest not only to researchers in frame theory but also to those in approximation theory, representation theory, functional analysis, and harmonic analysis.
## Selected Papers on Analysis and Differential Equations

This volume contains translations of papers that originally appeared in the Japanese journal ""Sugaku"". The papers range over a variety of topics, including nonlinear partial differential equations, $C^*$-algebras, and Schrodinger operators. The volume is suitable for graduate students and research mathematicians interested in analysis and differential equations.
## A Course on Rough Paths

Lyons’ rough path analysis has provided new insights in the analysis of stochastic differential equations and stochastic partial differential equations, such as the KPZ equation. This textbook presents the first thorough and easily accessible introduction to rough path analysis. When applied to stochastic systems, rough path analysis provides a means to construct a pathwise solution theory which, in many respects, behaves much like the theory of deterministic differential equations and provides a clean break between analytical and probabilistic arguments. It provides a toolbox allowing to recover many classical results without using specific probabilistic properties such as predictability or the martingale property. The study of stochastic PDEs has recently led to a significant extension – the theory of regularity structures – and the last parts of this book are devoted to a gentle introduction. Most of this course is written as an essentially self-contained textbook, with an emphasis on ideas and short arguments, rather than pushing for the strongest possible statements. A typical reader will have been exposed to upper undergraduate analysis courses and has some interest in stochastic analysis. For a large part of the text, little more than Itô integration against Brownian motion is required as background.
## Frames and Operator Theory in Analysis and Signal Processing

This volume contains articles based on talks presented at the Special Session Frames and Operator Theory in Analysis and Signal Processing, held in San Antonio, Texas, in January of 2006. Recently, the field of frames has undergone tremendous advancement. Most of the work in this field is focused on the design and construction of more versatile frames and frames tailored towards specific applications, e.g., finite dimensional uniform frames for cellular communication. In addition, frames are now becoming a hot topic in mathematical research as a part of many engineering applications, e.g., matching pursuits and greedy algorithms for image and signal processing. Topics covered in this book include: Application of several branches of analysis (e.g., PDEs; Fourier, wavelet, and harmonic analysis; transform techniques; data representations) to industrial and engineering problems, specifically image and signal processing. Theoretical and applied aspects of frames and wavelets. Pure aspects of operator theory emphasizing the connections to applied mathematics, frames, and signal processing. This volume will be equally attractive to pure mathematicians working on the foundations of frame and operator theory and their interconnections as it will to applied mathematicians investigating applications and to physicists and engineers employing these designs. It thus may appeal to a wide target group of researchers and may serve as a catalyst for cross-fertilization of several important areas of mathematics and the applied sciences.
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