Wavelet Methods for Pointwise Regularity and Local Oscillations of Functions
Author: Stéphane Jaffard
Publisher:
Published: 1996
Total Pages: 110
ISBN-13: 9781470401726
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Wavelet Methods for Pointwise Regularity and Local Oscillations of Functions
Author: Stphane Jaffard
Publisher: American Mathematical Soc.
Published: 1996-10-29
Total Pages: 132
ISBN-13: 9780821863107
DOWNLOAD EBOOK →Currently, new trends in mathematics are emerging from the fruitful interaction between signal processing, image processing, and classical analysis. One example is given by ``wavelets'', which incorporate both the know-how of the Calderon-Zygmund school and the efficiency of some fast algorithms developed in signal processing (quadrature mirror filters and pyramidal algorithms.) A second example is ``multi-fractal analysis''. The initial motivation was the study of fully developed turbulence and the introduction by Frisch and Parisi of the multi-fractal spectrum. Multi-fractal analysis provides a deeper insight into many classical functions in mathematics. A third example--``chirps''--is studied in this book. Chirps are used in modern radar or sonar technology. Once given a precise mathematical definition, chirps constitute a powerful tool in classical analysis. In this book, wavelet analysis is related to the 2-microlocal spaces discovered by J. M. Bony. The authors then prove that a wavelet based multi-fractal analysis leads to a remarkable improvement of Sobolev embedding theorem. In addition, they show that chirps were hidden in a celebrated Riemann series. Features: Provides the reader with some basic training in new lines of research. Clarifies the relationship between pointwise behavior and size properties of wavelet coefficents.
Wavelet Methods for Pointwise Regularity and Local Oscillations of Functions
Author: Stéphane Jaffard
Publisher: American Mathematical Soc.
Published: 1996
Total Pages: 127
ISBN-13: 0821804758
DOWNLOAD EBOOK →We investigate several topics related to the local behavior of functions: pointwise Hölder regularity, local scaling invariance and very oscillatory "chirp-like" behaviors. Our main tool is to relate these notions to two-microlocal conditions which are defined either on the Littlewood-Paley decomposition or on the wavelet transform. We give characterizations and the main properties of these two-microlocal spaces and we give several applications, such as bounds on the dimension of the set of Hölder singularities of a function, Sobolev regularity of trace functions, and chirp expansions of specific functions.
Wavelet Methods In Mathematical Analysis And Engineering
Author: Alain Damlamian
Publisher: World Scientific
Published: 2010-09-21
Total Pages: 190
ISBN-13: 9814464058
DOWNLOAD EBOOK →This book gives a comprehensive overview of both the fundamentals of wavelet analysis and related tools, and of the most active recent developments towards applications. It offers a state-of-the-art in several active areas of research where wavelet ideas, or more generally multiresolution ideas have proved particularly effective.The main applications covered are in the numerical analysis of PDEs, and signal and image processing. Recently introduced techniques such as Empirical Mode Decomposition (EMD) and new trends in the recovery of missing data, such as compressed sensing, are also presented. Applications range for the reconstruction of noisy or blurred images, pattern and face recognition, to nonlinear approximation in strongly anisotropic contexts, and to the classification tools based on multifractal analysis.
Analysis, Applications, and Computations
Author: Uwe Kähler
Publisher: Springer Nature
Published: 2023-12-01
Total Pages: 696
ISBN-13: 3031363752
DOWNLOAD EBOOK →This volume contains the contributions of the participants of the 13th International ISAAC Congress 2021, held in Ghent, Belgium. The papers, written by respected international experts, address recent results in mathematics, with a special focus on analysis. The volume provides to both specialists and non-specialists an excellent source of information on current research in mathematical analysis and its various interdisciplinary applications.
Scaling, Fractals and Wavelets
Author: Patrice Abry
Publisher: John Wiley & Sons
Published: 2013-03-01
Total Pages: 382
ISBN-13: 1118622901
DOWNLOAD EBOOK →Scaling is a mathematical transformation that enlarges or diminishes objects. The technique is used in a variety of areas, including finance and image processing. This book is organized around the notions of scaling phenomena and scale invariance. The various stochastic models commonly used to describe scaling — self-similarity, long-range dependence and multi-fractals — are introduced. These models are compared and related to one another. Next, fractional integration, a mathematical tool closely related to the notion of scale invariance, is discussed, and stochastic processes with prescribed scaling properties (self-similar processes, locally self-similar processes, fractionally filtered processes, iterated function systems) are defined. A number of applications where the scaling paradigm proved fruitful are detailed: image processing, financial and stock market fluctuations, geophysics, scale relativity, and fractal time-space.
Fractal Geometry and Applications: A Jubilee of Benoit Mandelbrot
Author: Michel Laurent Lapidus
Publisher: American Mathematical Soc.
Published: 2004
Total Pages: 592
ISBN-13: 0821836382
DOWNLOAD EBOOK →This volume offers an excellent selection of cutting-edge articles about fractal geometry, covering the great breadth of mathematics and related areas touched by this subject. Included are rich survey articles and fine expository papers. The high-quality contributions to the volume by well-known researchers--including two articles by Mandelbrot--provide a solid cross-section of recent research representing the richness and variety of contemporary advances in and around fractal geometry. In demonstrating the vitality and diversity of the field, this book will motivate further investigation into the many open problems and inspire future research directions. It is suitable for graduate students and researchers interested in fractal geometry and its applications. This is a two-part volume. Part 1 covers analysis, number theory, and dynamical systems; Part 2, multifractals, probability and statistical mechanics, and applications.
Multifractional Stochastic Fields: Wavelet Strategies In Multifractional Frameworks
Author: Ayache Antoine
Publisher: World Scientific
Published: 2018-09-24
Total Pages: 236
ISBN-13: 9814525677
DOWNLOAD EBOOK →Fractional Brownian Motion (FBM) is a very classical continuous self-similar Gaussian field with stationary increments. In 1940, some works of Kolmogorov on turbulence led him to introduce this quite natural extension of Brownian Motion, which, in contrast with the latter, has correlated increments. However, the denomination FBM is due to a very famous article by Mandelbrot and Van Ness, published in 1968. Not only in it, but also in several of his following works, Mandelbrot emphasized the importance of FBM as a model in several applied areas, and thus he made it to be known by a wide community. Therefore, FBM has been studied by many authors, and used in a lot of applications.In spite of the fact that FBM is a very useful model, it does not always fit to real data. This is the reason why, for at least two decades, there has been an increasing interest in the construction of new classes of random models extending it, which offer more flexibility. A paradigmatic example of them is the class of Multifractional Fields. Multifractional means that fractal properties of models, typically, roughness of paths and self-similarity of probability distributions, are locally allowed to change from place to place.In order to sharply determine path behavior of Multifractional Fields, a wavelet strategy, which can be considered to be new in the probabilistic framework, has been developed since the end of the 90's. It is somehow inspired by some rather non-standard methods, related to the fine study of Brownian Motion roughness, through its representation in the Faber-Schauder system. The main goal of the book is to present the motivations behind this wavelet strategy, and to explain how it can be applied to some classical examples of Multifractional Fields. The book also discusses some topics concerning them which are not directly related to the wavelet strategy.
Theory of Function Spaces III
Author: Hans Triebel
Publisher: Springer Science & Business Media
Published: 2006-09-10
Total Pages: 433
ISBN-13: 3764375825
DOWNLOAD EBOOK →This volume presents the recent theory of function spaces, paying special attention to some recent developments related to neighboring areas such as numerics, signal processing, and fractal analysis. Local building blocks, in particular (non-smooth) atoms, quarks, wavelet bases and wavelet frames are considered in detail and applied to diverse problems, including a local smoothness theory, spaces on Lipschitz domains, and fractal analysis.
Computation and Applied Mathematics
