Applied and Numerical Harmonic Analysis
Volume 68 / 2015
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Book Series
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Book
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Chapter and Conference Paper
A Review of Univariate and Multivariate Multifractal Analysis Illustrated by the Analysis of Marathon Runners Physiological Data
We review the central results concerning wavelet methods in multifractal analysis, which consists in analysis of the pointwise singularities of a signal, and we describe its recent extension to multivariate mu...
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Chapter
The Unreasonable Effectiveness of Haar Frames
Alex Grossmann used to point out the relevance of redundant frame decompositions as opposed to the economical setting supplied by orthonormal bases. We investigate here another unexpected occurrence of the rel...
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Book
New Trends in Applied Harmonic Analysis, Volume 2
Harmonic Analysis, Geometric Measure Theory, and Applications
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Article
Multifractal analysis of the Brjuno function
The Brjuno function B is a 1-periodic, nowhere locally bounded function, introduced by Yoccoz because it encapsulates a key information concerning analytic small divisor problems in dimension 1. We show that ...
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Chapter and Conference Paper
New Exponents for Pointwise Singularity Classification
We introduce new tools for pointwise singularity classification: We investigate the properties of the two-variable function which is defined at every point as the p-exponent of a fractional integral of order t; n...
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Chapter and Conference Paper
Multifractal Analysis Based on p-Exponents and Lacunarity Exponents
Many examples of signals and images cannot be modeled by locally bounded functions, so that the standard multifractal analysis, based on the Hölder exponent, is not feasible. We present a multifractal analysis...
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Chapter
Multivariate Davenport Series
We consider series of the form ∑a n {n ⋅x}, where n ∈ Z d and {x} is the sawtooth function. They are the natural multivar...
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Article
Multifractal analysis of Lévy fields
We study the pointwise regularity properties of the Lévy fields introduced by T. Mori; these fields are the most natural generalization of Lévy processes to the multivariate setting. We determine their spectru...
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Chapter and Conference Paper
Function Spaces Vs. Scaling Functions: Tools for Image Classification
We investigate the properties of several classes of new parameters issued from multifractal analysis and used in image analysis and classification. They share the following common characteristics: They are der...
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Chapter
Space-Filling Functions and Davenport Series
In this paper, we study the pointwise Hölder regularity of some spacefilling functions. In particular, we give some general results concerning the pointwise regularity of the Davenport series.
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Chapter
On the impact of the number of vanishing moments on the dependence structures of compound Poisson motion and fractional Brownian motion in multifractal time
From a theoretical perspective, scale invariance, or simply scaling, can fruitfully be modeled with classes of multifractal stochastic processes, designed from positive multiplicative martingales (or cascades)...
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Chapter and Conference Paper
Wavelet Decomposition of Measures: Application to Multifractal Analysis of Images
We show the relevance of multifractal analysis for some problems in image processing. We relate it to the standard question of the determination of correct function space settings. We show why a scale-invarian...
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Chapter and Conference Paper
Wavelet Leaders in Multifractal Analysis
The properties of several multifractal formalisms based on wavelet coefficients are compared from both mathematical and numerical points of view. When it is based directly on wavelet coefficients, the multifra...
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Chapter and Conference Paper
MULTIFRACTAL ANALYSIS OF IMAGES: NEW CONNEXIONS BETWEEN ANALYSIS AND GEOMETRY
Natural images can be modelled as patchworks of homogeneous textures with rough contours. The following stages play a key role in their analysis:
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Article
Wavelet Analysis of Fractal Boundaries. Part 2: Multifractal Analysis
This second part deals with the global analysis of the boundary of domains . We develop methods for determining the dimensions of the sets where the local behaviors introduced ...
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Article
Wavelet Analysis of Fractal Boundaries. Part 1: Local Exponents
Let be a domain of . In Part 1 of this paper, we introduce new tools in order to analyse the local behavior of the boundary of ...
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Article
Beyond Besov Spaces, Part 2: Oscillation Spaces
We study several extensions of Besov spaces; these extensions include the oscillation spaces Ops,s’which take into account correlations between the positions of large wavelet coefficients through the scales...
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Article
Beyond Besov Spaces Part 1: Distributions of Wavelet Coefficients
We determine which information can be extracted from the distributions of the wavelet coefficients of a function f at each scale, but does not depend on the particular wavelet basis which is chosen. This informat...
