Barral / Seuret Recent Developments in Fractals and Related Fields

1. Auflage 2010
ISBN: 978-0-8176-4888-6
Verlag: Birkhäuser Basel
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 419 Seiten

The Applied and Numerical Harmonic Analysis (ANHA) book series aims to provide the engineering, mathematical, and scienti?c communities with s- ni?cant developments in harmonic analysis, ranging from abstract harmonic analysis to basic applications. The title of the series re?ects the importance of applications and numerical implementation, but richness and relevance of applications and implementation depend fundamentally on the structure and depth of theoretical underpinnings. Thus, from our point of view, the int- leaving of theory and applications and their creative symbiotic evolution is axiomatic. Harmonic analysis is a wellspring of ideas and applicability that has ?o- ished, developed, and deepened over time within many disciplines and by means of creative cross-fertilizationwith diverse areas. The intricate and f- damental relationship between harmonic analysis and ?elds such as signal processing, partial di?erential equations (PDEs), and image processing is - ?ected in our state-of-the-art ANHA series. Our vision of modern harmonic analysis includes mathematical areas such as wavelet theory, Banach algebras, classical Fourier analysis, time-frequency analysis, and fractal geometry, as well as the diverse topics that impinge on them.

Autoren/Hrsg.

Barral, Julien

Seuret, Stéphane

Inhaltsverzeichnis

1;ANHA Series Preface
;7
2;Preface
;10
3;Contents
;12
4;List of Contributors
;15
5;Part I Geometric Measure Theory and Multifractals;19
5.1;Occupation Measure and Level Sets of the Weierstrass–Cellerier Function;20
5.1.1;1 Introduction;20
5.1.2;2 Notation and Preliminary Results;22
5.1.3;3 Main Results;24
5.1.4;4 Proof of Lemma 3;28
5.1.5;References;34
5.2;Space-Filling Functions and Davenport Series;36
5.2.1;1 Introduction;36
5.2.2;2 Definitions;36
5.2.2.1;2.1 Hölder Spaces and Hölder Exponents;37
5.2.2.2;2.2 Uniform Hölder Spaces and Strongly Monohölder Functions;38
5.2.3;3 The Peano Function;39
5.2.4;4 A Strong Monohölderianity Criterion;41
5.2.5;5 The Lebesgue Function;42
5.2.6;6 The Schoenberg Function;44
5.2.7;7 The Cantor Function;46
5.2.8;8 Hölder Exponent of p-adic Davenport Series;49
5.2.9;References;51
5.3;Dimensions and Porosities;52
5.3.1;1 Porous Sets;52
5.3.2;2 Porous Measures;54
5.3.3;3 Mean Porous Measures;56
5.3.4;References;59
5.4;On Upper Conical Density Results;61
5.4.1;1 Introduction;61
5.4.2;2 Notation and Preliminaries;61
5.4.3;3 Packing Type Measures;63
5.4.4;4 Measures with Positive Dimension;66
5.4.5;5 Purely Unrectifiable Measures;67
5.4.6;References;69
5.5;On the Dimension of Iterated Sumsets;71
5.5.1;1 Introduction and Statement of Results;71
5.5.2;2 Examples and Proof of the Main Result;73
5.5.2.1;2.1 Basic Facts;73
5.5.2.2;2.2 Examples for Hausdorff Dimension;74
5.5.2.3;2.3 Examples for Hausdorff and Box-Counting Dimensions;78
5.5.2.4;2.4 Proof of the Main Result;85
5.5.3;3 Plünnecke Estimates for Box-Counting Dimensions;86
5.5.4;References;88
5.6;Geometric Measures for Fractals;89
5.6.1;1 Introduction;89
5.6.2;2 Minkowski Content;90
5.6.3;3 Sequences of Signed Measures;92
5.6.4;4 Curvature Measures and Fractal Curvatures;94
5.6.4.1;4.1 Curvature Measures;94
5.6.4.2;4.2 Fractal Curvature Measures;96
5.6.5;5 Curvature Measures for Self-Similar Sets;97
5.6.6;6 Proof of Theorem 5.1;101
5.6.7;7 Generalizations;103
5.6.8;References;104
6;Part II Harmonic and Functional Analysis and, Signal Processing;106
6.1;A Walk from Multifractal Analysis to Functional Analysis with S Spaces, and Back;107
6.1.1;1 Introduction;107
6.1.1.1;1.1 Where do the Spaces S Come from?;107
6.1.1.2;1.2 Heuristic Description;108
6.1.1.3;1.3 Outline of this Chapter;108
6.1.2;2 S Metrizable Topology;109
6.1.2.1;2.1 Definitions and Notation;109
6.1.2.2;2.2 More Functional Analysis: p-Convexity;110
6.1.2.3;2.3 More Functional Analysis: Typical Properties;111
6.1.2.3.1;Case p0 = 1;111
6.1.2.3.2;Case p0<1;112
6.1.2.4;2.4 More Functional Analysis: Properties of the Dual;114
6.1.3;3 Contributions to Multifractal Analysis;115
6.1.3.1;3.1 Prevalent Properties;115
6.1.3.2;3.2 Quasi-Sure Properties;116
6.1.3.3;3.3 On the Maximal Spectrum in Locally p-Convex Spaces;117
6.1.3.4;3.4 Open Questions;119
6.1.4;References;120
6.2;Concentration of the Integral Norm of Idempotents;121
6.2.1;1 Introduction and Statement of Results;121
6.2.2;2 Uniform p-Concentration;125
6.2.3;3 Failure of Uniform 1-Concentration on Zq;130
6.2.4;4 The 2-Concentration on Measurable Sets;131
6.2.5;5 Improvement of Constants for p not an Even Integer;136
6.2.6;References;142
6.3;Le calcul symbolique dans certaines algèbres de type Sobolev;144
6.3.1;1 Généralités sur le calcul symbolique;144
6.3.1.1;1.1 Le cas des algèbres de fonctions;145
6.3.1.2;1.2 Le cas des algèbres régulières;145
6.3.1.3;1.3 Vers un calcul symbolique maximal?;147
6.3.1.4;1.4 Le cas des algèbres sur le cercle;147
6.3.2;2 Calcul symbolique dans les espaces de Sobolev;148
6.3.2.1;2.1 Définition et théorème principal;148
6.3.2.2;2.2 Reformulation du théorème principal;149
6.3.2.3;2.3 Preuve des corollaires 7 et 8;150
6.3.2.4;2.4 Preuve du théorème 2.3;150
6.3.2.4.1;Préliminaires;150
6.3.2.4.2;Normes équivalentes dans l'espace de Sobolev;151
6.3.2.4.3;La 2-variation d'une fonction ;151
6.3.2.4.4;Les détails de la preuve;152
6.3.3;3 Diverses Extensions;153
6.3.3.1;3.1 Le cas des espaces de Besov et de Lizorkin-Triebel;153
6.3.3.2;3.2 Le cas des espaces définis sur Rn;153
6.3.3.3;3.3 Le cas des espaces de Sobolev à valeurs vectorielles;155
6.3.4;4 Régularité du calcul symbolique;155
6.3.4.1;4.1 Calcul symbolique borné;155
6.3.4.2;4.2 Calcul symbolique de classe Cr;156
6.3.5;References;156
6.4;Lp-Norms and Fractal Dimensions of Continuous Function Graphs;158
6.4.1;1 Introduction;158
6.4.2;2 Operator Norms;161
6.4.3;3 Application to Box Dimension;166
6.4.4;4 Application to Several Fractal Indices;168
6.4.5;5 Examples;172
6.4.5.1;5.1 Riemann Functions;172
6.4.5.2;5.2 Weierstrass Functions;174
6.4.5.3;5.3 Fractal Sums of Pulses;174
6.4.6;References;176
6.5;Uncertainty Principles, Prolate Spheroidal Wave Functions, and Applications;178
6.5.1;1 Introduction;178
6.5.2;2 Uncertainty Principles and Prolate Spheroidal Wave Functions;180
6.5.2.1;2.1 Uncertainty Principles;180
6.5.2.2;2.2 Properties of the PSWFs and Their Eigenvalues;182
6.5.3;3 Computation of the PSWFs and Their Eigenvalues;184
6.5.3.1;3.1 Mathematical Preliminaries;184
6.5.3.2;3.2 A Classical Computational Method;186
6.5.3.3;3.3 Matrix Representation of Qc and PSWFs;187
6.5.3.4;3.4 A Quadrature Method for the Computation of the PSWFs;190
6.5.3.5;3.5 Examples;195
6.5.4;4 Computation of the Spectrum of High Frequency PSWFs;197
6.5.5;5 Applications of the PSWFs;199
6.5.5.1;5.1 PSWFs and Quality of Approximation;199
6.5.5.2;5.2 Exact Reconstruction of Band-Limited Functions by the PSWFs;200
6.5.5.3;5.3 Examples;201
6.5.6;References;202
6.6;2-Microlocal Besov Spaces;204
6.6.1;1 Introduction and Preliminaries;204
6.6.2;2 Characterization with Wavelets;208
6.6.3;3 The Local Spaces Bs,s'p,q(U)loc;210
6.6.3.1;3.1 Definition and Wavelet Characterization;210
6.6.3.2;3.2 Embeddings;211
6.6.3.3;3.3 The 2-Microlocal Domain;212
6.6.3.3.1;Acknowledgment;213
6.6.4;References;213
6.7;Refraction on Multilayers;215
6.7.1;1 Precise Measurement;215
6.7.2;2 Fuzzy Measurements and Algebra;216
6.7.3;References;218
6.8;Wavelet Shrinkage: From Sparsity and Robust Testing to Smooth Adaptation;219
6.8.1;1 Introduction;219
6.8.2;2 WaveShrink: Estimation by Sparse Transform and Thresholding;222
6.8.2.1;2.1 Background;222
6.8.2.2;2.2 WaveShrink by Soft Thresholding;223
6.8.3;3 From Non-Parametric Statistical Decision to Sparsity;224
6.8.3.1;3.1 Motivation;224
6.8.3.2;3.2 Detection of Random Signals with Unknown Distribution and Prior in White Gaussian Noise;225
6.8.3.3;3.3 Sparse Sequences and Detection Thresholds;227
6.8.3.4;3.4 Application to WaveShrink by Soft Thresholding and the Universal Detection Threshold;228
6.8.4;4 Smooth Adapted WaveShrink with Adapted Detection Thresholds;230
6.8.4.1;4.1 SSBS Functions;231
6.8.4.2;4.2 Adapted Detection Thresholds;235
6.8.4.3;4.3 Experimental Results;236
6.8.5;5 Conclusion;240
6.8.6;References;243
7;Part III Dynamical Systems and Analysis on Fractals;245
7.1;Simple Infinitely Ramified Self-Similar Sets;246
7.1.1;1 Introduction;246
7.1.1.1;1.1 A Few Remarks on Fractal Analysis;246
7.1.1.2;1.2 The Advantage of Self-Similarity;247
7.1.1.3;1.3 Contents of the Chapter;249
7.1.2;2 Self-Similar Sets of Low Complexity;249
7.1.2.1;2.1 Basic Definitions;249
7.1.2.2;2.2 Counting Neighbor Types to Measure Complexity;250
7.1.3;3 Symmetric Examples;252
7.1.3.1;3.1 Carpet and Gasket Constructions;252
7.1.3.2;3.2 Fractal m-gons;253
7.1.4;4 The Boundary Structure;255
7.1.4.1;4.1 Self-Similarity of the Intersection Sets;255
7.1.4.2;4.2 Intersection Sets as Minimal Cuts;256
7.1.5;5 Examples with Exact Overlap of Pieces;258
7.1.6;References;259
7.2;Quantitative Uniform Hitting in Exponentially Mixing Systems;261
7.2.1;1 Introduction;261
7.2.2;2 Probabilistic Setting;263
7.2.3;3 On the Exponentially Mixing Property;264
7.2.4;4 Weighted Borel–Cantelli Lemma;265
7.2.5;5 Fundamental Inequalities;266
7.2.5.1;5.1 Basic Inequalities;266
7.2.6;6 Proofs of Theorems;271
7.2.6.1;6.1 Proof of Theorem 2.1;271
7.2.6.2;6.2 Proof of Theorem 1.1;272
7.2.7;7 Gibbs Measures on Subshifts of Finite Type;272
7.2.8;References;275
7.3;Some Remarks on the Hausdorff and Spectral Dimension of V-Variable Nested Fractals;277
7.3.1;1 Introduction;277
7.3.2;2 The Model: Two Shapes and Their Random Mixing;278
7.3.2.1;2.1 Classical Models;278
7.3.2.2;2.2 The V-Variable Model;280
7.3.3;3 Hausdorff Dimension;281
7.3.4;4 Spectral Dimension;283
7.3.4.1;4.1 The Deterministic Case;283
7.3.4.2;4.2 The V-Variable Case;284
7.3.4.3;4.3 Construction of the Energy Form;284
7.3.4.4;4.4 Spectral Asymptotics;286
7.3.5;5 Generalization: V-Variable Nested Fractals;288
7.3.6;References;291
7.4;Cantor Boundary Behavior of Analytic Functions;293
7.4.1;1 Introduction;293
7.4.2;2 The Basic Setup;294
7.4.3;3 The Cantor Boundary Behavior;295
7.4.4;4 The Complex Weierstrass Functions;298
7.4.5;5 Cauchy Transform on Sierpinski Gasket ;300
7.4.6;References;304
7.5;Measures of Full Dimension on Self-Affine Graphs;305
7.5.1;1 Introduction;305
7.5.2;2 Notation and Background;308
7.5.3;3 Classical Self-Affine Graphs and Their Sofic Coding;310
7.5.3.1;3.1 McMullen–Bedford (1984);310
7.5.3.2;3.2 Przytycki and Urbanski (1989) and Urbanski (1990);311
7.5.3.3;3.3 Kamae (1986);314
7.5.4;References;317
8;Part IV Stochastic Processes and Random Fractals;319
8.1;A Process Very Similar to Multifractional Brownian Motion;320
8.1.1;1 Introduction and Statement of the Main Results;320
8.1.2;2 The Main Ideas of the Proofs;325
8.1.3;3 Some Technical Lemmas;332
8.1.4;References;334
8.2;Gaussian Fields Satisfying Simultaneous Operator Scaling Relations;336
8.2.1;1 Introduction;336
8.2.2;2 A First Example of a Field Satisfying Simultaneous Operator Scaling Properties;339
8.2.3;3 A General Approach;341
8.2.4;4 Existence Results;343
8.2.4.1;4.1 Real Diagonalizable Part of a Matrix;343
8.2.4.2;4.2 Assumptions on Matrices E1,…,Em;343
8.2.5;5 Reformulation of the Problem in Terms of Group Self-Similarity;344
8.2.6;6 Definition of the Desired Gaussian Field;346
8.2.6.1;6.1 Definition of a Suitable Spectral Density;346
8.2.6.2;6.2 Definition of Renormalization Directions;347
8.2.6.3;6.3 Proof of Theorem 1;349
8.2.7;References;350
8.3;On Randomly Placed Arcs on the Circle;351
8.3.1;1 Introduction and Statement of the Results;351
8.3.1.1;1.1 Size Properties of the Set E;353
8.3.1.2;1.2 Large Intersection Properties of the Set E;354
8.3.2;2 Proof of Theorems 1 and 2;356
8.3.3;3 Proof of Corollary 1;357
8.3.4;4 Proof of Corollary 2;357
8.3.5;5 Proof of Proposition 2;358
8.3.6;References;358
8.4;T-Martingales, Size Biasing, and Tree Polymer Cascades;360
8.4.1;1 Introduction;360
8.4.2;2 Background and Notation;361
8.4.2.1;2.1 Some Special Notation and Assumptions;363
8.4.3;3 T-Martingales and Size-Bias Theory;364
8.4.4;4 Asymptotic Polymer Path Free Energy-Type Calculations for Weak and Strong Disorder;367
8.4.5;5 Diffusive Limits Under Full Range of Weak Disorder;373
8.4.6;6 Vector Cascades;378
8.4.7;7 Related Directions in T-Martingale Theory ;385
8.4.7.1;Acknowledgments;386
8.4.8;References;386
9;Part V Combinatorics on Words;388
9.1;Univoque Numbers and Automatic Sequences;389
9.1.1;1 Introduction;389
9.1.2;2 A Class of Sequences Belonging to strict;390
9.1.3;3 Proof of Theorem 1;392
9.1.3.1;3.1 Proof of the First Assertion;392
9.1.3.2;3.2 The Sequence limk r(k)((u 0)) is Not Periodic;393
9.1.4;4 Automatic Sequences and the Sets and strict;393
9.1.5;5 Alphabets with More Than Two Letters;395
9.1.6;6 Conclusion;396
9.1.7;References;396
9.2;A Crash Look into Applications of Aperiodic Substitutive Sequences;398
9.2.1;1 Into the Past: The ``Gang of Five";398
9.2.2;2 Non-Bragg Diffraction in Systems with Aperiodic Order and Lebesgue Classification;399
9.2.3;3 Substitutive Sequences as Sources for Discrete Laplacians: Spectrum, Band Structure, and Eigenstates in Localization Studies;401
9.2.4;References;402
9.3;Invertible Substitutions with a Common Periodic Point;405
9.3.1;1 Introduction;405
9.3.2;2 Some Known Results Concerning Invertible Substitutions;406
9.3.2.1;2.1 Frequency of a Substitution, the Generating Matrix;406
9.3.2.2;2.2 Sturmian Words;407
9.3.3;3 Rauzy Fractals of Invertible Substitutions;408
9.3.4;4 Characterization of Rauzy Fractals Stepped Surface;409
9.3.4.1;4.1 The Stepped Surface;409
9.3.4.2;4.2 A Tiling Associated with the Stepped Surface;409
9.3.4.3;4.3 Invertible Substitutions with a Given Incidence Matrix;410
9.3.5;5 Proof of Theorem 1;411
9.3.6;References;412
9.4;Some Studies on Markov-Type Equations;414
9.4.1;1 Introduction;414
9.4.2;2 Proof of Theorem 1;418
9.4.3;3 The Structure of Solutions of Equation (1);421
9.4.4;References;422