WAVELETS, MULTISCALE SYSTEMS AND HYPERCOMPLEX ANALYSIS
CONTENTS TEODORESCU TRANSFORM DECOMPOSITION OF MULTIVECTOR FIELDS ON FRACTAL HYPERSURFACES 1 INTRODUCTION 2 PRELIMINARIES 2 1 CLI?ORD ALGEBRAS AND MULTIVECTORS 2 2 CLI?ORD ANALYSIS AND HARMONIC MULTIVECTOR ?ELDS 2 3 FRACTAL DIMENSIONS AND WHITNEY EXTENSION THEOREM 3 JUMP PROBLEM AND MONOGENIC EXTENSIONS 4 K-MULTIVECTORIAL CASE DYNKIN PROBLEM AND HARMONIC EXTENSION 5 EXAMPLE 5 1 THE CURVE OF B KATS 5 2 THE SURFACE ?? 5 3 THE FUNCTION U? 5 4 PROOF OF PROPERTIES A) E) REFERENCES METRIC DEPENDENT CLI?ORD ANALYSIS WITH APPLICATIONS TO WAVELET ANALYSIS 1 INTRODUCTION 2 THE METRIC DEPENDENT CLI?ORD TOOLBOX 2 1 TENSORS 2 2 FROM GRASSMANN TO CLI?ORD 2 3 EMBEDDINGS OF RM 2 4 FISCHER DUALITY AND FISCHER DECOMPOSITION 2 5 THE EULER AND ANGULAR DIRAC OPERATORS 2 6 SOLID G-SPHERICAL HARMONICS 2 7 THE G-FOURIER TRANSFORM 3 METRIC INVARIANT INTEGRATION THEORY 3 1 THE BASIC LANGUAGE OF CLI?ORD DI?ERENTIAL FORMS 3 2 ORTHOGONAL SPHERICAL MONOGENICS 3 2 1 THE CAUCHY-POMPEIU FORMULA 3 2 2 SPHERICAL MONOGENICS 4 THE RADIAL G-CLI?ORD-HERMITE POLYNOMIALS AND ASSOCIATED CCWT 4 1 THE RADIAL G-CLI?ORD-HERMITE POLYNOMIALS 4 2 THE G-CLI?ORD-HERMITE WAVELETS 4 3 THE G-CLI?ORD-HERMITE CONTINUOUS WAVELET TRANSFORM A HIERARCHICAL SEMI-SEPARABLE MOORE-PENROSE EQUATION SOLVER 1 INTRODUCTION 2 HSS REPRESENTATIONS 3 PRELIMINARIES 4 HSS ROW ABSORPTION PROCEDURE COMPLEXITY CALCULATION 5 AN HSS MOORE-PENROSE REDUCTION METHOD 6 DISCUSSION AND CONCLUSIONS ACKNOWLEDGEMENTS METHODS FROM MULTISCALE THEORY AND WAVELETS APPLIED TO NONLINEAR DYNAMICS 1 INTRODUCTION 2 CONNECTION TO SIGNAL PROCESSING AND WAVELETS 3 MOTIVATING EXAMPLES, NONLINEARITY MRAS IN GEOMETRY AND OPERATOR THEORY 3 1 SPECTRUM AND GEOMETRY: WAVELETS, TIGHT FRAMES, AND HILBERT SPACES ON JULIA SETS 3 1 1 BACKGROUND 3 1 2 WAVELET ?LTERS IN NONLINEAR MODELS 3 2 MULTIRESOLUTION ANALYSIS (MRA) 3 2 1 PYRAMID ALGORITHMS AND GEOMETRY 3 3 JULIA SETS FROM COMPLEX DYNAMICS 4 MAIN RESULTS 4 1 SPECTRAL DECOMPOSITION OF COVARIANT REPRESENTATIONS: PROJECTIVE LIMITS 5 REMARKS ON OTHER APPLICATIONS NONCOMMUTATIVE TRIGONOMETRY 1 INTRODUCTION 2 THE ?RST (ACTIVE) PERIOD 1966–1972 3 THE SECOND (INTERMITTENT) PERIOD 1973–1993 4 THE THIRD (MOST ACTIVE) PERIOD 1994–2005 5 RELATED WORK: DISCUSSION 6 NONCOMMUTATIVE TRIGONOMETRY: OUTLOOK 6 1 EXTENSIONS TO MATRIX AND OPERATOR ALGEBRAS 6 2 MULTISCALE SYSTEM THEORY, WAVELETS, ITERATIVE METHODS 6 3 QUANTUM MECHANICS STATIONARY RANDOM FIELDS OVER GRAPHS AND RELATED STRUCTURES 1 INTRODUCTION 2 SECOND-ORDER RANDOM ?ELDS 2 1 BASIC NOTIONS 2 2 SPATIAL RANDOM ?ELDS WITH ORTHOGONAL INCREMENTS 2 3 THE KARHUNEN REPRESENTATION 3 STATIONARITY OF RANDOM ?ELDS 3 1 GRAPHS, BUILDINGS AND THEIR ASSOCIATED POLYNOMIAL STRUCTURES 3 1 1 DISTANCE-TRANSITIVE GRAPHS AND CARTIER POLYNOMIALS 3 1 2 TRIANGLE BUILDINGS AND CARTWRIGHT POLYNOMIALS 3 2 STATIONARY RANDOM ?ELDS OVER HYPERGROUPS 3 3 ARNAUD-LETAC STATIONARITY MATRIX REPRESENTATIONS AND NUMERICAL COMPUTATIONS OF WAVELET MULTIPLIERS 1 WAVELET MULTIPLIERS 2 THE LANDAU-POLLAK-SLEPIAN OPERATOR 3 FRAMES IN HILBERT SPACES 4 MATRIX REPRESENTATIONS OF WAVELET MULTIPLIERS 5 NUMERICAL COMPUTATIONS OF WAVELET MULTIPLIERS THAN 2-DIMENSIONAL EUCLIDEAN GROUP WITH DILATIONS 1 INTRODUCTION 2 CLI?ORD ALGEBRA-VALUED ADMISSIBLE WAVELET TRANSFORM 3 EXAMPLES OF CLI?ORD ALGEBRA-VALUED ADMISSIBLE WAVELETS ACKNOWLEDGEMENT CLI?ORD ALGEBRA-VALUED ADMISSIBLE WAVELETS ASSOCIATED TO MORE