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Four Short Courses on Harmonic Analysis : Wavelets, Frames, Time-Frequency Methods, and Applications to Signal and Image Analysis / edited by Brigitte Forster, Peter Massopust
Contributor(s): Resource type: Ressourcentyp: Buch (Online)Book (Online)Language: English Series: Applied and Numerical Harmonic Analysis | SpringerLink BücherPublisher: Boston, MA : Birkhäuser Boston, 2010Description: Online-Ressource (XVIII, 249p. 36 illus, digital)ISBN:- 9780817648916
- 9781280391125
- 128039112X
- 515.785
- 515.2433 23
- QA403-403.3
- QA403.3
Contents:
Summary: Introduction: Mathematical Aspects of Time-Frequency Analysis -- B-Spline Generated Frames -- Continuous and Discrete Reproducing Systems That Arise from Translations. Theory and Applications of Composite Wavelets -- Wavelets on the Sphere -- Wiener’s Lemma: Theme and Variations. An Introduction to Spectral Invariance and Its Applications.Summary: This state-of-the-art textbook examines four research directions in harmonic analysis and features some of the latest applications in the field, including cosmic microwave background analysis, human cortex image denoising, and wireless communication. The work is the first one that combines spline theory (from a numerical or approximation-theoretical view), wavelets, frames, and time-frequency methods leading up to a construction of wavelets on manifolds other than Rn. Written by internationally renowned mathematicians, the interdisciplinary chapters are expository by design, enabling the reader to understand the theory behind modern image and signal processing methodologies. The main emphasis throughout the book is on the interdependence of the four modern research directions covered. Each chapter ends with exercises that allow for a more in-depth understanding of the material and are intended to stimulate the reader toward further research. A comprehensive index completes the work. Topics covered: * Frames and bases in mathematics and engineering * Wavelets with composite dilations and their applications * Wavelets on the sphere and their applications * Wiener's Lemma: theme and variations Four Short Courses on Harmonic Analysis is intended as a graduate-level textbook for courses or seminars on harmonic analysis and its applications. The work is also an excellent reference or self-study guide for researchers and practitioners with diverse mathematical backgrounds working in different fields such as pure and applied mathematics, image and signal processing engineering, mathematical physics, and communication theory.PPN: PPN: 1648759521Package identifier: Produktsigel: ZDB-2-SEB | ZDB-2-SXMS | ZDB-2-SMA
""ANHA Series Preface""; ""Preface""; ""Contents""; ""List of Contributors""; ""Chapter 1 Introduction: Mathematical Aspects of Time-Frequency Analysis""; ""1.1 Aims of Time-Frequency Analysis""; ""1.1.1 Signal and Model""; ""1.1.2 Transforms""; ""1.1.2.1 Fourier Series""; ""1.1.2.2 Convolution""; ""1.1.2.3 Haar System""; ""1.1.2.4 Rademacher System""; ""1.1.3 Signal Manipulations�Filters""; ""1.1.4 Why Discretizing? Techniques, Challenges, Pitfalls""; ""1.2 Basic Methods of Time-Frequency Analysis: Orthonormal Bases and Generalized Fourier Series""
""1.2.1 Schauder Bases in Banach Spaces""""1.2.1.1 Biorthogonality""; ""1.2.1.2 Unconditional Convergence""; ""1.2.2 Generalized Fourier Series""; ""1.3 The Fourier Integral Transform""; ""1.3.1 Definition and Properties""; ""1.3.2 The Plancherel Transform""; ""1.3.3 The Theorem of Paley�Wiener""; ""1.3.4 Discretization: The Poisson Summation Formula and the Sampling Theorem""; ""1.4 Windowed Fourier Transforms""; ""1.4.1 The Short-Time Fourier Transform (STFT)""; ""1.4.2 The Gabor Transform""
""1.4.2.1 The Gabor Transform as a Fourier and Plancherel Transformâ€?From the View point of the Variable Ï?""""1.4.2.2 The Gabor Transform from the View point of the Variable b : Window Translation""; ""1.4.2.3 The Window Parameter s and Time-Frequency Localization""; ""1.4.3 The Heisenberg Uncertainty Principle""; ""1.4.4 Discretization: Gabor Frames""; ""1.4.5 Shortcomings of the Windowed Fourier Transform""; ""1.5 The Wavelet Transform""; ""1.5.1 Definition and Properties""; ""1.5.2 Scale Discretizationâ€?The Dyadic Wavelet Transform""; ""1.5.3 Multiresolution Analyses""
""1.6 Other Multiscale Transforms""""1.6.1 Tensor Product Wavelets in 2D""; ""1.6.2 Some Wavelet-Type Transforms""; ""1.6.3 Moving to Other Manifolds�Wavelets on the Sphere""; ""Exercises""; ""Chapter 2 B-Spline Generated Frames""; ""2.1 Introduction""; ""2.2 Bessel Sequences in Hilbert Spaces""; ""2.3 General Bases and Orthonormal Bases""; ""2.4 Riesz Bases""; ""2.5 Frames and Their Properties""; ""2.6 Frames and Riesz Bases""; ""2.7 B-Splines""; ""2.8 Frames of Translates""; ""2.9 Basic Gabor Frame Theory""; ""2.10 Tight Gabor Frames""; ""2.11 The Duals of a Gabor Frame""
""2.12 Explicit Construction of Dual Gabor Frame Pairs""""2.13 Wavelets and the Unitary Extension Principle""; ""Exercises""; ""Chapter 3 Continuous and Discrete Reproducing Systems That Arise from Translations. Theory and Applications of Composite Wavelets""; ""3.1 Introduction""; ""3.2 Unified Theory of Reproducing Systems""; ""3.2.1 Unified Theorem for Reproducing Systems""; ""3.3 Continuous Wavelet Transform""; ""3.3.1 Admissible Groups""; ""3.3.2 Wave Packet Systems""; ""3.4 Affine Systems with Composite Dilations""; ""3.4.1 Affine System with Composite Dilations""
""3.4.2 Other Examples""
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