Fourier Analysis and Distributions: A First Course with Applications

Description

This comprehensive book offers an accessible introduction to Fourier analysis and distribution theory, blending classical mathematical theory with a wide range of practical applications. Designed for undergraduate and beginning Master's students in mathematics and engineering. Key Features: Balanced Approach: The book is structured to include both theoretical and application-based chapters, providing readers with a solid understanding of the fundamentals alongside real-world scenarios. Diverse Applications: Topics include Fourier series, ordinary differential equations, AC circuit calculations, heat and wave equations, digital signal processing, and image compression. These applications demonstrate the versatility of Fourier analysis in solving complex problems in engineering, physics, and computational sciences. Advanced Topics: The text covers convolution theorems, linear filters, the Shannon Sampling Theorem, multi-carrier transmission with OFDM, wavelets, and a first insight into quantum mechanics. It also introduces readers to the finite element method (FEM) and offers an elementary proof of the Malgrange-Ehrenpreis theorem, showcasing advanced concepts in a clear and approachable manner. Practical Insights: Includes a detailed discussion of Hilbert spaces, orthonormal systems, and their applications to topics like the periodic table in chemistry and the structure of water molecules. The book also explores continuous and discrete wavelet transforms, providing insights into modern data compression and denoising techniques. Comprehensive Support: Appendices cover essential theorems in function theory and Lebesgue integration, complete with solutions to exercises, a reference list, and an index. With its focus on practical applications, clear explanations, and a wealth of examples, Fourier Analysis and Distributions bridges the gap between classical theory and modern computational methods. This text will appeal to students and practitioners looking to deepen their understanding of Fourier analysis and its far-reaching implications in science and engineering. Preface Contents List of Symbols and Physical Quantities 1 Introduction 1.1 Preliminary Remarks on History 1.2 The Problem of the Force-Free Vibrating String 2 Trigonometric Polynomials and Fourier Coefficients 2.1 Representation of Trigonometric Polynomials 2.2 Fourier Coefficients of Trigonometric Polynomials Computation of Fourier Coefficients Equality of Trigonometric Polynomials Real-Valued Trigonometric Polynomials and Complex Amplitudes Number of Zeros of Trigonometric Polynomials 2.3 Dirichlet Kernels 2.4 Summary on Trigonometric Polynomials 3 Fourier Series 3.1 The First Fourier Series Approximation Errors and Pointwise and Uniform Convergence An Initial Idea to Study the Series k=1∞ eepicsin(kt)k Study of the Series k=1 ∞ eepic sin (k t) k The Gibbs Phenomenon for the Sawtooth Function 3.2 Basic Theorems on Fourier Series First Explanations of the Theorems Pointwise Convergence in the Theorem of Dirichlet On Fejér's Theorem On the Gibbs Phenomenon 3.3 The Spectrum of Periodic Functions Significance of the Discrete Spectrum Further Examples of Fourier Series 1. Explicit Computation of a Fourier Series Representation 2. Fourier Series Expansion with the Use of an Already Known Series 3.4 Exercises 4 Calculating with Fourier Series 4.1 Symmetry Properties, Linearity, and Similarity 4.2 Translations in Time and Frequency Domains 4.3 Derivatives of Fourier Series 4.4 Integration of Fourier Series 4.5 Decrease of Fourier Coefficients andRiemann-Lebesgue Lemma The Riemann-Lebesgue Lemma Order of Magnitude of Fourier Coefficients and Smoothness of f 4.6 Spectrum and Power and Parseval Equation 4.7 Exercises 5 Application Examples for Fourier Series 5.1 Best Approximation in Quadratic Mean Geometric Interpretation Convergence in Quadratic Mean 5.2 Periodic Convolution and Application to Linear Systems The Fourier Series of a Periodic Convolution Application to Asymptotically Stable Time-Invariant Linear Systems Mechanical Systems of Second Order with Periodic Forces 5.3 The Potential Equation on a Circular Disk Solution by Fourier Series Expansion for Given Boundary Values The Poisson Integral Formula Smoothness and Uniqueness of the Solution and Maximum Principle Uniqueness of the Solution Illustration of the Solution for a Dirichlet Boundary Value Problem 5.4 Solution for the Problem of the Force-Free Vibrating String On Differentiability of the Solution D'Alembert's Solution for the Force-Free Vibrating String Uniqueness of the Solution Meaning of the Solution 5.5 The Approximation Theorem of Weierstrass 5.6 The 1/f-Theorem of Wiener 5.7 Exercises 6 Discrete Fourier Transforms, First Applications 6.1 Finite Discrete Fourier Transform (DFT) Consequences for Applications of the DFT Alias Effect and Frequency Assignment with Undersampling The Inverse Discrete Fourier Transform (IDFT) Properties and Calculation Rules for the Discrete Fourier Transform 6.2 Trigonometric Interpolation 6.3 The Discrete Cosine Transform DCT I 6.4 Shifted Nodes, Discrete Cosine Transform DCT II 6.5 Numerical Integration by Clenshaw-Curtis Quadrature 6.6 Approximation and Interpolation by Chebyshev Polynomials Interpolation with Chebyshev Polynomials An Extremal Property of Chebyshev Polynomials Useful in Filter Design Chebyshev Lowpass Filters 6.7 Further Application Examples for the DFT Discrete Linear Filters Time Series Analysis 6.8 The Basic Principle of the Fast Fourier Transform (FFT) FFT Algorithm 6.9 DCT-2D 6.10 Exercises 7 Convergence of Fourier Series 7.1 The Theorem of Dirichlet 7.2 The Theorem of Fejér, Convergence by Smoothing Uniform Convergence of Fejér Means for Continuous Functions Convergence of Fejér Means for Piecewise Continuous Functions Convergence of Fourier Series of Piecewise Continuous Functions Completeness of the Trigonometric System Fourier Series of Piecewise Continuously Differentiable Functions Vanishing of the Gibbs Phenomenon in Fejér Means 7.3 The Parseval Equation Continuity of Periodic Convolutions of Piecewise Continuous Functions The Parseval Equation for Piecewise Continuous Periodic Functions 7.4 Fourier Series for Functions of Several Variables A Dirichlet Boundary Value Problem for a Rectangle Membrane A Warning Example 7.5 Reasons for the Transition to Distributions A Reviewing Summary Transition to Distributions and Lebesgue Integral 7.6 Exercises 8 Fundamentals of Distribution Theory 8.1 Characterizing Functions by Their Means Pointwise Reconstruction of Continuous Functions by Means 8.2 The Space of Test Functions Convergence of Test Functions 8.3 The Dirac Impulse Impulses in Electrical Engineering Definition of δ-Impulses Evaluation of Dirac Impulses, δ as Sampling Functional Dirac Distributions as Generalized Density Functions The δ-Impulse as Derivative of the Unit Step Function 8.4 Distributions Definition of Distributions Basic Examples of Distributions 8.5 Calculating with Distributions Differentiation of Distributions Further Examples Primitives of Distributions Convergence of Sequences of Distributions Coordinate Transformations for Distributions 8.6 Test Functions and Distributions with Several Variables Characterization of Continuity of Distributions 8.7 Tensor Product and Convolution The Tensor Product of Distributions The Support of a Distribution The Convolution of Distributions Sufficient Conditions for the Existence of Convolutions Properties of Convolutions Examples of Convolutions Approximations of Distributions by Smooth Functions The Spaces E' and DR, Continuity of Convolution Operators 8.8 Exercises 9 Application Examples for Distributions 9.1 Periodic Distributions are Generalized Fourier Series Fourier Series as Distributions Periodic Distributions Are Generalized Fourier Series The Impulse Method for Calculating Fourier Coefficients 9.2 Linear Differential Equations with Constant Coefficients Fundamental Solutions The Causal Fundamental Solution Impulse Response, Step Response of Time-Invariant Linear Systems Linear Initial Value Problems of n-th Order with Constant Coefficients Initial Value Problems on Half-Lines, Suppression of the Past Causal Linear First-Order Systems with Constant Coefficients The Malgrange-Ehrenpreis Theorem 9.3 Application to Linear Electrical Networks 9.4 3D Potential Problems Examples Approaches for Solving Boundary Value Problems 9.5 The Basic Idea of Finite Elements The Ritz-Galerkin Method The Linear System of Equations for a Ritz-Galerkin Solution Finite Elements Triangulation of the Domain, Choice of Basis Functions, Linear Elements Setting up the Linear System of Equations Graphical Representation of an Approximate Solution 9.6 Distributional Solution of the 1D Wave Equation 9.7 Summary 9.8 Exercises 10 The Fourier Transform 10.1 Representation of Functions by Harmonic Oscillations The Fourier Inversion Theorem for Piecewise Continuously Differentiable Functions 10.2 Fourier Transform of Real-Valued Functions Examples of Spectral Functions 10.3 Gibbs Phenomenon and Smoothing 10.4 Calculations with Fourier Transforms 10.5 The Fourier Transform of Tempered Distributions The Fourier Transform of Rapidly Decreasing Functions Continuity of the Fourier Transform on S Tempered Distributions The Fourier Transform on S' Inverse Fourier Transform on S' Calculating with Fourier Transforms in S' Examples of Tempered Distributions and Their Fourier Transforms 10.6 Fourier Transform of Convolutions Examples 10.7 Fourier Transform of Square-Integrable Functions 10.8 The Fourier Transform for Functions of Several Variables The Jordan Inversion Formula The Fourier Transform for Tempered Distributions on Rp Summary 10.9 Exercises 11 Basics of Linear Filters 11.1 Signals 11.2 Translation-Invariant Linear Systems 11.3 Analog Linear Filters, Continuity, and Causality Automatic Continuity of Causal Time-Invariant Linear Systems The Frequency Response of Analog Linear Filters Butterworth Lowpass Filter 11.4 Analog Filters with Rational Frequency Responses Common Linear Factors of the Polynomials P and Q Frequency Response and Transfer Function of the Causal System Stability of the Causal System Realization of the Causal System 11.5 Periodic Signals and Stationary Filter Response Periodization in the Time Domain and Sampling in the Frequency Domain Numerical Approximations for Fourier Transforms The Poisson Summation Formula Application Examples 11.6 Discrete Linear Filters and z-Transform Automatic Continuity of Causal Linear Discrete Systems Continuous, Causal, and Stable Translation-Invariant Linear Systems That Cannot Be Represented as Convolutions Stability and Realizability of Discrete Linear Filters Frequency Response and Transfer Function of Discrete Linear Filters and z-Transforms of Discrete Signals Basic Properties of the z-Transform First Application Examples Causal Filters with Rational Transfer Function and Difference Equations Realization of Filters with Rational Transfer Function Causality and Stability of Filters with Rational Transfer Functions Stable Inverse Filters and Stable Signal Reconstruction Amplitude Response, Phase Response, and Group Delay Filter Examples and Filter Design Causal FIR Filters with Real Coefficients and Linear Phase Design of FIR Filters by Approximation with a Window Function Design of IIR Filters Using the Bilinear Transformation Notes on Applications of Noncausal Discrete Filters 11.7 Exercises 12 Further Applications of the Fourier Transform 12.1 Shannon's Sampling Theorem Shannon's Sampling Theorem for Bandlimited Functions 12.2 Sampling as the Basis of Digital Transmission Technology Sampling, Critical Sampling, Over-, and Undersampling The Scheme of Digital Transmission in Practice Modulation with Nyquist Pulses 12.3 The Basic Idea of Multi-Carrier Transmission with OFDM Mathematical Components of an OFDM Transmission System 12.4 Heisenberg's Uncertainty Principle Examples Uncertainty Principle for the Time-Bandwidth Product Application Examples Heisenberg's Uncertainty Principle in Quantum Mechanics 12.5 Time-Frequency Analysis, Windowed Fourier Transforms Windowed Fourier Transforms, Gabor Transform Reconstruction of a Signal from Its Windowed Fourier Transform Signal Processing with Windowed Fourier Transforms Discrete Windowed Fourier Transform 12.6 Time Windows with the Discrete Fourier Transform Truncation Effects in the Discrete Fourier Transform Selection of Time Windows in the Discrete Fourier Transform 12.7 Initial Value Problems for Stable LTI Systems 12.8 Initial Value Problems for 3D Wave and Heat Equations The Initial Value Problem for the 3D Homogeneous Wave Equation The Initial Value Problem for the 2D Homogeneous Wave Equation The Initial Value Problem for the Homogeneous Heat Equation Inhomogeneous Boundary Value Problems for the Heat Equation 12.9 Exercises 13 The Malgrange-Ehrenpreis Theorem 13.1 Preliminaries 13.2 The Malgrange-Ehrenpreis Theorem Appendix: Technical Resources 14 Outlook on Further Concepts 14.1 Hilbert Spaces and Special Complete Orthogonal Systems Schematically Hilbert Spaces Examples of Some Hilbert Spaces Complete Orthonormal Systems in Hilbert Spaces Examples of Specific Complete Orthonormal Systems in Hilbert Spaces Atoms with Multiple Electrons and the Periodic Table of Elements The H2O Water Molecule 14.2 Wavelets Time-Frequency Analysis with the Windowed Fourier Transform Time-Scale Analysis with the Wavelet Transform Some Fundamental Properties of the Wavelet Transform The Haar Wavelet Pointwise Inversion Formula for the Wavelet Transform Discrete Wavelet Transform and Multiscale Analysis Multiscale Analysis with the Haar Wavelet The Vector Spaces Generated by the Haar Wavelet Multiscale Analysis over the Scale of Vector Spaces Vn Scaling Function and Scaling Equation of Multiscale Analysis Fast Wavelet Transform with the Haar Wavelet Mallat's Reconstruction Algorithm Multiscale Analysis with Other Wavelets Daubechies Wavelets Image Data Processing and Two-Dimensional Multiscale Analysis Example for Denoising Further Areas of Application At the End A The Residue Theorem and the Fundamental Theorem of Algebra The Residue Theorem Analytic Functions and the Fundamental Theorem of Algebra On Bounds for Roots of Polynomials Partial Fraction Decomposition of Rational Functions Calculation of a Partial Fraction Decomposition B Tools from Integration Theory Measures, Null Sets, and Integrals Examples Fundamental Theorems of Integration Theory Examples Integration over a Spherical Surface Measures with Densities Lp-Spaces and Convolutions Convolutions in Sequence Spaces The Sobolev Space H01(Ω) and the Poincaré-Friedrichs Inequality C Solutions to the Exercises Exercises of Chap.3 Exercises of Chap.4 Exercises of Chap.5 Exercises of Chap.6 Exercises of Chap.7 Exercises of Chap.8 Exercises of Chap.9 Exercises of Chap.10 Exercises of Chap.11 Exercises of Chap.12 References Index