1 Periodic Functions.- 1.1 The Characters.- 1.2 Some Tools of the Trade.- 1.3 Fourier Series: Lp Theory.- 1.4 Fourier Series: L2 Theory.- 1.5 Fourier Analysis of Measures.- 1.6 Smoothness and Decay of Fourier Series.- 1.7 Translation Invariant Operators.- 1.8 Problems.- 2 Hardy Spaces.- 2.1 Hardy Spaces and Invariant Subspaces.- 2.2 Boundary Values of Harmonic Functions.- 2.3 Hardy Spaces and Analytic Functions.- 2.4 The Structure of Inner Functions.- 2.5 The H1 Case.- 2.6 The Szegö-Kolmogorov Theorem.- 2.7 Problems.- 3 Prediction Theory.- 3.1 Introduction to Stationary Random Processes.- 3.2 Examples of Stationary Processes.- 3.3 The Reproducing Kernel.- 3.4 Spectral Estimation and Prediction.- 3.5 Problems.- 4 Discrete Systems and Control Theory.- 4.1 Introduction to System Theory.- 4.2 Translation Invariant Operators.- 4.3 H?Control Theory.- 4.4 The Nehari Problem.- 4.5 Commutant Lifting and Interpolation.- 4.6 Proof of the Commutant Lifting Theorem.- 4.7 Problems.- 5 Harmonic Analysis in Euclidean Space.- 5.1 Function Spaces on Rn.- 5.2 The Fourier Transform on L1.- 5.3 Convolution and Approximation.- 5.4 The Fourier Transform: L2 Theory.- 5.5 Fourier Transform of Measures.- 5.6 Bochner's Theorem.- 5.7 Problems.- 6 Distributions.- 6.1 General Distributions.- 6.2 Two Theorems on Distributions.- 6.3 Schwartz Space.- 6.4 Tempered Distributions.- 6.5 Sobolev Spaces.- 6.6 Problems.- 7 Functions with Restricted Transforms.- 7.1 General Definitions and the Sampling Formula.- 7.2 The Paley-Wiener Theorem.- 7.3 Sampling Band-Limited Functions.- 7.4 Band-Limited Functions and Information.- 7.5 Problems.- 8 Phase Space.- 8.1 The Uncertainty Principle.- 8.2 The Ambiguity Function.- 8.3 Phase Space and Orthonormal Bases.- 8.4 The Zak Transform and the Wilson Basis.- 8.5 AnApproximation Theorem.- 8.6 Problems.- 9 Wavelet Analysis.- 9.1 Multiresolution Approximations.- 9.2 Wavelet Bases.- 9.3 Examples.- 9.4 Compactly Supported Wavelets.- 9.5 Compactly Supported Wavelets II.- 9.6 Problems.- A The Discrete Fourier Transform.- B The Hermite Functions.
Inhaltsverzeichnis
1 Periodic Functions. - 1. 1 The Characters. - 1. 2 Some Tools of the Trade. - 1. 3 Fourier Series: Lp Theory. - 1. 4 Fourier Series: L2 Theory. - 1. 5 Fourier Analysis of Measures. - 1. 6 Smoothness and Decay of Fourier Series. - 1. 7 Translation Invariant Operators. - 1. 8 Problems. - 2 Hardy Spaces. - 2. 1 Hardy Spaces and Invariant Subspaces. - 2. 2 Boundary Values of Harmonic Functions. - 2. 3 Hardy Spaces and Analytic Functions. - 2. 4 The Structure of Inner Functions. - 2. 5 The H1 Case. - 2. 6 The Szegö-Kolmogorov Theorem. - 2. 7 Problems. - 3 Prediction Theory. - 3. 1 Introduction to Stationary Random Processes. - 3. 2 Examples of Stationary Processes. - 3. 3 The Reproducing Kernel. - 3. 4 Spectral Estimation and Prediction. - 3. 5 Problems. - 4 Discrete Systems and Control Theory. - 4. 1 Introduction to System Theory. - 4. 2 Translation Invariant Operators. - 4. 3 H? Control Theory. - 4. 4 The Nehari Problem. - 4. 5 Commutant Lifting and Interpolation. - 4. 6 Proof of the Commutant Lifting Theorem. - 4. 7 Problems. - 5 Harmonic Analysis in Euclidean Space. - 5. 1 Function Spaces on Rn. - 5. 2 The Fourier Transform on L1. - 5. 3 Convolution and Approximation. - 5. 4 The Fourier Transform: L2 Theory. - 5. 5 Fourier Transform of Measures. - 5. 6 Bochner s Theorem. - 5. 7 Problems. - 6 Distributions. - 6. 1 General Distributions. - 6. 2 Two Theorems on Distributions. - 6. 3 Schwartz Space. - 6. 4 Tempered Distributions. - 6. 5 Sobolev Spaces. - 6. 6 Problems. - 7 Functions with Restricted Transforms. - 7. 1 General Definitions and the Sampling Formula. - 7. 2 The Paley-Wiener Theorem. - 7. 3 Sampling Band-Limited Functions. - 7. 4 Band-Limited Functions and Information. - 7. 5 Problems. - 8 Phase Space. - 8. 1 The Uncertainty Principle. - 8. 2 The Ambiguity Function. - 8. 3 Phase Space and Orthonormal Bases. - 8. 4 The Zak Transform and the Wilson Basis. - 8. 5 AnApproximation Theorem. - 8. 6 Problems. - 9 Wavelet Analysis. - 9. 1 Multiresolution Approximations. - 9. 2 Wavelet Bases. - 9. 3 Examples. - 9. 4 Compactly Supported Wavelets. - 9. 5 Compactly Supported Wavelets II. - 9. 6 Problems. - A The Discrete Fourier Transform. - B The Hermite Functions.
