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• Preface
• Table of Contents
• 1 Introduction
• 1.1 Preliminaries, Definitions and Notations
• 1.2 Basic Problems and Outlook
• 1.3 Approximation Methods for Data Analysis
• 1.4 Hints on Classical and More Recent Literature
• 2 Basic Methods and Numerical Algorithms
• 2.1 Linear Least Squares Approximation
• 2.2 Regularization Methods
• 2.3 Interpolation by Algebraic Polynomials
• 2.4 Divided Differences and the Newton Representation
• 2.5 Error Estimates and Optimal Interpolation Points
• 2.6 Interpolation by Trigonometric Polynomials
• 2.7 The Discrete Fourier Transform
• 3 Best Approximations
• 3.1 Existence
• 3.2 Uniqueness
• 3.3 Dual Characterization
• 3.4 Direct Characterization
• 3.5 Exercises
• 4 Euclidean Approximation
• 4.1 Construction of Best Approximations
• 4.2 Orthogonal Bases and Orthogonal Projections
• 4.3 Fourier Partial Sums
• 4.4 Orthogonal Polynomials
• 4.5 Exercises
• 5 Chebyshev Approximation
• 5.1 Approaches to Construct Best Approximations
• 5.2 Strongly Unique Best Approximations
• 5.3 Haar Spaces
• 5.4 The Remez Algorithm
• 5.5 Exercises
• 6 Asymptotic Results
• 6.1 The Weierstrass Theorem
• 6.2 Complete Orthogonal Systems and Riesz Bases
• 6.3 Convergence of Fourier Partial Sums
• 6.4 The Jackson Theorems
• 6.5 Exercises
• 7 Basic Concepts of Signal Approximation
• 7.1 The Continuous Fourier Transform
• 7.2 The Fourier Transform on L2(R)
• 7.3 The Shannon Sampling Theorem
• 7.4 The Multivariate Fourier Transform
• 7.5 The Haar Wavelet
• 7.6 Exercises
• 8 Kernel-based Approximation
• 8.1 Multivariate Lagrange Interpolation
• 8.2 Native Reproducing Kernel Hilbert Spaces
• 8.3 Optimality of the Interpolation Method
• 8.4 Orthonormal Systems, Convergence, and Updates
• 8.5 Stability of the Reconstruction Scheme
• 8.6 Kernel-based Learning Methods
• 8.7 Exercises
• 9 Computerized Tomography
• 9.1 The Radon Transform
• 9.2 The Filtered Back Projection
• 9.3 Construction of Low-Pass Filters
• 9.4 Error Estimates and Convergence Rates
• 9.5 Implementation of the Reconstruction Method
• 9.6 Exercises
• References
• Subject Index
• Name Index