Approximation Theory and Algorithms for Data Analysis

Höfundur: Armin Iske (Útgáfa: 0)
Approximation Theory and Algorithms for Data Analysis

Kaup valmöguleikar

This textbook offers an accessible introduction to the theory and numerics of approximation methods, combining classical topics of approximation with recent advances in mathematical signal processing, and adopting a constructive approach, in which the development of numerical algorithms for data analysis plays an important role. The following topics are covered: * least-squares approximation and regularization methods * interpolation by algebraic and trigonometric polynomials * basic results on best approximations * Euclidean approximation * Chebyshev approximation * asymptotic concepts: error estimates and convergence rates * signal approximation by Fourier and wavelet methods * kernel-based multivariate approximation * approximation methods in computerized tomography Providing numerous supporting examples, graphical illustrations, and carefully selected exercises, this textbook is suitable for introductory courses, seminars, and distance learning programs on approximation for undergraduate students.

Nánar um bókina

Útgefandi
Springer Nature
ISBN
9783030052287
Print ISBN
9783030052270
Format
Page Fidelity (PDF)
Útgáfa
0
Höfundar
Armin Iske
Tungumál
English
Útgefið
2018-12-14
Prent takmörkun á líftíma
100
Prent takmörkun
2
Afritunar takmörkun
2

Kaflar

  • 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