Book contents
- Frontmatter
- Contents
- Preface
- Acknowledgements
- Symbols
- Part I Numerical Linear Algebra
- Part II Constructive Approximation Theory
- 9 Polynomial Interpolation
- 10 Minimax Polynomial Approximation
- 11 Polynomial Least Squares Approximation
- 12 Fourier Series
- 13 Trigonometric Interpolation and the Fast Fourier Transform
- 14 Numerical Quadrature
- Part III Nonlinear Equations and Optimization
- Part IV Initial Value Problems for Ordinary Differential Equations
- Part V Boundary and Initial Boundary Value Problems
- Appendix A Linear Algebra Review
- Appendix B Basic Analysis Review
- Appendix C Banach Fixed Point Theorem
- Appendix D A (Petting) Zoo of Function Spaces
- References
- Index
11 - Polynomial Least Squares Approximation
from Part II - Constructive Approximation Theory
Published online by Cambridge University Press: 29 September 2022
Book contents
- Frontmatter
- Contents
- Preface
- Acknowledgements
- Symbols
- Part I Numerical Linear Algebra
- Part II Constructive Approximation Theory
- 9 Polynomial Interpolation
- 10 Minimax Polynomial Approximation
- 11 Polynomial Least Squares Approximation
- 12 Fourier Series
- 13 Trigonometric Interpolation and the Fast Fourier Transform
- 14 Numerical Quadrature
- Part III Nonlinear Equations and Optimization
- Part IV Initial Value Problems for Ordinary Differential Equations
- Part V Boundary and Initial Boundary Value Problems
- Appendix A Linear Algebra Review
- Appendix B Basic Analysis Review
- Appendix C Banach Fixed Point Theorem
- Appendix D A (Petting) Zoo of Function Spaces
- References
- Index
Summary
The best polynomial approximation in a weighted least squares sense is studied in this chapter. The essential notion of orthogonal polynomials, and their properties are analyzed. These are, first of all, used to show existence, uniqueness, and convergence of a least squares best approximation.This motivates the introduction of generalized Fourier series. The issue of uniform convergence of least squares approximations for smooth functions is then studied.
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- Information
- Classical Numerical AnalysisA Comprehensive Course, pp. 300 - 319Publisher: Cambridge University PressPrint publication year: 2022