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
12 - Fourier Series
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
This chapter is a close companion to the previous one. Here we study the best least squares approximation to periodic functions via trigonometric polynomials. Many of the ideas and results of the previous chapter are repeated in this scenario. They are then expanded to deal with merely square integrable functions. The Fourier transform of periodic functions, and its inverse, is then introduced and studied. Uniform convergence of trigonometric series, under several different smoothness assumptions is then discussed.Trigonometric approximation in periodic Sobolev spaces is then discussed.
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- Information
- Classical Numerical AnalysisA Comprehensive Course, pp. 320 - 344Publisher: Cambridge University PressPrint publication year: 2022