This volume studies approximation of functions and data by simpler objects such as polynomials, splines, and rational functions.
This volume studies approximation of functions and data by simpler objects such as polynomials, splines, and rational functions. It develops both theoretical bounds and computational methods.
Part I. Foundations of Approximation
Chapter 1. Approximation Problems
1.1 Function approximation setup 1.2 Norms and error measures 1.3 Best approximation 1.4 Existence and uniqueness 1.5 Examples
Chapter 2. Normed Function Spaces
2.1 Metric and normed spaces 2.2 Uniform norm 2.3 Lp norms 2.4 Completeness 2.5 Examples
Chapter 3. Basic Approximation Theorems
3.1 Weierstrass approximation theorem 3.2 Stone–Weierstrass theorem 3.3 Density results 3.4 Applications 3.5 Examples
Part II. Polynomial Approximation
Chapter 4. Interpolation
4.1 Polynomial interpolation 4.2 Lagrange form 4.3 Newton form 4.4 Error analysis 4.5 Examples
Chapter 5. Least Squares Approximation
5.1 Orthogonality principle 5.2 Normal equations 5.3 Approximation in L2 5.4 Applications 5.5 Examples
Chapter 6. Chebyshev Approximation
6.1 Uniform approximation 6.2 Chebyshev polynomials 6.3 Minimax approximation 6.4 Alternation theorem 6.5 Examples
Part III. Rational Approximation
Chapter 7. Rational Functions
7.1 Definitions 7.2 Approximation properties 7.3 Advantages over polynomials 7.4 Examples 7.5 Applications
Chapter 8. Padé Approximation
8.1 Definition 8.2 Construction 8.3 Convergence 8.4 Applications 8.5 Examples
Chapter 9. Continued Fractions
9.1 Definitions 9.2 Convergence 9.3 Approximation properties 9.4 Applications 9.5 Examples
Part IV. Splines and Piecewise Approximation
Chapter 10. Spline Functions
10.1 Definitions 10.2 Piecewise polynomials 10.3 Continuity conditions 10.4 Examples 10.5 Applications
Chapter 11. Interpolation with Splines
11.1 Cubic splines 11.2 Boundary conditions 11.3 Error analysis 11.4 Applications 11.5 Examples
Chapter 12. Approximation with Splines
12.1 Least squares splines 12.2 Smoothing splines 12.3 Adaptive methods 12.4 Applications 12.5 Examples
Part V. Fourier and Expansion Methods
Chapter 13. Fourier Series
13.1 Orthogonal expansions 13.2 Convergence properties 13.3 Approximation quality 13.4 Applications 13.5 Examples
Chapter 14. Orthogonal Expansions
14.1 General orthogonal systems 14.2 Legendre and Chebyshev expansions 14.3 Convergence behavior 14.4 Applications 14.5 Examples
Chapter 15. Wavelets (Overview)
15.1 Basic concepts 15.2 Multiresolution analysis 15.3 Applications 15.4 Examples 15.5 Connections
Part VI. Approximation in Function Spaces
Chapter 16. Approximation in Lp Spaces
16.1 Definitions 16.2 Convergence 16.3 Density 16.4 Applications 16.5 Examples
Chapter 17. Nonlinear Approximation
17.1 Adaptive methods 17.2 Sparse approximation 17.3 Greedy algorithms 17.4 Applications 17.5 Examples
Chapter 18. Approximation of Operators
18.1 Operator approximation 18.2 Spectral approximation 18.3 Numerical methods 18.4 Applications 18.5 Examples
Part VII. Computational Methods
Chapter 19. Numerical Approximation
19.1 Floating-point considerations 19.2 Stability 19.3 Error bounds 19.4 Applications 19.5 Examples
Chapter 20. High-Dimensional Approximation
20.1 Curse of dimensionality 20.2 Sparse grids 20.3 Low-rank approximations 20.4 Applications 20.5 Examples
Chapter 21. Software and Algorithms
21.1 Libraries and tools 21.2 Implementation strategies 21.3 Performance optimization 21.4 Visualization 21.5 Applications
Part VIII. Research Directions
Chapter 22. Advanced Topics
22.1 Approximation theory in PDEs 22.2 Machine learning connections 22.3 Compressed sensing 22.4 Modern developments 22.5 Emerging areas
Chapter 23. Open Problems
23.1 Optimal approximation rates 23.2 High-dimensional challenges 23.3 Stability questions 23.4 Computational limits 23.5 Future directions
Chapter 24. Historical and Conceptual Notes
24.1 Development of approximation theory 24.2 Key contributors 24.3 Evolution of methods 24.4 Cross-disciplinary impact 24.5 Summary
Appendix
A. Common approximation formulas B. Error bound reference C. Proof techniques checklist D. Algorithm templates E. Cross-reference to other MSC branches
This volume develops approximation as a central tool for analysis and computation. It emphasizes error control, convergence, and efficient representation.