TREFETHEN'S NYU LECTURES ON
  _APPROXIMATION THEORY AND APPROXIMATION PRACTICE_

<longer index>

**LECTURE 1**
CHAPTER 1. INTRODUCTION [4:40]
   History of approximation theory [5:00]
   Matlab, Chebfun, Publish, ATAP [13:30]
CHAPTER 2. CHEBYSHEV POINTS AND INTERPOLANTS [22:00]
   Chebfun demo [28:20]
   Clustering at the boundary and point charges [37:20]
CHAPTER 3. CHEBYSHEV POLYNOMIALS AND SERIES [45:20]
   Fourier, Laurent, and Chebyshev [46:00]
   Chebyshev polynomials [56:40]
   Thm 3.1: Chebyshev series [1:01:50]
   Examples [1:07:40]
   Chebfun demos [1:10:20]
   How Chebfun resolves a function [1:16:20] <video failure>

**LECTURE 2**
   Review of last lecture [0:00]
CHAPTER 4. INTERPOLANTS, TRUNCATIONS, AND ALIASING [7:30]
   Chebfun demo [15:30]
   Periodic functions and 'trig' [21:20]
   Aliasing [26:00]
CHAPTER 5. BARYCENTRIC INTERPOLATION FORMULA [44:30]
   Chebfun demo [55:20]
   A big error in "Spectral Methods in Matlab" [58:40]
   How not to evaluate an interpolant [1:01:30]
   Thm 5.1: barycentric formulas [1:06:20]
   History of "Lagrange interpolation" [1:15:30]
   1st and 2nd kind barycentric formulas [1:25:30]
   Chebfun demo: random interpolation points [1:29:00]
   "Happy birthday to the barycentric interpolation formula" [1:32:40]

**LECTURE 3**
CHAPTER 6. WEIERSTRASS APPROXIMATION THEOREM [0:00]
   Weierstrass's proof [5:30]
   Chebfun demo of Weierstrass's proof [11:00]
   Lebesgue's proof [14:30]
   Fejer's first proof [20:00]
   Fejer's second proof [25:00]
   Bernstein's proof [30:00]
CHAPTER 7. CONVERGENCE FOR DIFFERENTIABLE FUNCTIONS [39:30]
   Central dogma of approximation theory [40:30]
   Chebfun demos [43:30]
   Smoothness assumptions for Thms 7.1 and 7.2 [55:30]
   Thm 7.1: decay of Chebyshev coefficients [1:01:00]
   Thm 7.2: accuracy of approximations [1:02:00]
   Analytic functions and Bernstein ellipses [1:18:00]
   "Bernstein ellipses and beyond" [1:21:00] <video failure>

**LECTURE 4**
CHAPTER 8. CONVERGENCE FOR ANALYTIC FUNCTIONS [0:00]
   Discussion of analytic functions [0:00]
   Smoothness assumptions for Thms 8.1 and 8.2 [5:30]
   Thm 8.1: decay of Chebyshev coefficients [6:50]
   Thm 8.2: accuracy of approximations [7:20]
CHAPTER 9. GIBBS PHENOMENON [29:40]
   Gibbs phenomenon theorems and Chebfun demo [29:40]
   Splines vs. global polynomial approximations [39:40]
CHAPTER 10. BEST APPROXIMATION [46:30]
   Thm 10.1: equioscillation [56:10]
   Chebfun demos [1:17:00] <video failure>

**LECTURE 5**
   Review of interpolation, projection, best approximation [0:30]
CHAPTER 11. HERMITE INTEGRAL FORMULA [9:00]
   Thm 11.1: Hermite intgral formula [16:30]
   Applying the theorem, following Runge [36:00]
   Chebfun demo [47:10]
   A walk through Runge's great paper [52:40]
CHAPTER 13. EQUISPACED POINTS, RUNGE PHENOMENON [57:40]
   Chebfun demo [59:40]
CHAPTER 12: POTENTIAL THEORY [1:06:10]

**LECTURE 6**
CHAPTER 14. DISCUSSION OF HIGHER-ORDER POLYNOMIAL INTERPOLATION [0:00]
   Faber's theorem of 1914 [0:30]
   Confusion caused by ill-conditioned problem + unstable algorithms [6:40]
   Quotes from some numerical analysis texts [10:30]
CHAPTER 15. LEBESGUE CONSTANTS [17:00]
   Thm 15.1: Lebesgue constants and best approximation [23:20]
   Thm 15.2: size of Lebesgue constants for various grids [26:20]
   Chebfun demo [30:50]
   Proof of Faber's theorem [39:40]
   "Six myths of polynomial interpolation and quadrature" [45:40]
   Quotes from Yogi Berra (who died a few days ago) [51:40]

**LECTURE 7**
   Chebfun 'turbo' mode for coefficients below 1e-16 [0:10]
CHAPTER 16. BEST AND NEAR-BEST [3:30]
   Thm 16.1: near-best approximation [4:50]
CHAPTER 17. ORTHOGONAL POLYNOMIALS [6:50]
   Where do 3-term recurrence relations come from? [25:20]
   Chebfun demo of orthogonal polynomials [33:50]
   Fast algorithms exploiting Legendre <-> Cheb. coeffs. [38:00]
CHAPTER 18. POLYNOMIAL ROOTS AND COLLEAGUE MATRICES [44:30]
   Chebfun demo of roots and extrema [45:00]
   Companion matrices and colleague matrices [50:10]
   Jack Good, inventor of proxy rootfinding and much more [1:03:40]
   Numerical rootfinding [1:08:10]
   Chebfun demos of rootfinding [1:20:40] <video failure>

**LECTURE 8**
   Chebfun rootfinding demo [0:00]
CHAPTER 19. CLENSHAW-CURTIS AND GAUSS QUADRATURE [5:30]
   Newton-Cotes quadrature [10:40]
   Clenshaw-Curtis and Gauss quadrature [13:10]
   Thm 19.1: polynomial degrees of exactness [16:00]
   Implementation of Clenshaw-Curtis quadrature [24:00]
   Implementation of Gauss quadrature [29:40]
   O(n) algorithms un-dreamed-of by Golub and Welsch [39:40]
   Chebfun demo [44:00]
   Demo of accuracy of Gauss vs. Clenshaw-Curtis [48:50]
   Thm 19.3: comparative accuracy [50:50]
   Chebfun demos [1:03:10]
   "Is Gauss quadrature better than Clenshaw-Curtis?" [1:14:30]

**LECTURE 9**
CHAPTER 20. CARATHEODORY-FEJER APPROXIMATION [0:00]
   Chebfun demo [0:30]
   Derivation of CF approximation [7:10]
   Thm 20.1: extended CF approximation [14:50]
   Chebfun demo [29:20]
   Demo for rational approximation, best vs CF [30:30]
   "CF approximation 30 years later" [32:40]
CHAPTER 21. SPECTRAL METHODS [40:00]
   Chebyshev and Fourier spectral methods [40:40]
   Thm 21.1: accuracy of spectral derivatives [44:10]
   Proof by series [48:00]
   Proof by contour integrals [52:10]
   Differentiation matrices [59:00]
   Chebgui and other demos [1:02:30] <microphone failure>

**LECTURE 10**
   Chebfun demo of remez vs cf [0:00]
CHAPTER 22. LINEAR APPROXIMATIONS: BEYOND POLYNOMIALS [4:00]
   "Beating Gauss quadrature" [4:40]
CHAPTER 23. NONLINEAR APPROXIMATIONS: WHY RATIONAL FUNCTIONS? [24:30]
   How do polynomial and rtional approximations compare? [34:30]
   Chebfun demo [38:50]
   Twelve applications of rational functions [43:20]
   Application 12: one-way wave equations [52:30]
   What's the use of approximation theory? [1:03:20]
CHAPTER 24. RATIONAL BEST APPROXIMATION [1:09:20]
   Thm 24.1: equioscillation [1:14:40]
   Chebfun demos [1:29:00]

**LECTURE 11**
CHAPTER 25. TWO FAMOUS PROBLEMS [0:00]
   Thm 25.1: approximation of abs(x) on [-1,1] [1:40]
   Derivation of near-optimal approximation by trapezoidal rule [12:50]
   Thm 25.2: approximation of exp(x) on [-inf,0] [25:00]
   Derivation of near-optimal approximation by trapezoidal rule [29:00]
   "Talbot quadratures, rational approximations, and stiff PDEs" [36:20]
CHAPTER 26. RATIONAL INTERPOLATION AND LINEARIZED LEAST-SQUARES [41:40]
   Examples of nonexistence, nonuniquenss, and ill-posedness [44:20]  
   Chebfun demo of rational interpolation [49:50]
   Froissart doublets = poles with very small residue [53:00]
   Robust computation via regularization [1:02:50]
   Chebfun demo of robust rational interpolation [1:18:00]
   

**LECTURE 12**
CHAPTER 27. PADÉ APPROXIMATION [1:00]
   Pade table for exp(z) [5:20]
   Thm 27.1: existence and uniqueness of Pade approximations [11:40]
   Square blocks in the Pade table [16:40]
   Thm 27.2: square blocks
   Chebfun demo
   "Robust Pade approximation via SVD" [39:00]
CHAPTER 28. ANALYTIC CONTINUATION AND CONVERGENCE ACCELERATION [45:40]
   Jentzsch's theorem on zeros clustered on a circle [47:50]
   Walsh's theorem on zeros clustered on an ellipse [50:00]
   de Montessus de Ballore thm on convergence on rows of Pade table [52:40]
   Nuttall-Pommerenke theorem on convergence on rows of Pade table [52:40]
   Branch points and Stahl's theorem [56:00]
   Extrapolation of sequences and series [1:02:40]
   Aitken delta-square extrapolation [1:06:30]
   Eta and epsilon algorithms [1:07:30]
   Chebfun demo of slowly convergent and divergent series [1:09:00]