Approximation Theory III/IV

Outline and Notes

Complete lecture notes (in one pdf)

Notes for each lecture are made available in advance subject to the disclaimer that they may not precisely match the lecture itself. Please let me know if you spot any typos.

Interesting Links

• Six Myths of Polynomial Interpolation and Quadrature (L.N. Trefethen).
• History of Approximation Theory with list of people and links to seminal papers.
• B-splines and geometric design (SIAM News).
• There are nice online applets to demonstrate Bernstein polynomial approximation and minimax approximation in this article on The Chebyshev Equioscillation Theorem.
• From Bézier to Bernstein (AMS feature column).
• Explained: The Discrete Fourier Transform (MIT News).
• Gauss and the History of the Fast Fourier Transform.
• Image Compression: How Math Led to the JPEG2000 Standard (SIAM).

Header image

The error curves $f(x)-p(x)$ for two different polynomial approximations of degree 100 to the function $f(x)=|x-\frac{1}{4}|$. The first is the ``minimax'' polynomial approximation (red) and the second is the interpolant at the Chebyshev nodes (blue).