Lecture 32 Summary

Derived convergence rate of Fourier series via integration by parts (as in the handout from last lecture), and showed that a function whose k-th derivative is piecewise continuous and bounded has Fourier coefficients c_m that decay as O(m^−(k+1)). It follows that a smooth function has Fourier coefficients that decay faster than any polynomial in 1/m; sketched contour-integration proof that the convergence rate is exponential for analytic functions with poles a finite distance from the real axis.

Related this convergence rate to trapezoidal rule, and explained why there is an additional cancellation if the discontinuities occur only at the endpoints, and hence the N-point trapezoidal rule always converges as an even power of 1/N.

Explained the idea of Clenshaw–Curtis quadrature as a change of variables + a cosine series to turn the integral of any function into the integral of periodic functions. This way, functions only need to be analytic on the interior of the integration interval in order to get exponential convergence. (See Wikipedia handout.)

Also explained (as in the handout) how to precompute the weights in terms of a discrete cosine transform, rather than cosine-transforming the function values every time one needs an integral, via a simple transposition trick.

Discussed the importance of nested quadrature rules for a posteriori error estimation and adaptive quadrature. Discussed p-adaptive vs. h-adaptive adaptive schemes.