FoCM 2014 conference
Workshop B1 - Approximation Theory
December 17, 14:35 ~ 15:25 - Room A12
Stable reconstruction from Fourier samples
Alexei Shadrin
University of Cambridge, UK - A.Shadrin@dampt.cam.ac.uk
For an analytic and periodic function $f$, the $m$-th partial sums of its Fourier series converge exponentially fast in $m$, but such rapid convergence is destroyed once periodicity is no longer present (because of the Gibbs phenomenon at the end-points).
We can restore higher-order convergence, e.g., by reprojecting the slowly convergent Fourier series onto a suitable basis of orthogonal algebraic polynomials, however, all exponentially convergent methods appear to suffer from some sort of ill-conditioning, whereas methods that recover $f$ in a stable manner have a much slower approximation rate.
We give to these observations a definite explanation in terms of the following fundamental stability barrier: the best possible convergence rate for a stable reconstruction from the first $m$ Fourier coefficients is root-exponential in $m$.