Seminar: Optimal function reconstruction in the worst-case setting

Time:

Venue/Location: Phòng C101, VIASM

Báo cáo viên: Felix Bartel (Chemnitz University of Technology, Germany)

Abstract: In this lecture we approximate functions from reproducing kernel Hilbert spaces and measure the error in $L_2$. We introduce the sampling width, which is the best possible approximation based on function evaluations for a given class of functions, and compare it with the linear width, which is the best possible for arbitrary linear functionals. Using the $L_2$-Marcinkiewicz-Zygmund inequalities from the first lecture we will show that both quantities coincide, i.e., function evaluations are as powerfull as arbitrary linear functionals. Astonishingly, the algorithm achieving the optimal bound is the least squares method form around 1800 used in a clever way. Further we will have a look at the example of optimal function reconstruction from subsampled rank-1 lattices, which makes it possible to further utilize fast Fourier algorithms, improving the approximation performance.