NONUNIFORM SAMPLING AND MULTISCALE COMPUTATION∗
Christina Frederick · 2016
In homogenization theory and multiscale modeling, typical functions satisfy the scaling law $f^{\epsilon}(x)=f(x,x/\epsilon)$, where $f$ is periodic in the second variable and $\epsilon$ is the smallest relevant wavelength, $0<\epsilon\ll1$. Our main result is a new $L^{2}$-stability estimate for the reconstruction of bandlimited multiscale functions $f^{\epsilon}$ from periodic nonuniform samples. The goal of this paper is to demonstrate the close relation between sampling strategies developed in information theory and computational grids in multiscale modeling. This connection is of much interest because numerical simulations often involve discretizations by means of sampling, and the proposed sampling sets are of optimal rate according to the minimal sampling requirements of Landau [Proc. IEEE, 55 (1967), pp. 1701--1706].