Error Analysis for Multi-Dimensional Shannon Sampling Expansion

摘要

Let $B^p_{bf v}({Bbb R}^d)$, $1leq p< infty,$ be the space of all bounded bandlimited functions from $L_p({Bbb R}^d)$. The uniform bounds for truncated multi-dimensional Whittaker-Shannon series based on local sampling are derived for signal functions $fin B^p_{bf v}({Bbb R}^d)$ without decay assumption. Then the optimal bounds of aliasing and truncation errors for non-bandlimited signal functions from Sobolev classes $ {cal U}(W^r_{p }( {Bbb R}^d))$ with $rgeq d$ are obtained up to a logarithmic factor. Our results show that for the smoothness non-bandlimited signal functions, Shannon sampling series provide good approximations.