Error Analysis for Multi-Dimensional Shannon Sampling Expansion

Let B_v^p(R^d),1 ≤ p <; ∞, be the space of all bounded bandlimited functions from L_p(R^d). The uniform bounds for truncated multi-dimensional Whittaker-Shannon series based on local sampling are derived for signal functions f ∈B_v^p(R^d) without decay assumption. Then the optimal bounds of aliasing and truncation errors for non-bandlimited signal functions from Sobolev classes U(W_p^r(R^d)) with r ≥ 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.