Multivariate sampling-type approximation

Abstract

Approximation properties of the expansions [Formula: see text], where [Formula: see text] is a linear differential operator and [Formula: see text] is a matrix dilation, are studied. The sampling expansions are a special case of such differential expansions. Error estimations in [Formula: see text]-norm, [Formula: see text], are given in terms of the Fourier transform of [Formula: see text]. The approximation order depends on the smoothness of [Formula: see text], the order of [Formula: see text], the order of Strang–Fix condition for [Formula: see text] and [Formula: see text]. A wide class of [Formula: see text] including both band-limited and compactly supported functions is considered, but a special condition of compatibility [Formula: see text] with [Formula: see text] is required. Such differential expansions may be useful for engineers.