The Exponentials in the Span of the Multiinteger Translates of a Compactly Supported Function; Quasiinterpolation and Approximation Order

Given a compactly supported function tp: W -*• C and the space 5 spanned by its integer translates, we study quasiinterpolants which reproduce (entirely or in part) the space H of all exponentials in S. We do this by imitating the action on H of the associated semi-discrete convolution operator *' by a convolution A*, A being a compactly supported distribution, and inverting A*|H by another local convolution operator fi*. This leads to a unified theory for quasiinterpolants on regular grids, showing that each specific construction now in the literature corresponds to a special choice of A and fi. The natural choice A = is singled out, and the interrelation between $*' and * is analysed in detail. We use these observations in the conversion of the approximation order at zero of an exponential space H into approximation rates from any space which contains H and is spanned by the /iZ'-translates of a single compactly supported function . The bounds obtained are attractive in the sense that they rely only on H and the basic quantities diamsupp0 and A'll^H/^CO)