THE EXPONENTIALS IN THE SPAN OF THE MULTIINTEGER TRANSLATES OF A COMPACTLY SUPPORTED FUNCTION - QUASIINTERPOLATION AND APPROXIMATION ORDER

Given a compactly supported function phi:R(s) --> C and the space S 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 phi*' by a convolution lambda*, lambda being a compactly supported distribution, and inverting lambda*\H by another local convolution operator mu*. 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 lambda and mu. The natural choice lambda = phi is singled out, and the interrelation between phi*' and phi* 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 hZ(s)-translates of a single compactly supported function phi. The bounds obtained are attractive in the sense that they rely only on H and the basic quantities diam supp phi and h(s)\\phi\\infinity/phi(0).