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

Given a compactly supported function ' : IR 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 ' 0 by a convolution , being a compactly supported distribution, and inverting jH by another local convolution operator ¯. 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 and ¯. The natural choice = ' is singled out, and the interrelation between ' 0 and ' is analyzed in detail. We use these observations in the conversion of the local approximation order of an exponential space H into approximation rates from any space which contains H and is spanned by the hZZ s - translates of a single compactly supported function '. The bounds obtained are attractive in the se..