The quasi-interpolant to a function f : Rn→R on an infinite regular grid of spacing h can be defined by where ψ: Rn→R is a function which decays quickly for large argument. In the case of radial basis functions ψ has the form where φ : R+→R is known as a radial basis function and, in general, µj ε R (j = 1,…,m) and xj ε Rn (j = 1,…,m), though here only the particular case xj ε Ζn (j = 1,…, m) is considered. This paper concentrates on the case φ(r) = r, a generalization of linear interpolation, although some of the analysis is more general. It is proved that, if n is odd, then there is a function ψ such that the maximum difference between a sufficiently smooth function and its quasi-interpolant is bounded by a constant multiple of hn+1. This is done by first showing that such a quasi-interpolation formula can reproduce polynomials of degree n.