[For the entire collection see Zbl 0643.00021.]
The approximations that are considered are composed of functions of the form \(\{\psi (x)=\sum_{i}\lambda_ i\phi (\| x-x_ i\|_ 2):x\in {\mathcal R}\) \(n\}\), where \(\phi\) is a fixed function from \({\mathcal R}\) \(+\) to \({\mathcal R}\), where each \(\lambda_ i\) is a real coefficient and where \(\{x_ i\}\) is a finite subset of the infinite integer lattice \({\mathcal Z}^ n.\) These parameters should make \(\psi\) close to a cardinal function of interpolation on the lattice, because the paper studies some properties of the quasi-interpolant \(\{s(x)=\sum_{y}f(y)\psi (x-y):x\in {\mathcal R}\) \(n\}\) is an approximation to a given function \(\{\) f(x):x\(\in {\mathcal R}\) \(n\}\), where this sum is over \(y\in {\mathcal Z}^ n.\) In particular, we ask whether \(f\in \Pi_ k\) can imply \(s\in \Pi_ k\), where \(\Pi_ k\) is the space of polynomials of total degree at most k. Both \(\psi\) and k must be such that the sum over y is absolutely convergent, which allows several useful choices of \(\phi\) that have the property \(\phi\) (r)\(\to \infty\) as \(r\to \infty\). We also require \(\int \psi (x)dx=1\), the range of integration being \({\mathcal R}^ n,\) because this condition is necessary for \(s=f\) when \(f\in \Pi_ 0\). It is established that \(f\in \Pi_ k\) implies \(s\in \Pi_ k\) provided that \(\phi\) is sufficiently smooth, and that \(s=f\) when \(f\in \Pi_ 0\), which gives a new proof of uniform convergence for radial basis function approximation on lattices whose mesh sizes tend to zero. An important recent result of I. R. H. Jackson in the case \(\{\phi (r)=r:r\in {\mathcal R}\) \(+\}\) is mentioned, namely that \(s=f\) can be achieved for all \(f\in \Pi_ n\) provided that n is odd.