Multivariate cardinal interpolation with radial-basis functions

For a radial-basis function?∶?→? we consider interpolation on an infinite regular lattice , tof∶? ⁿ→?, whereh is the spacing between lattice points and the cardinal function , satisfiesX(j)=δ _oj for allj∈? ⁿ. We prove existence and uniqueness of such cardinal functionsX, and we establish polynomial precision properties ofI _h for a class of radial-basis functions which includes (varphi (r) = r^{2q + 1} ) , (varphi (r) = r^{2q} log r,varphi (r) = sqrt {r^2 + c^2 } ) , and (varphi (r) = 1/sqrt {r^2 + c^2 } ) whereq∈? ₊. We also deduce convergence orders ofI _hf to sufficiently differentiable functionsf whenh→0.