Wiener–Hopf Difference Equations and Semi-Cardinal Interpolation with Integrable Convolution Kernels

Abstract

Let Hsubset {mathbb {Z}}^d be a half-space lattice, defined either relative to a fixed coordinate (e.g. H = {mathbb {Z}}^{d-1}!times !{mathbb {Z}}_+), or relative to a linear order preceq on {mathbb {Z}}^d, i.e. H = {jin {mathbb {Z}}^d: 0preceq j}. We consider the problem of interpolation at the points of H from the space of series expansions in terms of the H-shifts of a decaying kernel phi . Using the Wiener–Hopf factorization of the symbol for cardinal interpolation with phi on {mathbb {Z}}^d, we derive some essential properties of semi-cardinal interpolation on H, such as existence and uniqueness, Lagrange series representation, variational characterization, and convergence to cardinal interpolation. Our main results prove that specific algebraic or exponential decay of the kernel phi is transferred to the Lagrange functions for interpolation on H, as in the case of cardinal interpolation. These results are shown to apply to a variety of examples, including the Gaussian, Matérn, generalized inverse multiquadric, box-spline, and polyharmonic B-spline kernels.