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

Abstract

AbstractLet$$H\subset {\mathbb {Z}}^d$$H⊂Zdbe a half-space lattice, defined either relative to a fixed coordinate (e.g.$$H = {\mathbb {Z}}^{d-1}\!\times \!{\mathbb {Z}}_+$$H=Zd-1×Z+), or relative to a linear order$$\preceq $$⪯on$${\mathbb {Z}}^d$$Zd, i.e.$$H = \{j\in {\mathbb {Z}}^d: 0\preceq j\}$$H={j∈Zd:0⪯j}. We consider the problem of interpolation at the points ofHfrom the space of series expansions in terms of theH-shifts of a decaying kernel$$\phi $$ϕ. Using the Wiener–Hopf factorization of the symbol for cardinal interpolation with$$\phi $$ϕon$${\mathbb {Z}}^d$$Zd, we derive some essential properties of semi-cardinal interpolation onH, 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$$\phi $$ϕis transferred to the Lagrange functions for interpolation onH, 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.