Wiener-Hopf difference equations and semi-cardinal interpolation with integrable convolution kernels

TL;DR

This paper develops a theoretical framework for semi-cardinal interpolation on half-space lattices using Wiener-Hopf factorization, demonstrating how decay properties of kernels influence interpolation functions, with applications to various kernels like Gaussian and B-splines.

Contribution

It introduces a Wiener-Hopf based approach to analyze semi-cardinal interpolation on half-space lattices, establishing existence, uniqueness, and decay transfer properties of Lagrange functions.

Findings

Decay properties of kernels transfer to Lagrange functions

Results apply to Gaussian, Matérn, and B-spline kernels

Provides explicit interpolation formulas and convergence analysis

Abstract

Let $H\subset\mathbb{Z}^d$ 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.\ $H = \{j\in\mathbb{Z}^d : 0\preceq j\}$. 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, Mat\'{e}rn, generalized inverse multiquadric, box-spline, and polyharmonic B-spline kernels.