Construction of irregular complete interpolation sets for shift-invariant spaces

TL;DR

This paper characterizes and constructs irregular complete interpolation sets for shift-invariant spaces, using Toeplitz operators, recurrence relations for exponential splines, and analysis of zeta functions, to identify conditions for complete interpolation.

Contribution

It introduces a new characterization of complete interpolation sets for shift-invariant spaces and determines specific conditions on parameters for their existence.

Findings

Identifies all  for which the sample set forms a complete interpolation set.

Develops a new recurrence relation for exponential splines and analyzes their zeros.

Shows that certain irregular sets are complete interpolation sets if and only if ||<1/2.

Abstract

For several shift-invariant spaces, there exists a real number $a\in\mathbb{R}$ such that the set $a+\mathbb{Z}$ is a complete interpolation set. In this paper, we characterize the complete interpolation property of the set $(a+\mathbb{N}_0)\cup(\alpha+a+\mathbb{N}^{-})$ for shift-invariant spaces using Toeplitz operators. Using this characterization, we determine all $\alpha$ for which the sample set $\mathbb{N}_0\cup\alpha+\mathbb{N}^{-}$ forms a complete interpolation set for transversal-invariant spaces. We introduce a new recurrence relation for exponential splines, examines the zeros of these splines, and explores the zero-free region of the doubly infinite Lerch zeta function. Consequently, we demonstrate that $\left\langle\frac{m}{2}\right\rangle+\mathbb{N}_0\cup\alpha+\left\langle\frac{m}{2}\right\rangle+\mathbb{N}^{-}$ is a complete interpolation set for a shift-invariant spline space of order $m\geq 2$ if and only if $|\alpha|<1/2$.