Construction of irregular complete interpolation sets for shift-invariant spaces

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 $⟨\frac{m}{2}⟩+{\mathbb{N}}_0\cup \alpha +⟨\frac{m}{2}⟩+{\mathbb{N}}^{-}$ is a complete interpolation set for a shift-invariant spline space of order $m\ge 2$ if and only if $|\alpha |<1/2$.

30 pages