Construction of irregular complete interpolation sets for shift-invariant spaces

For several shift-invariant spaces, there exists a real number a such that the set a+Z is a complete interpolation set. In this paper, we characterize the complete interpolation property of the set (a+N₀)∪(α+a+N⁻) for shift-invariant spaces using Toeplitz operators. Using this characterization, we determine all α for which the sample set N₀∪α+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 /2+N₀∪α+/2+N⁻ is a complete interpolation set for a shift-invariant spline space of order m≥ 2 if and only if |α|<1/2.