Separation of zeros and a Hermite interpolation based frame algorithm for band limited functions

TL;DR

This paper establishes a zero separation property for band-limited functions with multiple zeros and introduces a Hermite interpolation-based frame algorithm for reconstructing such functions from nonuniform samples under certain gap conditions.

Contribution

It provides a novel zero separation result for band-limited functions with multiple zeros and develops a frame reconstruction method using Hermite interpolation under maximum gap constraints.

Findings

Existence of a zero gap greater than a specific bound for functions with double zeros.

A frame algorithm for function reconstruction from nonuniform samples.

Reconstruction guaranteed under a maximum gap condition related to Wirtinger-Sobolev constants.

Abstract

It is shown that if a non-zero function $f\in B_\sigma$ has infinitely many double zeros on the real axis, then there exists at least one pair of consecutive zeros whose distance apart is greater than $\dfrac{\pi}{\sigma}\tau^{1/4}$, $\tau\approx5.0625$. A frame algorithm is provided for reconstructing a function $f\in B_\sigma$ from its nonuniform samples $\{f^{(j)}(x_i):j=0,1,\dots, k-1, i\in\mathbb{Z}\}$ with maximum gap condition, $\sup\limits_i(x_{i+1}-x_i)=\delta<\dfrac{1}{\sigma}c_k^{1/2k}$, where $c_k$ is a Wirtinger-Sobolev constant, using Hermite interpolation.