Existence and uniqueness of spline reconstruction from local weighted average samples

Abstract

The sampling theorem is one of the most powerful and useful results in signal analysis. However, most of the results on sampling theorem assumes that the distances between the consecutive sampling points are strictly less than one. In this paper, we consider the case when the distance between the consecutive sampling points is always one and study the reconstruction of cardinal spline functions from their weighted local average samples \(y_{n}=f\star h(n),n\in {\mathbb {Z}}\), where the weight function \(h(t)\) has support in \([-\frac{1}{2}, \frac{1}{2}].\) We prove under certain restrictive conditions on the average function \(h\) that for a given \(y_{n}\), there exists a unique cardinal spline \(f(t)\) of degree d satisfying \(y_{n}=f\star h(n),n\in {\mathbb {Z}}.\)