Reconstruction of splines from local average samples

We study the reconstruction of cardinal splines f(t)f(t) from their average samples yn=f∗h(n)yn=f∗h(n), n∈Zn∈Z, when the average function h(t)h(t) has support in [−1/2,1/2][−1/2,1/2]. We investigate the existence and uniqueness of the solution of the following problem: For given dates ynyn, find a cardinal spline f(t)f(t), of a given degree, satisfying  yn=f∗h(n)yn=f∗h(n), n∈Zn∈Z.