Reconstruction of splines from local average samples

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