Reconstruction of L-splines of polynomial growth from their local weighted average samples

In this paper, we study the reconstruction of cardinal L-spline functions from their weighted local average samples yn=(fh)(n),n∈Z,yn=(fh)(n),n∈Z, where the weight function h(t  ) has support in [−12,12]. It is shown that there are infinitely many L-spline functions which are solutions to the problem: For the given data yn   and given d∈N,d∈N, find a cardinal L-spline f(t)∈Cd−1(R)f(t)∈Cd−1(R) satisfying yn=(fh)(n),n∈Z.yn=(fh)(n),n∈Z. Further, it is shown that for d=1,2d=1,2 and for every nonnegative h   supported in [−12,12], there is a unique solution to this problem if both the samples and the L-splines are of polynomial growth. Moreover, for d > 2, it is shown that for every sample of polynomial growth, the above problem has a unique solution of polynomial growth when the weight function h   supported in [−12,12] is positive definite.