Histopolating splines

Given an integrable function f  , we are concerned with the construction of a spline Hn(f)Hn(f) of degree ⩽n⩽n with prescribed knots t=(tj)j∈Zt=(tj)j∈Z that satisfies the histopolation conditions∫tjsn+1t(j+1)sn+1Hn(f)(x)dx=∫tjsn+1t(j+1)sn+1f(x)dx,j∈Zfor some fixed sn+1∈Nsn+1∈N. Additionally, the resulting spline operator should be local and reproduce   all polynomials of degree ⩽n⩽n. Our approach of generating such a histospline is based on a local spline interpolation   operator that is exact for all polynomials of degree ⩽n⩽n.