Histopolating splines

Given an integrable function f, we are concerned with the construction of a spline H n ( f ) of degree ⩽ n with prescribed knots t = ( t j ) j ∈ Z that satisfies the histopolation conditions ∫ t js n + 1 t ( j + 1 ) s n + 1 H n ( f ) ( x ) d x = ∫ t js n + 1 t ( j + 1 ) s n + 1 f ( x ) d x , j ∈ Z for some fixed s n + 1 ∈ N . Additionally, the resulting spline operator should be local and reproduce all polynomials of degree ⩽ 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 .