Local interpolation with optimal polynomial exactness in refinement spaces

Abstract:

A constructive existence result for a class of polynomial identities is established and applied in two related contexts. First, an algorithm is developed for the explicit construction of a sequence of local interpolation operators mapping the space of continuous functions on the real line into the nested sequence of refinement spaces generated by the shifts of a given refinable function, and where optimal polynomial exactness, as governed by the order of the sum-rule condition satisfied by the corresponding refinement sequence, is achieved. The above algorithm requires as input only the values at the integers of the refinable function, and we proceed, secondly, to derive sufficient conditions for the existence of a refinable function with prescribed values at the integers. As our main examples, we consider the cardinal $B$-spline case, as well as refinable functions with normalized binomial coefficient values at the integers.