Exponentials reproducing subdivision schemes

This paper is concerned with a family of nonstationary, interpolatory subdivision schemes that have the capability of reproducing functions in a finite-dimensional subspace of exponential polynomials. We give conditions for the existence and uniqueness of such schemes, and analyze their convergence and smoothness. It is shown that the refinement rules of an even-order exponentials reproducing scheme converge to the Dubuc-Deslauriers interpolatory scheme of the same order, and that both schemes have the same smoothness. Unlike the stationary case, the application of a nonstationary scheme requires the computation of a different rule for each refinement level. We show that the rules of an exponentials reproducing scheme can be efficiently derived by means of an auxiliary orthogonal scheme , using only linear operations. The orthogonal schemes are also very useful tools in fitting an appropriate space of exponential polynomials to a given data sequence.