Non-stationary Schemes Reproducing Exponential Polynomials

In this paper we develop a novel approach to construct non-stationary subdivision schemeswith a tensor control parameter which can reproduce functions in a finite-dimensionalsubspace of exponential polynomials. The construction process is mainly implemented bysolving linear systems for primal and dual subdivision schemes respectively, based ondifferent parameterizations. The schemes constructed in this paper contain most of theinterpolatory subdivision schemes reproducing polynomials. The rule at resolution level k,operates on values taken from an exponential polynomial from the aforesaid space,reproducing values of the same exponential polynomial at refinement level k+1. Compute themask after giving the system according to the reproduction condition. Also, the existence anduniqueness of the solution is analyzed. The space of exponential polynomials is derived fromconstant differential equation and the discrete values of the same function can be obtainedfrom the corresponding difference equation. Since the refinement is operated on the values ofequidistant parameters, there is strong connection between mask and difference equation andsome specific formations of the mask are mainly based on this fact. Laurent polynomial as apowerful tool for subdivision representation can give a great favor when deducing someuseful systems. In this paper, the mask we construct for polynomials is easy to compute andsome uniform expression is given. As a necessary step, the convergence and smoothness areanalyzed and there are stationary subdivision schemes which are asymptotically equivalent tothe non-stationary schemes in this paper. And hence the smoothness order can be given basedon classical theorem. As a special case, the conic curves can be reproduced easily by ourconstruction method. Since there are tensor control parameters in our schemes, we can selectthe parameters to make the schemes reproduce polynomials.