Exponential polynomial reproducing property of non-stationary symmetric subdivision schemes and normalized exponential B-splines

One of the important capabilities for a subdivision scheme is the reproducing property of circular shapes or parts of conics that are important analytical shapes in geometrical modelling. In this regards, the first goal of this study is to provide necessary and sufficient conditions for a non-stationary subdivision to have the reproducing property of exponential polynomials. The result in fact extends the work of Dyn et al. [12], where the conditions for algebraic polynomial reproduction are discussed, to the case of non-stationary schemes. Then, we provide the approximation order of a non-stationary scheme reproducing a certain set of exponential polynomials. Next, we find that an exponential B-spline generates exponential polynomials in the associated spaces, but it may not reproduce any exponential polynomials. Thus, we present normalized exponential B-splines that reproduce certain sets of exponential polynomials. One interesting feature is that depending on the normalization factor, the set of exponential polynomials to be reproduced is varied. This provides us with the necessary accuracy and flexibility in designing target curves and surfaces. Some numerical results are presented to support the advantages of the normalized scheme by comparing them to the results without normalization.