An annihilator-based strategy for the automatic detection of exponential polynomial spaces in subdivision

Abstract

Exponential polynomials are essential in subdivision for the reconstruction of specific families of curves and surfaces, such as conic sections and quadric surfaces. It is well known that if a linear subdivision scheme is able to reproduce a certain space of exponential polynomials, then it must be level-dependent, with rules depending on the frequencies (and eventual multiplicities) defining the considered space. This work discusses a general strategy that exploits annihilating operators to locally detect those frequencies directly from the given data and therefore to choose the correct subdivision rule to be applied. This is intended as a first step towards the construction of self-adapting subdivision schemes able to locally reproduce exponential polynomials belonging to different spaces. An application of the proposed strategy is shown explicitly on an example involving the classical butterfly interpolatory scheme. This particular example is the generalization of what has been done for the univariate case in Donat and López-Ureña (2019), which inspired this work.