Exponential splines and pseudo-splines: generation versus reproduction of exponential polynomials

Subdivision schemes are iterative methods for the design of smooth curves and surfaces. Any linear subdivision scheme can be identified by a sequence of Laurent polynomials, also called subdivision symbols, which describe the linear rules determining successive refinements of coarse initial meshes. The goal of this work is to derive explicit expressions of the subdivision symbols of exponential pseudo-splines reproducing exponential polynomials and to study their symmetry properties. The $k$-level subdivision symbol of an exponential pseudo-spline is defined by a suitable polynomial ``correction'' of the $k$-level subdivision symbol of an exponential B-spline. The polynomial correction is such that the resulting subdivision symbol is the one of minimal support that fulfills the conditions for reproduction of the space of exponential polynomials generated by the exponential B-spline, or a subset of it. Reproduction of exponential polynomials is indeed an important property in many applicative areas ranging from computer--aided geometric design to signal and image processing.