Exponential Splines and Pseudo-Splines: Generation versus reproduction of exponential polynomials

TL;DR

This paper introduces exponential pseudo-splines, a family of subdivision schemes capable of reproducing exponential polynomials, and analyzes their explicit symbols, symmetry, convergence, and regularity properties.

Contribution

It derives explicit formulas for exponential pseudo-spline subdivision symbols and examines their mathematical properties, extending the pseudo-spline framework to exponential polynomials.

Findings

Explicit formulas for exponential pseudo-spline symbols are provided.

Exponential pseudo-splines exhibit desirable symmetry and convergence properties.

The regularity of exponential pseudo-splines is characterized and analyzed.

Abstract

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. One important property of subdivision schemes is their capability of exactly reproducing in the limit specific types of functions from which the data is sampled. Indeed, this property is linked to the approximation order of the scheme and to its regularity. When the capability of reproducing polynomials is required, it is possible to define a family of subdivision schemes that allows to meet various demands for balancing approximation order, regularity and support size. The members of this family are known in the literature with the name of pseudo-splines. In case reproduction of exponential polynomials instead of polynomials is requested, the resulting family turns out to be the non-stationary counterpart of the one of pseudo-splines, that we here call the family of exponential pseudo-splines. The goal of this work is to derive the explicit expressions of the subdivision symbols of exponential pseudo-splines and to study their symmetry properties as well as their convergence and regularity.