On the reproduction properties of non-stationary subdivision schemes

TL;DR

This paper investigates the algebraic conditions necessary for non-stationary subdivision schemes to reproduce exponential polynomial spaces, extending polynomial reproduction theory and enabling the construction of schemes with specific reproduction capabilities.

Contribution

It provides algebraic conditions for exponential polynomial reproduction in non-stationary schemes and demonstrates how to construct schemes with desired reproduction properties.

Findings

Derived algebraic conditions for reproduction of exponential polynomials.

Extended polynomial reproduction theory to non-stationary schemes.

Enabled construction of new schemes with specific reproduction properties.

Abstract

We present an accurate investigation of the algebraic conditions that the symbols of a convergent, univariate, binary, non-stationary subdivision scheme should fulfill in order to reproduce spaces of exponential polynomials. A subdivision scheme is said to possess the property of reproducing exponential polynomials if, for any initial data uniformly sampled from some exponential polynomial function, the scheme yields the same function in the limit. The importance of this property is due to the fact that several functions obtained as combinations of exponential polynomials (such as conic sections, spirals or special trigonometric and hyperbolic functions) are of great interest in graphical and engineering applications. Since the space of exponential polynomials trivially includes standard polynomials, the results in this work extend the recently developed theory on polynomial reproduction to the non-stationary context. A significant application of the derived algebraic conditions on the subdivision symbols is the construction of new non-stationary subdivision schemes with specified reproduction properties.