Polynomial reproduction for univariate subdivision schemes of any arity

Abstract

In this paper, we study the ability of convergent subdivision schemes to reproduce polynomials in the sense that for initial data, which is sampled from some polynomial function, the scheme yields the same polynomial in the limit. This property is desirable because the reproduction of polynomials up to some degree d implies that a scheme has approximation order d + 1. We first show that any convergent, linear, uniform, and stationary subdivision scheme reproduces linear functions with respect to an appropriately chosen parameterization. We then present a simple algebraic condition for polynomial reproduction of higher order. All results are given for subdivision schemes of any arity m ≥ 2 and we use them to derive a unified definition of general m-ary pseudo-splines. Our framework also covers non-symmetric schemes and we give an example where the smoothness of the limit functions can be increased by giving up symmetry.

Exponential pseudo-splines: Looking beyond exponential B-splines

“…To conclude the proof we observe that, when θ ℓ = 0, the reproduction of the pair {1, x} is obtained by setting p = 0 if the k-level symbol is odd-symmetric and p = − 1 2 if it is even-symmetric, as shown in [15]. We continue by analyzing useful algebraic properties fulfilled by symmetric subdivision symbols.…”

“…In particular, when p ∈ Z the parametrization is termed primal, whereas if p ∈ Z 2 it is called dual. For a complete discussion concerning the choice of the parametrization in the analysis of the polynomial reproduction properties of stationary subdivision schemes, we refer the reader to [5,15,26].…”

“…, M − 1 for reproduction of polynomials up to degree M − 1. Its actual degree of polynomial reproduction is thus min{N − 1, M − 1} (see [15,24]). …”

“…Binary, primal and dual (polynomial) pseudo-splines have been more recently generalized to any arity and to arbitrary parametrizations by [15]. Both primal and dual (polynomial) pseudo-splines are obtained by means of stationary subdivision schemes whose symbols can be read as a suitable polynomial "correction" of the polynomial B-spline symbol.…”

A constructive algebraic strategy for interpolatory subdivision schemes induced by bivariate box splines

“…For the corresponding interpolatory subdivision schemes a convergence and smoothness analysis is also conducted together with a polynomial reproduction investigation. For all related theoretical results and definitions we refer the reader to [4] (multivariate extension of the results contained in the papers [8], [9]). …”