Nonlinear stationary subdivision schemes reproducing hyperbolic and trigonometric functions

Abstract

In this paper we define a family of nonlinear, stationary, interpolatory subdivision schemes with the capability of reproducing conic shapes including polynomials upto second order. Linear, nonstationary, subdivision schemes do also achieve this goal, but different conic sections require different refinement rules to guarantee exact reproduction. On the other hand, with our construction, exact reproduction of different conic shapes can be achieved using exactly the same nonlinear scheme. Convergence, stability, approximation and shape preservation properties of the new schemes are analyzed. In addition, the conditions to obtain C 1 limit functions are also studied.

Annihilation Operators for Exponential Spaces in Subdivision

“…. Moreover, when applied to a sequence of function values sampled from F ∈ E Γ , i.e., F k := {F (2 −k α), α ∈ Z 2 }, k ∈ N, the equations in (11) allow the identification of γ and γ exactly as done in the univariate case (see (2) and for details [5]). The latter is crucial to succeed in designing bivariate subdivision rules reproducing the space E Γ without any preliminary knowledge of γ and γ.…”

“…This is also true in the non-linear case where the derivation of subdivision rules that guarantee the preservation of exponential-polynomials rely on the definition of an annihilation operator (also called annihilator ) whose kernel is consisting of them. Indeed, in [5], the construction of an interpolatory, non-linear, stationary subdivision scheme capable of reproducing functions in the space E Γ := span{1, exp(γz), exp(−γz), z ∈ R} with γ ∈ R >0 ∪ ı(0, π), is based on an annihilation operator for the space E Γ . For f : R −→ C and γ ∈ R >0 ∪ ı(0, π), this annihilator is obtained by the repeated application of the differential operator…”

“…Section 2 characterizes exponential spaces as the kernel of an annihilation operator based on the repeated application of a particular differential operator. Section 3 shows how this differential operator can be replaced by a discrete operator, which is the generalization to the bivariate setting of the finite difference operator in (1), proposed in [5] for univariate functions.…”