From approximating to interpolatory non-stationary subdivision schemes with the same generation properties

In this paper we describe a general, computationally feasible strategy to deduce a family of interpolatory non-stationary subdivision schemes from a symmetric non-stationary, non-interpolatory one satisfying quite mild assumptions. To achieve this result we extend our previous work [C.Conti, L.Gemignani, L.Romani, Linear Algebra Appl. 431 (2009), no. 10, 1971-1987] to full generality by removing additional assumptions on the input symbols. For the so obtained interpolatory schemes we prove that they are capable of reproducing the same exponential polynomial space as the one generated by the original approximating scheme. Moreover, we specialize the computational methods for the case of symbols obtained by shifted non-stationary affine combinations of exponential B-splines, that are at the basis of most non-stationary subdivision schemes. In this case we find that the associated family of interpolatory symbols can be determined to satisfy a suitable set of generalized interpolating conditions at the set of the zeros (with reversed signs) of the input symbol. Finally, we discuss some computational examples by showing that the proposed approach can yield novel smooth non-stationary interpolatory subdivision schemes possessing very interesting reproduction properties

Hermite-interpolatory subdivision schemes

Stationary interpolatory subdivision schemes for Hermite data that consist of function values and first derivatives are examined. A general class of Hermite-interpolatory subdivision schemes is proposed, and some of its basic properties are stated. The goal is to characterise and construct certain classes of nonlinear (and linear) Hermite schemes. For linear Hermite subdivision, smoothness conditions known from the literature are discussed. In order to allow a simpler construction of suitable nonlinear Hermite subdivision schemes, these conditions are posed as assumptions. For linear Hermite subdivision, explicit schemes that satisfy sufficient conditions for $C^2$-convergence are constructed. This leads to larger classes of $C^2$ schemes than known from the literature. Finally, convexity preserving Hermite-interpolatory subdivision is examined, and some explicit rational schemes that generate $C^1$ limit functions are presented

Stability of subdivision schemes

The stability of stationary interpolatory subdivision schemes for univariate data is investigated. If the subdivision scheme is linear, its stability follows from the convergence of the scheme, but for nonlinear subdivision schemes one needs stronger conditions and the stability analysis of nonlinear schemes is more involved. Apart from the fact that it is natural to demand that subdivision schemes are stable, it also has an advantage in a theoretical sense: is it shown that the approximation properties of stable schemes can very easily be determined

Scalar and Hermite subdivision schemes

AbstractA criterion of convergence for stationary nonuniform subdivision schemes is provided. For periodic subdivision schemes, this criterion is optimal and can be applied to Hermite subdivision schemes which are not necessarily interpolatory. For the Merrien family of Hermite subdivision schemes which involve two parameters, we are able to describe explicitly the values of the parameters for which the Hermite subdivision scheme is convergent

On a-ary Subdivision for Curve Design III. 2m-Point and (2m + 1)-Point Interpolatory Schemes

In this paper, we investigate both the 2m-point a-ary for any a ≥ 2 and (2m + 1)-point a-ary for any odd a ≥ 3 interpolatory subdivision schemes for curve design. These schemes include the extended family of the classical 4- and 6-point interpolatory a-ary schemes and the family of the 3- and 5-point a-ary interpolatory schemes, both having been established in our previous papers (Lian [9]) and (Lian [10])

Shape preserving C2C^2 interpolatory subdivision schemes

Stationary interpolatory subdivision schemes which preserve shape properties such as convexity or monotonicity are constructed. The schemes are rational in the data and generate limit functions that are at least $C^2$. The emphasis is on a class of six-point convexity preserving subdivision schemes that generate $C^2$ limit functions. In addition, a class of six-point monotonicity preserving schemes that also leads to $C^2$ limit functions is introduced. As the algebra is far too complicated for an analytical proof of smoothness, validation has been performed by a simple numerical methodology