Totally positive refinable functions with general dilation M

We construct a new class of approximating functions that are M-refinable and provide shape preserving approximations. The refinable functions in the class are smooth, compactly supported, centrally symmetric and totally positive. Moreover, their refinable masks are associated with convergent subdivision schemes. The presence of one or more shape parameters gives a great flexibility in the applications. Some examples for dilation M=4and M=5are also given

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])

Circular Nonlinear Subdivision Schemes for Curve Design

Two new families of nonlinear 3-point subdivision schemes for curve design are introduced. The first family is ternary interpolatory and the second family is binary approximation. All these new schemes are circular-invariant, meaning that new vertices are generated from local circles formed by three consecutive old vertices. As consequences of the nonlinear schemes, two new families of linear subdivision schemes for curve design are established. The 3-point linear binary schemes, which are corner-cutting depending on the choices of the tension parameter, are natural extensions of the Lane-Riesenfeld schemes. The four families of both nonlinear and linear subdivision schemes are implemented extensively by a variety of examples

On a-ary Subdivision for Curve Design: I. 4-Point and 6-Point Interpolatory Schemes

The classical binary 4-point and 6-point interpolatery subdivision schemes are generalized to a-ary setting for any integer a greater than or equal to 3. These new a-ary subdivision schemes for curve design are derived easily from their corresponding two-scale scaling functions, a notion from the context of wavelets