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In this paper we present a non-stationary 4-point ternary interpolatory subdivision scheme which provides the user with a tension parameter that, when increased within its range of definition, can generate C 2 -continuous limit curves showing considerable variations of shape.As a generalization we additionally propose a locally-controlled C 2 -continuous subdivision scheme, which allows a different tension value to be assigned to every edge of the original control polygon.

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Though a Hermite subdivision scheme is non-stationary by nature, its non-stationarity can be of two types, making useful the distinction between Inherently Stationary (I.S.) and Inherently Non-Stationary (I.N.S.) Hermite subdivision schemes. This paper focuses on the class of inherently stationary, dual non-interpolatory Hermite subdivision schemes that can be obtained from known Hermite interpolatory ones, by applying a generalization of the de Rham corner cutting strategy. Exploiting specific tools for the analysis of inherently stationary Hermite subdivision schemes we show that, giving up the interpolation condition, the smoothness of the associated basic limit function can be increased by one, while its support width is only enlarged by one. To accomplish the analysis of de Rham-type Hermite subdivision schemes two new theoretical results are derived and the new notion of HC ℓ -convergence is introduced. It allows the construction of Hermite-type subdivision schemes of order d + 1 with the first element of the vector valued limit function having regularity ℓ ≥ d.

a b s t r a c tIn this work we construct three novel families of approximating subdivision schemes that generate piecewise exponential polynomials and we show how to convert these into interpolating schemes of great interest in curve design for their ability to reproduce important analytical shapes and to provide highly smooth limit curves with a controllable tension.In particular, throughout this paper we will focus on the derivation of 6-point interpolating schemes that turn out to be unique in combining vital ingredients like C 2 -continuity, simplicity of definition, ease of implementation, user independency, tension control and ability to reproduce salient trigonometric and transcendental curves.

We study scalar multivariate non-stationary subdivision schemes with a general dilation matrix. We characterize the capability of such schemes to reproduce exponential polynomials in terms of simple algebraic conditions on their symbols. These algebraic conditions provide a useful theoretical tool for checking the reproduction properties of existing schemes and for constructing new schemes with desired reproduction capabilities and other enhanced properties. We illustrate our results with several examples.

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In this paper we present a general strategy to deduce a family of interpolatory masks from a symmetric Hurwitz non-interpolatory one. This brings back to a polynomial equation involving the symbol of the non-interpolatory scheme we start with. The solution of the polynomial equation here proposed, tailored for symmetric Hurwitz subdivision symbols, leads to an efficient procedure for the computation of the coefficients of the corresponding family of interpolatory masks. Several examples of interpolatory masks associated with classical approximating masks are given.