Abstract
In this paper, by suitably using the so-called push-back operation, a connection between the approximating and interpolatory subdivision, a new family of nonstationary subdivision schemes is presented. Each scheme of this family is a quasi-interpolatory scheme and reproduces a certain space of exponential polynomials. This new family of schemes unifies and extends quite a number of the existing interpolatory schemes reproducing exponential polynomials and noninterpolatory schemes like the cubic exponential B-spline scheme. For these new schemes, we investigate their convergence, smoothness, and accuracy and show that they can reach higher smoothness orders than the interpolatory schemes with the same reproduction property and better accuracy than the exponential B-spline schemes. Several examples are given to illustrate the performance of these new schemes.
Highlights
Subdivision schemes are efficient tools to generate smooth curves/surfaces from a given set of discrete control points
It is known that stationary schemes generate algebraic polynomials. e nonstationary schemes, can generate the richer function spaces, i.e., the exponential polynomial spaces and special curves/surfaces such as hyperbolas/spheres, which cannot be done using stationary schemes. erefore, there have been continuous works on nonstationary subdivision schemes generating exponential polynomials
Conti et al [13] transformed the nonstationary approximating schemes into interpolatory ones with the same generation property. All of these works can be seen as performed using the polynomial correction, which operates by taking the convex combination of the approximating subdivision masks to derive new subdivision masks, including the interpolatory ones
Summary
Subdivision schemes are efficient tools to generate smooth curves/surfaces from a given set of discrete control points. Unlike the polynomial correction, most of the works related to the push-back operation, except the work in [24], are restricted to the stationary case, and the reproduction property of the obtained interpolatory schemes depends largely on that of the original approximating schemes [20]. Since interpolatory schemes are usually less smooth than the approximating ones, in this paper, we aim to give a different try on the use of the pushback operation to construct new nonstationary subdivision schemes with better properties such as reproduction of exponential polynomials with higher smoothness orders. Us, a nonstationary quasiinterpolatory scheme in this paper is a nonstationary scheme with a good approximation order In this way, these new schemes can own higher smoothness orders than the interpolatory ones with the same reproduction property and better accuracy than the exponential B-spline schemes.
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