论文标题
二线半线的细分方案
Subdivision schemes on a dyadic half-line
论文作者
论文摘要
在本文中，考虑了用于函数近似和曲线生成的细分方案。在经典情况下，对于实际线上定义的函数，由于在建设性近似理论，信号处理以及生成分形曲线和表面的多个应用中，细分方案的理论广为人知。二线半线的细分方案是一个正期，配备了标准的Lebesgue度量和DigitWise二进制添加操作，其中WALSH函数扮演了指数的作用，并进行了研究。就矩阵的光谱特性以及相应的细化方程溶液的平滑度而言，细分方案的必要和足够收敛条件已证明。还研究了细分方案与非负系数的收敛问题。获得了带有四个系数的细分方案的显式收敛标准。作为辅助结果，定义了二线半线上的分形曲线，并证明了其平滑度的公式。本文包含各种插图和数值结果。
In this paper subdivision schemes, which are used for functions approximation and curves generation, are considered. In classical case, for the functions defined on the real line, the theory of subdivision schemes is widely known due to multiple applications in constructive approximation theory, signal processing as well as for generating fractal curves and surfaces. Subdivision schemes on a dyadic half-line, which is a positive half-line, equipped with the standard Lebesgue measure and the digitwise binary addition operation, where the Walsh functions play the role of exponents, are defined and studied. Necessary and sufficient convergence conditions of the subdivision schemes in terms of spectral properties of matrices and in terms of the smoothness of the solution of the corresponding refinement equation are proved. The problem of the convergence of subdivision schemes with non-negative coefficients is also investigated. Explicit convergence criterion of the subdivision schemes with four coefficients is obtained. As an auxiliary result fractal curves on a dyadic half-line are defined and the formula of their smoothness is proved. The paper contains various illustrations and numerical results.
