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.

中文翻译：

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二元半线上的细分方案

本文考虑了用于函数逼近和曲线生成的细分方案。在经典情况下，对于在实线上定义的函数，由于在构造逼近理论、信号处理以及生成分形曲线和曲面中的多种应用，细分方案理论广为人知。定义并研究了二元半线的细分方案，它是一条正半线，配备了标准 Lebesgue 测度和数字二进制加法运算，其中 Walsh 函数扮演指数的角色。证明了细分方案在矩阵谱性质和相应细化方程解的平滑性方面的充分必要收敛条件。还研究了具有非负系数的细分方案的收敛问题。得到了具有四个系数的细分方案的显式收敛准则。作为辅助结果，定义了二进半线上的分形曲线，并证明了它们的光滑度公式。该论文包含各种插图和数值结果。