编码分段多项式的有限自动机
Finite Automata Encoding Piecewise Polynomials
AI总结：
针对有限自动机仅能编码线性可微函数的局限，本文基于分层张量积B样条框架提出一种新表示方法，实现了对任意平滑度分段多项式函数的编码，并证明其在无界支撑集下解决计算问题的有效性。
摘要：
有限自动机被用于编码几何图形、函数，并可用于图像压缩与处理。原始方法是将$\mathbb{R}^n$中图形的每个点表示为以某种进制写成的其$n$个坐标的卷积。如果与图形中点相对应的卷积集合被有限自动机接受，则称该图形被编码为有限自动机。以这种方式能编码为有限自动机的唯一可微函数是线性函数。在本文中，我们提出一种表示方法，能够编码具有任意平滑度的分段多项式函数，这实质性地扩展了可编码为有限自动机的函数族。这种表示自然源自分层张量积B样条框架，该框架是数值计算几何中广泛使用的分段多项式。我们证明，当函数的支撑集无界时，有限自动机为解决该框架中出现的计算问题提供了合适的工具。
英文摘要：
Finite automata are used to encode geometric figures, functions and can be used for image compression and processing. The original approach is to represent each point of a figure in $\mathbb{R}^n$ as a convolution of its $n$ coordinates written in some base. Then a figure is said to be encoded as a finite automaton if the set of convolutions corresponding to the points in this figure is accepted by a finite automaton. The only differentiable functions which can be encoded as a finite automaton in this way are linear. In this paper we propose a representation which enables to encode piecewise polynomial functions with arbitrary degrees of smoothness that substantially extends a family of functions which can be encoded as finite automata. Such representation naturally comes from the framework of hierarchical tensor product B-splines, which are piecewise polynomials widely utilized in numerical computational geometry. We show that finite automata provide a suitable tool for solving computational problems arising in this framework when the support of a function is unbounded.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
