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.

翻译：有限自动机可用于编码几何图形和函数，并能应用于图像压缩与处理。原始方法是将$\mathbb{R}^n$空间中图形的每个点表示为其$n$个坐标在特定进制下展开的卷积。若某图形中所有点对应的卷积集合能被有限自动机接受，则称该图形可由有限自动机编码。通过此方式可编码的唯一可微函数仅为线性函数。本文提出一种表示方法，能够编码具有任意光滑度的分段多项式函数，从而显著扩展了可由有限自动机编码的函数族。该表示方法自然源于层次张量积B样条的框架，这类分段多项式在数值计算几何领域已被广泛应用。我们证明，当函数的支撑集无界时，有限自动机为此框架中产生的计算问题提供了合适的求解工具。