Finite Automata Encoding Piecewise Polynomials

Abstract: 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.