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Finite automata are used to encode geometric figures, functions and can beused for image compression and processing. The original approach is torepresent each point of a figure (a graph of a function) in $\mathbb{R}^n$ as aconvolution of its $n$ coordinates written in some base. Then a figure is saidto be encoded as a finite automaton if the set of convolutions corresponding topoints in this figure is accepted by a finite automaton. Jurgensen, Staiger andYamasaki showed that the only continuously differentiable functions which canbe encoded as a finite automaton in this way are linear. In this paper wepropose a representation which enables to encode piecewise polynomial functionswith arbitrary degrees of smoothness that substantially extends a family offunctions which can be encoded as finite automata. This representationnaturally comes from the framework of hierarchical tensor product B-splinesutilized in numerical computational geometry. We show that finite automataprovide a simple tool suitable for solving computational problem arising inthis framework including the case when the support of a function is unbounded.

Finite Automata Encoding Functions: A Representation Using B-splines