Surface-tracing approximation by basis functions and its applications to neural networks

By a constructive method, it is proved that a linear sum of rather a small number of differentiable basis functions can well trace any surface defined by a polynomial in several variables on any compact sets. The linear sum is obtained in an explicit form. When the basis function is sufficiently many times differentiable, the tracing is so well done that the surfaces defined by derivatives of the polynomial can be simultaneously traced. This capability of a smooth basis function is partly based on the fact that it has various features of curvature. The radial basis function in two variables, for example, can trace any kinds of quadratic surfaces. This is different from the case of approximation by sigmoid functions. Since the proofs in this paper are concrete, they can be used as algorithms in applications