Gaussian Functions Combined with Kolmogorov’s Theorem as Applied to Approximation of Functions of Several Variables

Abstract

A special class of approximations of continuous functions of several variables on the unit coordinate cube is investigated. The class is constructed using Kolmogorov’s theorem stating that functions of the indicated type can be represented as a finite superposition of continuous single-variable functions and another result on the approximation of such functions by linear combinations of quadratic exponentials (also known as Gaussian functions). The effectiveness of such a representation is based on the author’s previously proved assertion that the Mexican hat mother wavelet on any fixed bounded interval can be approximated as accurately as desired by a linear combination of two Gaussian functions. It is proved that the class of approximations under study is dense everywhere in the class of continuous multivariable functions on the coordinate cube. For the case of continuous functions of two variables, numerical results are presented that confirm the effectiveness of approximations of the studied class.