Approximation of continuous functions of several variables by an arbitrary nonlinear continuous function of one variable, linear functions, and their superpositions

Abstract

Linear spaces of continuous functions of real variables closed under the superposition operation are considered. It has been proved that when such a space contains constants, linear functions, and at least one nonlinear function, it is dense in the space of all continuous functions in the topology of uniform convergence on compact sets. So, the approximation of continuous functions of several variables by an arbitrary nonlinear continuous function of one variable, linear functions, and their superpositions is possible.