Superposition, reduction of multivariable problems, and approximation

TL;DR

This paper develops explicit reduction schemes for multivariable functions into superpositions of single-variable functions, applicable in finite and infinite dimensions, using transform-based methods like Fourier, Radon, and Shannon interpolation.

Contribution

It introduces constructive, explicit reduction schemes for multivariable problems into superpositions of single-variable functions, extending to infinite-dimensional settings.

Findings

Explicit reduction schemes for finite variables

Extension to infinite-dimensional functions

Use of Fourier, Radon, and Shannon transforms

Abstract

We study reduction schemes for functions of "many" variables into system of functions in one variable. Our setting includes infinite-dimensions. Following Cybenko-Kolmogorov, the outline for our results is as follows: We present explicit reductions schemes for multivariable problems, covering both a finite, and an infinite, number of variables. Starting with functions in "many" variables, we offer constructive reductions into superposition, with component terms, that make use of only functions in one variable, and specified choices of coordinate directions. Our proofs are transform based, using explicit transforms, Fourier and Radon; as well as multivariable Shannon interpolation.