  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.
