ON REAL UNIVERSALITY IN THE BIRKHOFF SENSE

Abstract

In this paper we present a proof of the following statement: there is a C∞ function f on ℝn such that any other C∞ function on ℝn is the uniform limit, on the compact subsets of ℝn, of translations of f by natural numbers. This is a real version of the well-known Birkhoff’s result on the existence of a function with a similar property in the space of entire functions. Afterwards, we show that the technique used in our proof allows us to create 2ℵ0 linearly independent real C∞ universal functions. We also demonstrate that we may even obtain real analytic universal functions (in the sense of translations) by using Whitney’s Approximation Theorem.