ON REAL UNIVERSALITY IN THE BIRKHOFF SENSE

Abstract

In this paper we present a proof of the following statement: there is a ${\mathcal{C}}^{\infty }$ function $f$ on ${\mathrm{ℝ}}^n$ such that any other ${\mathcal{C}}^{\infty }$ function on ${\mathrm{ℝ}}^n$ is the uniform limit, on the compact subsets of ${\mathrm{ℝ}}^n$, 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 $2^{{\aleph }_0}$ linearly independent real ${\mathcal{C}}^{\infty }$ 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.