Approximation by Entire Functions in the Construction of Order-Isomorphisms and Large Cross-Sections

Abstract

A theorem of Hoischen states that given a positive continuous function $\varepsilon :\mathbb {R}^t\to \mathbb {R}$, a sequence U ₁ ⊆ U ₂ ⊆… of open sets covering $\mathbb {R}^t$ and a closed discrete set $T\subseteq \mathbb {R}^t$, any C ^∞ function $g:\mathbb {R}^t\to \mathbb {R}$ can be approximated by an entire function f so that for k = 1, 2, …, for all $x\in \mathbb {R}^t\setminus U_k$ and for each multi-index α such that |α|≤ k,

• (a) |(D ^α f)(x) − (D ^α g)(x)| < ε(x);

• (b) (D ^α f)(x) = (D ^α g)(x) if x ∈ T.

This theorem has been useful in helping to analyze the existence of entire functions restricting to order-isomorphisms of everywhere non-meager subsets of $\mathbb {R}$, analogous to the Barth-Schneider theorem, which gives entire functions restricting to order-isomorphisms of countable dense sets, and the existence of entire functions f determining cross-sections f ∩ A through everywhere non-meager subsets A of $\mathbb {R}^{t+1}\cong \mathbb {R}^t\times \mathbb {R}$ whose projection $\{x\in \mathbb {R}^t:(x,f(x))\in A\}$ onto $\mathbb {R}^t$ is everywhere non-meager, analogous to the Kuratowski-Ulam theorem which gives for residual sets A in $\mathbb {R}^{t+1}$, points $c\in \mathbb {R}$ so that the horizontal section of A determined by c has a residual projection $\{x\in \mathbb {R}^t:(x,c)\in A\}$ in $\mathbb {R}^t$. The insights gained from this work have also led to variations on the Hoischen theorem that incorporate the ability to require the values of the derivatives on a countable set to belong to given dense sets or to choose the approximating function so that the graphs of its derivatives cut a small section through a given null set or a given meager set. We discuss these results.