Summer Conference on Topology and Its Applications
 

Document Type

Topology + Algebra and Analysis

Publication Date

6-2017

Publication Source

32nd Summer Conference on Topology and Its Applications

Abstract

A theorem of Hoischen states that given a positive continuous function ε:Rn→R, an unbounded sequence 0 ≤ c1 ≤ c2 ≤ ... and a closed discrete set T ⊆ Rn, any C∞ function g:Rn→R can be approximated by an entire function f so that for k=0, 1, 2, ..., for all x ∈ Rn such that |x| ≥ ck, 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.

We show that if C ⊆ Rn+1 is meager, A ⊆ Rn is countable and disjoint from T, and for each multi-index α and p ∈ A we are given a countable dense set Ap, α ⊆ R, then we can require also that

        (c) (D α f)(p) ∈ A p, α for p ∈ A and α any multi-index;
          (d) if x ∉ T, q=(D α f)(x) and there are values of p ∈ A arbitrarily close to x for which q ∈ A p, α, then there are values of p ∈ A arbitrarily close to x for which q=(D α f)(p)
          (e) for each α, {x ∈

Rn

          : (x, (D α f)(x)) ∈ C} is meager in

Rn

        .

Clause (d) is a surjectivity property which can be strengthened to allow for finding solutions in A to equations of the form q=h*(x, (Dα f)(x)) under similar assumptions, where h(x, y)=(x, h*(x, y)) is one of countably many given fiber-preserving homeomorphisms of open subsets of Rn+1 ≅ Rn×R.

We also prove a weaker corresponding result with "meager" replaced by "Lebesgue null." In this context, the approximating function is C∞ rather than entire, and we do not know whether it can be taken to be entire.

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