 #### Document Type

Topology + Algebra and Analysis

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:RnR, an unbounded sequence 0 ≤ c1 ≤ c2 ≤ ... and a closed discrete set T ⊆ Rn, any C function g:RnR 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, αR, 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+1Rn×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.

This document is available for download with the permission of the presenting author and the organizers of the conference. Permission documentation is on file.

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