#### 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:**R**^{n}→**R**, an unbounded sequence 0 ≤ c_{1} ≤ c_{2} ≤ ... and a closed discrete set T ⊆ **R**^{n}, any C^{∞} function g:**R**^{n}→**R** can be approximated by an entire function f so that for k=0, 1, 2, ..., for all x ∈ **R**^{n} such that |x| ≥ c_{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.

We show that if C ⊆ **R**^{n+1} is meager, A ⊆ **R**^{n} is countable and disjoint from T, and for each multi-index α and p ∈ A we are given a countable dense set A_{p, 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 ∈

**R**^{n }

- : (x, (D α f)(x)) ∈ C} is meager in

**R**^{n }

- .

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, (D^{α} 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 **R**^{n+1} ≅ **R**^{n}×**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.

#### Copyright

Copyright © 2017, the Author

#### eCommons Citation

Burke, Maxim R., "Generic Approximation and Interpolation by Entire Functions via Restriction of the Values of the Derivatives" (2017). *Summer Conference on Topology and Its Applications*. 44.

http://ecommons.udayton.edu/topology_conf/44

## Comments

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