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

Chapter
Part of the Fields Institute Communications book series (FIC, volume 81)

Abstract

A theorem of Hoischen states that given a positive continuous function \(\varepsilon :\mathbb {R}^t\to \mathbb {R}\), a sequence U1 ⊆ U2 ⊆… 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,
  1. (a)

    |(D α f)(x) − (D α g)(x)| < ε(x);

     
  2. (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.

Keywords

Complex approximation Interpolation Hoischen’s theorem Order-isomorphism Piecewise monotone Kuratowski-Ulam theorem Sup-measurable Oracle-cc forcing 

1991 Mathematics Subject Classification.

Primary 30E10; Secondary 54E52 28A35 26B35 

Notes

Acknowledgements

Research supported by NSERC. The author thanks the Fields Institute and the organizers of New Trends in Approximation Theory: A Conference in Memory of André Boivin, as well as the organizers of the special session on Complex Analysis and Operator Theory at the 2015 Canadian Mathematical Society Winter Meeting for their support. He also thanks Paul Gauthier for helpful discussions and correspondence.

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Authors and Affiliations

  1. 1.School of Mathematical and Computational SciencesUniversity of Prince Edward IslandCharlottetownCanada

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