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Extensions and Smooth Approximations of Definable Functions in O-minimal Structures


In 1934, H. Whitney presented a series of papers which discussed how to determine whether a function or a jet of order m is the restriction of a Cm-function on Rn. In the first paper of the series, Whitney's Extension Theorem was proved. In the latter, Whitney answered special cases of the following question

Question(Whitney's Extension Theorem, WEPn,m) Let f be a continuous function from a closed subset E of Rn to R.

How can we determine whether f is the restriction of a Cm-function on Rn?

In this dissertation, we work in o-minimal expansions of real closed ordered fields. Definable versions of Whitney's Extension Theorem and Whitney's Extension Problems will be discussed in this context. Definable set-valued maps are also studied; a definable version of Michael's Selection Theorem will be proved and used, in combination with a definable version of Whitney's Extension Theorem, to give a positive answer to a definable version of WEPn,1.

In addition to the above problems, we also discuss smoothing problems. This is inspired by a series of papers by A. Fischer.

In this series, a construction of a definable Cm-approximation of a definable locally Lipschitz function is provided. Here, we also work in an o-minimal expansion of a real closed field and relax the condition further to just continuous.

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