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

Abstract

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 C m -function on R n . 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 R n to R .How can we determine whether f is the restriction of a C m -function on R n ? 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 C m -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.