I (very surprisingly) bumped into Charles Fefferman at Northwestern this afternoon…Hence we talked math for a little bit. Among other things I mentioned that I’ve been trying to extend functions to the disc volume-preservingly. After trying on the board for a while, he laughs out loud when he saw that this may be obtained applying his favorite Whitney’s extension theorem. (I’ll discuss what he did later in the poster)
Mean while, it’s a pity that I’ve never written a post on Whitney’s extension theorem, hence here it is~
Given a compact subset in and a function , when can we extend it to a function on the whole ?
First we note that there are obvious cases for which this can’t be done: for example, if we take to be a segment in and a one-variable function of lower regularity than , then of course there are no way to find a extension.
Hence it’s only reasonable to restrict our attention to those that has ‘candidate derivatives’ of all orders no larger than at all points in .
i.e. For any -fold subscript with (we will denote , there is a continuous function with the following property:
For all , where as and is uniform in .
i.e. The functions are compatible as Taylor coefficients of some function on , which is absolutely necessary for a extension to exist.
Whitney’s extension theorem: (classical version)
Suppose a set of functions with all multi-index satisfying the above Taylor condition at all points in $K$. Then there is a function s.t. and for all , . Furthermore, can be taken real analytic on .
This is indeed the best one could hope for. i.e. there is a extension whenever possible, furthermore the extension is at worst at the points which it is given to be only and much better (analytic) everywhere else.
However, sometimes we would like to control the norm of the resulting function in terms of the norm of the function on .
For any , there exists such that the extension where the norm is the norm.