Whitney extension theorem

$\def\a{\alpha} \def\b{\beta} \def\p{\partial}$

A deep theorem from the real analysis, showing which data are required to extent a real-valued function from a compact subset in $\R^n$ to its open neighborhood in a $C^m$-smooth or $C^\infty$-smooth way.

Jets and terminology

If $U$ is an open subset in $\R^n$ and $f:U\to\R$ is a smooth function, then one can define its partial derivatives to any order not exceeding the smoothness: in the multi-index notation the collection of all derivatives $$ f^{(\a)}=\p^\a f\in C^{m-|\a|}(U),\qquad 0\le |\a|\le m,\ f^{(0)}=f, $$ is called an $m$-jet of the function $f\in C^m(U)$.

The different derivatives are related by the obvious formulas $\p^\b f^{(a)}=f^{\a+\b)}$ as long as $|\a|+|\b|\le m$. This allows to compare them using the Taylor expansion. For each point $a\in U$ and each derivative $f^{(\a)}$ one can form the Taylor polynomial of order $r\le m-|\a|$ centered at $a$, $$ \Big(T_a^r f^{(\a)}\Big)(x)=\sum_{|\b|\le r}\frac1{\b!}\Big(\p^\b f^{(\alpha)}(a)\Big)\cdot(x-a)^{\b}= \sum_{|\b|\le r}\frac1{\b!}f^{(\a+\b)}(a)\cdot(x-a)^{\b}.\tag T $$ The difference between $f^{(\a)}(x)$ and the value provided by the Taylor polynomial $\Big(T_a^r f^{(\a)}\Big)(x)$ should be small together with $|x-a|$: $$ \Big|f^{(\a)}(x)-\Big(T_a^r f^{(\a)}\Big)(x)\Big|=o\Big(|x-a|^{r-|\a|}\Big). $$ These asymptotic conditions are necessary for the functions $f^{(\a)}$ to be partial derivatives of a smooth function.

Whitney data

Let $K\Subset U$ be a compact subset of $U$. The Whitney data (or "smooth function in the sense of Whitney") is the collection of continuous functions $$ \mathbf f=\{f^\a:K\to\R,\ |\a|\le m\}, $$ which satisfies the compatibility condition that were established above for the partial derivatives: for each multiindex $\a$ and natural $r$ the differences $$ R_m^\a(a,x)=f^\a(x)-\sum_{|\b|\le m-|\a|}\frac1{\b!}f^{\a+\b}(a)\cdot(x-a)^\b,\qquad x,a\in K, $$ should be small as specified, $$ |R^\a_m(a,x)|=o\Big(|x-a|^{r-|\a|}\Big),\qquad x,a\in K,\ |x-a|\to0. $$

References

[a1]	H. Whitney, "Analytic extensions of differentiable functions defined in closed sets" Trans. Amer. Math. Soc. , 36 (1934) pp. 63–89 MR1501735 Zbl 0008.24902 Zbl 60.0217.01
[a2]	B. Malgrange, "Ideals of differentiable functions" , Oxford Univ. Press (1966) pp. Chapt. I MR2065138 MR0212575 Zbl 0177.17902
[a3]	J.C. Tougeron, "Ideaux de fonction différentiables" , Springer (1972) pp. Chapt. IV MR0440598
[a4]	M. Golubitsky, "Stable mappings and their singularities" , Springer (1973) pp. 108ff MR0341518 Zbl 0294.58004