Whitney extension theorem

Let (respectively, ) be the space of all times differentiable (respectively, smooth) real-valued functions on . Let be compact. For a multi-index , , let , , , and for . The vector space consists of all tuples of continuous functions on indexed by the multi-indices with . For instance, if is a single point, then consists of sequences of real numbers, where , and can be identified with the space of all polynomials of total degree in variables, and can be seen as the space of all power series in variables.

Let assign to the -jet of , i.e. the -tuple of continuous functions restricted to ; cf. also Jet. For each and , let be the polynomial

and let be the element

of with components . The space of functions differentiable in the sense of Whitney on consists of those such that

(*)

Of course, the elements of are not really functions, but that does no harm. If is a point, . The Whitney extension theorem now states that there exists a linear mapping such that for every and every ,

and such that is smooth on .

For it follows that for every power series at (in the variables ) there is a smooth function on whose Taylor series at is precisely this power series.

This results also (by induction on the number of variables) from the Borel extension lemma. Let be a series of smooth functions defined on a neighbourhood of . Then there is a smooth function defined on a neighbourhood of such that for all .

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