Difference between revisions of "Whitney extension theorem"
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− | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> | + | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> H. Whitney, "Analytic extensions of differentiable functions defined in closed sets" ''Trans. Amer. Math. Soc.'' , '''36''' (1934) pp. 63–89 {{MR|1501735}} {{ZBL|0008.24902}} {{ZBL|60.0217.01}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> B. Malgrange, "Ideals of differentiable functions" , Oxford Univ. Press (1966) pp. Chapt. I {{MR|2065138}} {{MR|0212575}} {{ZBL|0177.17902}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> J.C. Tougeron, "Ideaux de fonction différentiables" , Springer (1972) pp. Chapt. IV {{MR|0440598}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> M. Golubitsky, "Stable mappings and their singularities" , Springer (1973) pp. 108ff {{MR|0341518}} {{ZBL|0294.58004}} </TD></TR></table> |
Revision as of 17:02, 15 April 2012
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 |
Whitney extension theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Whitney_extension_theorem&oldid=11936