Dealing with an unexpected breach of my GMail address book kept me busy yesterday. But at least I get this one up before today’s activities.
Now we come to the heart of our partitions of unity: the bump functions. These are like smooth analogues of characteristic functions. A characteristic function 
is defined as 
on a set and 
off of it. We can use them (and have!) to “mask” off a function 
; multiply by and suddenly is supported on 
. But doing this introduces some nasty discontinuities.
A bump function 
fixes the problem by smoothly tailing off to zero between an inner set 
and an outer open set 
that contains the closure of . Then the product 
will be at least as smooth as the original function was, except in the case of analytic functions. On , is identically , and so 
for points 
. Outside of , 
.
So let 
be the open cube in 
consisting of those vectors with each of their components in the interval 
. We will start by constructing a bump function between 
and 
.
The real core here is the function
I leave it to you to verify that this function is, in fact, smooth at 
; show that each derivative of the function on the right is zero at this point. It’s clearly not analytic, though, since its Taylor series at this point sums to the zero function.
Now, consider the function 
. If 
or 
, one or the other factor is zero, and so the product is supported inside ![[-2,2]](http://s0.wp.com/latex.php?latex=%5B-2%2C2%5D&bg=e6e6e6&fg=333333&s=0 "[-2,2]")
. We can also write down 
, which is everywhere strictly greater than zero, meaning we can divide by it:
If 
then 
, while if 
then 
. So on the whole interval ![[-1,1]](http://s0.wp.com/latex.php?latex=%5B-1%2C1%5D&bg=e6e6e6&fg=333333&s=0 "[-1,1]")
, this quotient is exactly .
Therefore is a bump function between the intervals 
and 
in 
. For , just define
and we have a suitable bump function between the cubes and for any dimension.
March 12, 2011
Posted by John Armstrong |
Differential Topology, Topology |
4 Comments
