I'm not much of an analyst myself, but I've time and time again come across proofs which require knowledge of the existence of bump functions. However, I've never studied them, so I'm missing important facts such as what specifications one can make and still be guaranteed the existence of a bump function, and what spaces one can construct bump functions on. What sort of text/chapters should I read to get a quick introduction to the subject and a proof of some important existence theorems pertaining to this?
Asked
Active
Viewed 887 times
0
-
Here is a similar question/answer that might be of some use: http://math.stackexchange.com/questions/799567/where-can-we-find-examples-of-smooth-functions-with-compact-support-is-there-a/799580#799580 – abnry Jun 03 '14 at 17:12
-
Typically you only need one bump function to begin with. Then you can smooth out pretty much any function, like a step function, by convolving with a scaled (small) bump function. – abnry Jun 03 '14 at 17:14
-
One standard exercise to start with is to show that the function, $f(x) = 0$ if $|x| > 1$ and $f(x) = e^{-\frac{1}{1-x^2}}$ for $|x| \leq 1$ is infinitely differentiable and has support in $[-1, 1]$. – Mustafa Said Jun 03 '14 at 19:57
-
For a quick intro, Wikipedia suffices. – Jun 03 '14 at 21:05
1 Answers
0
In my case, the reference which turned out to me most useful to me is Lee's Introduction to Smooth Manifolds, 2nd ed., Chapter 2.
Alex G.
- 8,848