Math 642-600 Current Assignments — Spring 2010
Assignment 7 - Not to be turned in.
- Read sections 10.1-10.3
- Do the following problems.
- Let f be in Schwartz space, and let g be C∞ and
satisfy
|g(j)(x)| ≤
cj(1+x2)nj
for all nonnegative integers j. Here cj and nj
depend on g and j. Show that fg is in Schwartz space. Explain how to
define the product g(x)T(x), where T is a tempered distribution.
- Let T be a tempered distribution. Find the Fourier transforms for
T′, xT(x), T(x−a), and eibxT(x).
- Let T(x) := (1 - |x|)+, which is a "hat"
function. Explain why this is a tempered distribution. Find T′
and T′′, along with their Fourier transforms. Use these to
find the Fourier transform of T.
- Section 10.3: 1, 4, 7, 9, 11
Updated 5/2/2010 (fjn).