Math 642-600 Current Assignments — Spring 2008
Assignment 7 - Due Friday, April 25.
- Read sections 10.1 and 10.3.
- Do the following problems.
- Let Hn(x) be the degree n Hermite polynomial. Show
that Hn(x)e− x2/2 is an
eigenfunction of the Fourier transform. Find the corresponding
eigenvalue. (Hint: use the formula Hn(x) =
ex2(d/dx)n e−
x2 for the Hermite polynomials.)
- 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.
- Problem 3(d), page 333 (§ 7.5)
Updated 4/19/08 (fjn).