Math 642-600 Current Assignments — Spring 2008

Assignment 7 - Due Friday, April 25.

• Read sections 10.1 and 10.3.
• Do the following problems.
  1. Let H_n(x) be the degree n Hermite polynomial. Show that H_n(x)e^{− x2/2} is an eigenfunction of the Fourier transform. Find the corresponding eigenvalue. (Hint: use the formula H_n(x) = e^x2(d/dx)ⁿ e^{− x2} for the Hermite polynomials.)
  2. Let f be in Schwartz space, and let g be C^∞ and satisfy
     |g^(j)(x)| ≤ c_j(1+x²)^n_j
     for all nonnegative integers j. Here c_j and n_j 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.
  3. Let T be a tempered distribution. Find the Fourier transforms for T′, xT(x), T(x−a), and e^ibxT(x).
  4. 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.
  5. Problem 3(d), page 333 (§ 7.5)

Updated 4/19/08 (fjn).