## Math 642-600 Current Assignment — Spring 2015

Assignment 9 - Not to be handed in.

• Suggested problems.

1. Let $f$ be in Schwartz space, $\mathcal S$, and let $g$ be $C^\infty$ and satisfy $|g^(m)(x)| \le c_m(1+t^2)^{n_m}$ for all nonnegative integers m. Here $c_m$ and $n_m$ depend on $g$ and $m$. Show that $fg$ is in Schwartz space. Use this result to explain how to define the product $g(x)T(x)$, where $T$ is a tempered distribution.

2. Let T be a tempered distribution. Find the Fourier transforms for T′, xT(x), T(x−a), and eibxT(x).

3. (Problem 1, p. 7, Feldman.) Suppose that $f$ is locally Lebesgue integrable on $\mathbb R$ and that there is a polynomial $P(x) = \sum_{k=N_1}^{N_2} a_kx^k$, where $0\le N_1 < N_2 <\infty$, such that $|f(x)| \le P(x)$ a.e. in $\mathbb R$.
1. Show that there is a constant $C>0$ such that $|f(x)|(1+x^2) \le C(|x|^{N_1} +|x|^{N_2})$.
2. Show that for all $\phi\in \mathcal S$ we have $\big|\langle f, \phi\rangle \big| \le \pi C\big( \|\phi\|_{N_1,0} + \| \phi \|_{N_2+2,0} \big)$.

4. Problem 7, page 464 (§10.3)

5. Problem 9, page 465 (§10.3)

6. Problem 11, page 465 (§10.3)

Updated 5/7/2015 (fjn)