Math 642-600 Current Assignment — Spring 2017

Assignment 9 - Not to be handed in.

• Suggested problems (Fourier transform convention: $\hat f(\xi) = \int_{-\infty}^\infty f(x)e^{-i\xi x}dx$.)
  1. Consider the Green's function that you found in problem 6, Assignment 8, for the operator $Lu- -u''$, $D_L=\{u,u''\in L^2[0,\infty)\colon u'(0)=0\}$. Use it and Stone's formula for the spectral family of $L$ to derive the Fourier cosine transform pair: \[ F(\xi)= \frac{2}{\pi}\int_0^\infty f(x)\cos(\xi x)dx \ \text{and}\ f(x)= \int_{0}^\infty F(\xi)\cos(\xi x)d\xi \]
  2. Prove the following theorems:
     1. If $f,g \in L^1(\mathbb R) \cap L^2(\mathbb R)$, then $f\ast g\in L^1(\mathbb R) \cap L^\infty(\mathbb R)$.
     2. If $f\in L^1(\mathbb R)$, then its Fourier transform $\hat f$ is in $C_0(\mathbb R)$.

  3. Consider the one dimensional heat equation, $u_t = u_{xx}$, with $u(x,0) = f(x)$, where $-\infty < x < \infty$ and $0 < t < \infty$. You are given that ${\mathcal F}[e^{-x^2}] =\sqrt{\pi}e^{-\xi^2/4}$. By taking the Fourier transform in $x$, show that the solution $u(x,t)$ is given by \[ u(x,t) = \int_{\mathbb R}K(x-y,t)f(y)dy,\ K(\xi,t) = (4\pi t)^{-1/2} e^{-\xi^2/4t}. \] (The function $K(\xi,t)$ is the one-dimensional heat kernel.)

  4. Let T be a tempered distribution. Find the Fourier transforms for T′, xT(x), T(x−a), and e^ibxT(x).

  5. Let $f$ be in Schwartz space, $\mathcal S$, and let $g$ be $C^\infty$ and satisfy $|g^{(m)}(x)| \le c_m(1+x^2)^{n_m}$ 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.

  6. Section 10.3: 1, 3, 14.