# Homework 5
###### tags: `224a 2020`
Due 11/30, no updates.
$\newcommand{\R}{\mathbb{R}}$
1. Compute the Fourier transform of the following functions, viewed as tempered distributions:
(a) the Heavyside function $H(x)=\chi_{[0\infty)}(x)$.
(b) $xH(x)$.
(c ) $x^2\sin(x)$
2. ~~Compute the Fourier transform of $\log|x|$ in $\R^2$, viewed as a tempered distribution. Use this to find the fundamental solution of the Laplace equation in $\R^2$.~~ This was too hard. Instead, verify that $u(x)=-\frac{1}{2\pi}\log|x|$ is a fundamental solution of the Poisson equation $(-\partial_{x_1}^2-\partial_{x_2}^2)u=\delta(x)$ in $\R^2$.
4. Show that
$$T(\phi)=\sum_{n\in \mathbb{Z}} \phi(2\pi n)$$ is a tempered distribution. Show that its Fourier transform is the distribution
$$ \mathcal{F}T(\phi):=\frac{1}{\sqrt{2\pi}}\sum_{n\in \mathbb{Z}} \hat{f}(n).$$ (this is called the Poisson summation formula. Hint: consider the $2\pi$-periodization of $f$. Justify all interchanges of limits and sums.)
(11/18: a previous version of this question was missing some $2\pi$ and $\sqrt{2\pi}$ factors, this is the corrected version).
4. Suppose $p_n(x)$ are monic orthogonal polynomials on a finite interval $(-a,a)$ with even weight function $w(x)=w(-x)$. Show that the Jacobi coefficients $a_n=0$ for all $n$.
5. Prove that the Legendre polynomials normalized to have $p_n(1)=1$ satisfy the *Rodridgues Formula* $$p_n(x)=\frac{1}{2^nn!}\frac{d^n}{dx^n} (x^2-1)^n.$$
Use this to show that they satisfy the *Legendre Differential Equation* $$[(x^2-1)u']'=\lambda u,$$
with $\lambda=n(n+1)$.

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