# 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 $\lambda=n(n+1)$.