Problem set 5

Problem 1¶

Show that a Schwartz function is $L^2$ on $\mathbb{R}$, i.e. $\int_{\mathbb{R}}|f(x)|^2 dx< \infty$.

Problem 2¶

(1) Let $f\in \mathcal S$. Show that $\mathcal F \mathcal F f = f^{-}$, where $f^{-}$ is defined by $f^-(x) = f(-x)$.

(2) Generate other functions than Gaussian that are fixed by Fourier transform.

Problem 3¶

For $s>0$, define $\Gamma(s) = \int_{0}^{+\infty}x^{s-1}e^{-x}dx$.

(1) Show that $\Gamma(s)$ is well-defined for $s>0$, i.e. the improper integrals $\int_{0}^1 x^{s-1}e^{-x}dx$ and $\int_{1}^{+\infty}x^{s-1}e^{-x}$ converges.

(2) Show that $\Gamma(s+1) = s\Gamma(s)$ whenever $s>0$. Conclude that $\Gamma(n+1) = n!$.

(3) Show that $\Gamma(\frac{1}{2}) = \sqrt{\pi}$ and $\Gamma(\frac{3}{2}) = \frac{\sqrt{\pi}}{2}$.

Problem 4¶

The aim of this problem is to show that there exists a smooth function φ with compact support on $\mathbb{R}$ such that $\varphi(0)>0$.

(1) Let $f(t) = \begin{cases} e^{-\frac{1}{t}},t> 0\\ 0, t\leq 0 \end{cases}$. Show that $f$ is a smooth function on $\mathbb{R}$.

(2) Let $\varphi(x) = f(1-x^2)$. Show that $\varphi(x)$ satisfies the property.

Problem 5¶

Let $f,g$ be continuous functions on $\mathbb{R}$. If $\int f\varphi = \int g\varphi$ for every $\varphi \in \mathscr{C}_c^{\infty}(\mathbb{R})$, then $f = g$.

Remark. This property also holds for integrable functions.

Problem 6¶

Compare with problem 3 of problem set 3.

Let $\Lambda(x) = \begin{cases} 1-|x|, |x|\leq 1\\ 0, \text{else}\end{cases}$. In the lecture we have showed that the Fourier transform of Λ is $\frac{\sin^2(\pi s)}{(\pi s)^2} = \mathrm{sinc}^2(s)$.

(1) Prove $\sum_{n = -\infty}^{+\infty} \frac{1}{(n+\alpha)^2} = \frac{\pi^2}{\sin^2(\pi \alpha)}$ for $\alpha\in \mathbb{R}\setminus \mathbb{Z}$.

(2) Prove $\sum_{n=-\infty}^{+\infty}\frac{1}{n+\alpha} = \pi \cot(\pi \alpha)$ for $\alpha \in \mathbb{R}\setminus \mathbb{Z}$.

Problem 7¶

Prove the Riemann-Lebesgue lemma: If $f\in L^1(\mathbb{R})$, then $\lim_{|s|\to \infty}|\mathcal{F}f(s)| = 0$.

(1) First prove the case when $f\in \mathscr{C}_c^0(\mathbb{R})$, i.e. $f$ is continuous with compact support.

(2) Use $L^1$ approximation to reduce the $L^1$ case to continuous case.