Problem set 6 - fouriernew

Problem 1¶

(1) Prove the last lemma in lecture 12.

(2) Conclude that the Fourier transform $\mathcal F: \mathcal S\to \mathcal S$ , $f\mapsto \mathcal Ff$ is continuous with respect to the topology on $\mathcal S$ .

Problem 2¶

Let $K(x) = e^{-\pi x^2}$ be the Gaussian function. Similar to what we did with Λ and $\Lambda_\epsilon$ , we define $K_\delta(x) = \frac{1}{\sqrt \delta}e^{-\frac{\pi x^2}{\delta}}$ .

(1) Show that $K_\delta$ is a good kernel on $\mathbb{R}$ when $\delta \to 0$ , in the sense that $\int_{-\infty}^{+\infty}K_\delta(x)dx = 1; \int_{-\infty}^{+\infty}|K_\delta(x)|dx \leq M$ ; and for $\eta> 0$ , $\int_{|x|>\eta}|K_\delta(x)|dx \to 0$ when $\delta \to 0$ .

(2) If $f$ is continuous and bounded on $\mathbb{R}$ , then $f*K_\delta \to f$ uniformly when $\delta \to 0$ .

(3) Calculate the Fourier transform of $K_\delta$ .

(4) Let $T$ be an operator on $S$ given by $(T,f):=\lim_{\delta\to 0} (T_{K_\delta},f)$ , calculate $T$ .

(5) Show that $f(0) = \int_{-\infty}^{+\infty}\mathcal Ff(s)ds$ for $f\in \mathcal S$ .

(6) Give an alternative proof of inversion theorem by looking at $F(y) = f(y+x)$ .

Problem 3¶

(1) For $k\geq 0$ , show that for every $f\in \mathscr{C}_c^k(\mathbb{R})$ , there exists a sequence $f_n$ in $\mathscr{C}_c^{\infty}(\mathbb{R})$ such that $f_n\to f$ in $\mathscr{C}_c^k(\mathbb{R})$ .

(2) Let $T$ be a distribution of order $k$ . Show that $u$ extends uniquely to a continuous linear functional on $\mathscr{C}_c^k(\mathbb{R})$ . In other words, define $T(\varphi)$ for every $\varphi\in \mathscr{C}_K^k(\mathbb{R})$ and show that $|(T,\varphi)|\leq C\sum_{0\leq l\leq k}\|\varphi\|_{l,K}$ , where $C$ is a constant depending on $K$ .

Problem 4 (optional)¶

(1) Let $T:\mathcal S\to \mathcal S$ be a linear operator. Show that if $T$ commutes with both $\frac{d}{dx}$ and $2\pi i x$ , i.e. $T\frac{d}{dx}f = \frac{d}{dx}Tf$ , $T(xf) = x(Tf)$ , then $T = cI$ for some constant $c$ , i.e. $Tf = cf$ for every $f\in S$ .

Hint

Show that for $f\in \mathcal S$ with $f(x_0) =0$ , then there exists $\phi(x)\in \mathcal S$ such that $f(x) = x\phi(x)$ . Then $T$ commutes with $x$ $\implies$ $(Tf)(x_0) = 0$ .
- Find $\phi(x)\in \mathscr{C}^{\infty}(\mathbb{R})$ such that $f(x) = (x-x_0)\phi(x)$ . You may need the following fact from calculus: If $f(0) = 0$ , then $f(x) = \int_{0}^1 (\frac{d}{dt}f(tx))dt$ .
- If $\phi \in \mathcal S$ then we are done. But we only have $\phi \in \mathscr{C}^{\infty}$ . We can use the result of problem 5 to modify the ϕ to be a Schwartz function. Note that the function $\frac{f(x)}{x-x_0}$ is rapidly decreasing but has differentiablilty issue at $x_0$ , so you can glue the smooth ϕ with it to produce rapidly decreasing smooth function satisfying our need.
This implies $Tf = g\cdot f$ for some $g\in \mathcal S$ .
- For $y\in \mathbb{R}$ , let $e_y f(x):= f(x) - f(y)e^{-(x-y)^2}$ . Then $e_yf \in \mathcal S$ and $e_yf(y) = 0$ $\implies Te_y f(y) = 0$ . What can you conclude from this?
Show that $T$ commutes with $\frac{d}{dx}$ implies $g$ is a constant.

(2) Let $T:f\mapsto (\mathcal F^2f)^-$ . Show that $T$ satisfies (1) and determine $c$ for $T$ .

(3) Conclude that the Fourier transform $\mathcal F: \mathcal S\to S$ is an isomorphism without directly using the Fourier inversion theorem. (This is an alternative proof of the inversion theorem.)

Problem 5¶

Prove that $f\in S$ and $T\in S'$ then $f*T$ is a tempered distribution.