Partial Differential Equations

Assignment 7

1. [5 points] Is there an$f$in$L^1(\mathbb{R})$such that$f * f = f$? What about$L^2(\mathbb{R})$

2. [5 points] For$\delta > 0$, let$f_\delta(x) = f(\delta x)$. Compute the Fourier transform of$f$. Hence or otherwise show the following:

   • If$\|\hat{f}\|_q \leq \|f\|_p$for all$f \in L^p$, then$\frac{1}{p} + \frac{1}{q} = 1$.

   • If$\|\hat{f}\|_p \leq \|f\|_p$for all$f \in L^p$, then$p = 2$.

3. [5 points] Compute the Fourier transform of$\chi_{[-n,n]}$. Let$f_n(x) = \frac{\sin x \sin nx}{x^2}$. Show that$\|f_n\|_1 \to \infty$as$n \to \infty$. Hence or otherwise prove that the map$f \to \hat{f}$is not onto from$L^1(\mathbb{R})$to$C_0(\mathbb{R})$. Prove that the range of the Fourier transform is dense in$C_0(\mathbb{R})$.

4. [5 points] If$f, g \in C_c^\infty(\mathbb{R})$and$f * g = 0$, prove that either$f$or$g$is zero. Prove that there exist$f$and$g$in$S(\mathbb{R})$such that$f * g = 0$.