The Fourier Transform and Differentiation

Let $\mathcal{D} = \left\{ f \in L^2(\mathbb {R}) : f \text{ is absolutely continuous on every bounded interval in } \mathbb {R} \text{ and } f' \in L^2(\mathbb {R})\right\}$. (Recall that for $f$ in $\mathcal{D}$, letting $Af = if'$ makes $A$ self-adjoint.)

Show that for every $f$ in $\mathcal{D}$ there is a sequence $\left\{ f_n\right\}$ in $C_c^{(1)}(\mathbb {R})$ such that $f_n \to f$ and $f_n' \to f'$ in $L^2(\mathbb {R})$.

Show that the Schwartz space $\mathcal{S}$ with the seminorms $\left\{ \left\| \cdot \right\|_{m,n} : m, n \geq 0\right\}$ is a Fréchet space.

If $\phi$ is infinitely differentiable on $\mathbb {R}$, show that $\phi \in \mathcal{S}$ if and only if for every integer $n \geq 0$ and every polynomial $p$, $\phi^{(n)}(x)p(x) \to 0$ as $\left|x\right| \to \infty$.

If $f \in L^p(\mathbb {R})$, $1 \leq p \leq \infty$, and $g \in L^1(\mathbb {R})$, show that $f * g \in L^p(\mathbb {R})$ and $\left\| f * g\right\|_p \leq \left\| f\right\|_p \left\| g\right\|_1$.

Recall Proposition 6.13: if $\psi \in L^1(\mathbb {R})$ is such that $(2\pi )^{-1/2}\int_{\mathbb {R}} \psi (x) \, dx = 1$ and if for $\varepsilon > 0$, $\psi_\varepsilon (x) = \varepsilon^{-1}\psi (x/\varepsilon )$, then for every $f$ in $C_0(\mathbb {R})$, $\psi_\varepsilon * f(x) \to f(x)$ uniformly on $\mathbb {R}$.

Let $\psi$ and $\psi_\varepsilon$ be as above. If $f \in L^p(\mathbb {R})$, $1 \leq p < \infty$, show that $\left\| f * \psi_\varepsilon - f\right\|_p \to 0$ as $\varepsilon \to 0$. If $f \in L^{\infty }(\mathbb {R})$, show that $f * \psi_\varepsilon \to f$ (weak*).

If $f \in L^1(\mathbb {R})$ and $\hat{f} \in L^1(\mathbb {R})$, show that $f(x) = (2\pi )^{-1/2} \int_{\mathbb {R}} \hat{f}(t) e^{ixt} \, dt$ a.e.

If $\mathscr {F} : L^2(\mathbb {R}) \to L^2(\mathbb {R})$ is the Plancherel Transform and $f \in L^2(\mathbb {R})$, show that $(\mathscr {F}^{-1}f)(x) = (\mathscr {F}f)(-x)$.

Show that $\mathscr {F}^4 = 1$ but $\mathscr {F}^2 \neq 1$. What does this say about $\sigma (\mathscr {F})$?

Find the Fourier transform of the Hermite polynomials. What do you think?