Homework 4

Let for $j \in \mathbb{N}$,

$f_j(t) = \begin{cases} j, & \text{if } 0 \le t \le \frac{1}{j} \\ 0 & \text{else} \end{cases}$

a) Let $\mathcal{S}$ denote the space of Schwartz functions. For $g \in \mathcal{S}$, define

$\bar{f}_j(g) := \int_0^{\infty} f_j(t)g(t)dt.$

Show that $\bar{f}_j$ is a distribution on $\mathcal{S}$, for $j \in \mathbb{N}$.

b) Show that

$\lim_{j \to \infty} \int_0^{\infty} f_j(t)g(t)dt = \bar{\delta}(g) = g(0).$

Conclude that, the sequence of regular distributions $\bar{f}_j$, represented by a real-valued, integrable function $f_j$, converges to the Dirac delta distribution $\bar{\delta}$ on the space of test functions $\mathcal{S}$.

Let $\mathcal{S}$ denote the space of Schwartz signals. Is the expression

$\int_0^{\infty} f(t)\phi(t)dt$

a distribution on $\mathcal{S}$ for any given $f \in L_{\infty}(\mathbb{R}_+; \mathbb{R})$?

(a) Study Theorem 3.4.1 and its proof in the lecture notes.

(b) Study Theorem 3.4.3 and its proof in the lecture notes.

In class we observed that various approximate identity sequences exist and these can be used to define distributions that converge to $\bar{\delta}$.

Recall that the sequence

$\psi_n(x) = c_n(1 + \cos(x))^n 1_{\{|x| \le \pi\}}$

can be used to show that the complex harmonics of the form $\frac{1}{\sqrt{2\pi}} e^{ikx}, k \in \mathbb{Z}$, form a complete orthonormal sequence in $L_2([-\pi, \pi], \mathbb{C})$. Here, we take the normalizing coefficient $c_n$ to make $\int \psi_n(x)dx = 1$.

Show that

$\lim_{n \to \infty} \int_{|x| \ge \delta} \psi_n(x)dx = 0, \qquad \forall \delta > 0.$

One useful sequence, which does not satisfy the non-negativity property that we discussed in class, but that satisfies the convergence property (to $\bar{\delta}$) is the following sequence:

$\psi_n(x) = \frac{\sin(nx)}{\pi x}$

Show that for any $\phi \in \mathcal{S}$

$\lim_{n \to \infty} \int \psi_n(x)\phi(x)dx = \phi(0)$

Hint: You can use the following results and hints:

• $\lim_{R \to \infty} \int_{-R}^{R} \frac{\sin(x)}{x} dx = \pi$

• Riemann-Lebesgue Lemma: For any integrable function $g$, $\lim_{|f| \to \infty, f \in \mathbb{R}} \int g(x)e^{ifx} dx = 0$.

• We have $\lim_{|x| \to 0} \frac{\phi(x) - \phi(0)}{x} = h(x)$ for some smooth $h$.

• Express the integration as

$\int_{|x| \ge 1} \psi_n(x)\phi(x) + \int_{|x| \le 1} \psi_n(x)\phi(x)dx$

First, show that the first term goes to zero by the Riemann-Lebesgue Lemma. For the second term, observe that

$\int_{|x| \le 1} \psi_n(x)\phi(x) = \int_{|x| \le 1} \phi(0)\psi_n(x)dx + \int_{|x| \le 1} \frac{\phi(x) - \phi(0)}{\pi x} \sin(nx)dx.$

Show that the second term goes to zero through $\lim_{|x| \to 0} \frac{\phi(x) - \phi(0)}{x} = h(x)$ and the Riemann-Lebesgue lemma. Finally, for the first term, use that

$\int_{|x| \le 1} \frac{\sin(nx)}{\pi x} dx = \int_{|u| \le n} \frac{\sin(u)}{\pi u} du,$

and conclude the result.

The derivative of a distribution $\overline{f}$ is defined with the relation:

$\overline{Df}(\phi) = -\overline{f}(D\phi),$

where $D(.)$ denotes the derivative operator.

Let $u(t)$ denote the step function: that is $u(t) = 1_{\{t \ge 0\}}$ ($1_{\{.\}}$ being the indicator function). Define for $\phi \in \mathcal{S}$,

$\bar{u}(\phi) = \int_{\mathbb{R}} u(t)\phi(t)dt = \int_0^{\infty} \phi(t)dt.$

Given the definition of a distributional derivative, show that the distributional derivative of the distribution $\overline{u}$ represented by the unit step function is the Dirac delta distribution.