Test Functions and Distributions

Suppose $f$ is a complex continuous function in $\mathbb {R}^{n}$, with compact support. Prove that $\psi P_{j} \rightarrow f$ uniformly on $\mathbb {R}^{n}$, for some $\psi \in \mathscr {D}$ and for some sequence $\left\{ P_{j}\right\}$ of polynomials.

Show that the metrizable topology for $\mathscr {D}(\Omega )$ that was rejected in Section 6.2 is not complete for any $\Omega$.

If $E$ is an arbitrary closed subset of $\mathbb {R}^{n}$, show that there is an $f \in C^{\infty }\left(\mathbb {R}^{n}\right)$ such that $f(x) = 0$ for every $x \in E$ and $f(x) > 0$ for every other $x \in \mathbb {R}^{n}$.

Suppose $\Lambda \in \mathscr {D}^{\prime }(\Omega )$ and $\Lambda \phi \geq 0$ whenever $\phi \in \mathscr {D}(\Omega )$ and $\phi \geq 0$. Prove that $\Lambda$ is then a positive measure in $\Omega$ (which is finite on compact sets).

Prove that the numbers $c_{\alpha \beta }$ in the Leibniz formula are

Let $\Omega = (0, \infty )$. Define

Prove that $\Lambda$ is a distribution of infinite order in $\Omega$. Prove that $\Lambda$ cannot be extended to a distribution in $R$; that is, there exists no $\Lambda_{0} \in \mathscr {D}^{\prime }(R)$ such that $\Lambda_{0} = \Lambda$ in $(0, \infty )$.

Characterize all distributions whose supports are finite sets.

Suppose $\left\{ f_{i}\right\}$ is a sequence of locally integrable functions in $\Omega$ (an open set in $\mathbb {R}^{n}$) and

for every compact $K \subset \Omega$. Prove that then $D^{\alpha } f_{i} \rightarrow 0$ in $\mathscr {D}^{\prime }(\Omega )$, as $i \rightarrow \infty$, for every multi-index $\alpha$.

Suppose $\Omega$ is open in $\mathbb {R}^{2}$, and $\left\{ f_{i}\right\}$ is a sequence of harmonic functions in $\Omega$ that converges in the distribution sense to some $\Lambda \in \mathscr {D}^{\prime }(\Omega )$; explicitly, the assumption is that

Prove then that $\left\{ f_{i}\right\}$ converges uniformly on every compact subset of $\Omega$ and that $\Lambda$ is a harmonic function. Hint: If $f$ is harmonic, $f(x)$ is the average of $f$ over small circles centered at $x$.

Recall that $\delta$ (the Dirac measure) is the distribution defined by $\delta (\phi ) = \phi (0)$, for $\phi \in \mathscr {D}(R)$. For which $f \in C^{\infty }(R)$ is it true that $f \delta^{\prime } = 0$? Answer the same question for $f \delta^{\prime \prime }$. Conclude that a function $f \in C^{\infty }(R)$ may vanish on the support of a distribution $\Lambda \in \mathscr {D}^{\prime }(R)$ although $f \Lambda \neq 0$.

If $\phi \in \mathscr {D}(\Omega )$ and $\Lambda \in \mathscr {D}^{\prime }(\Omega )$, does either of the statements

imply the other?

Suppose $K$ is the closed unit ball in $\mathbb {R}^{n}, \Lambda \in \mathscr {D}^{\prime }\left(\mathbb {R}^{n}\right)$ has its support in $K$, and $f \in C^{\infty }\left(\mathbb {R}^{n}\right)$ vanishes on $K$. Prove that $f \Lambda = 0$. Find other sets $K$ for which this is true. (Compare with Exercise 12).

Suppose $K \subset V \subset \Omega , K$ is compact, $V$ and $\Omega$ are open in $\mathbb {R}^{n}, \Lambda \in \mathscr {D}^{\prime }(\Omega )$ has its support in $K$, and $\left\{ \phi_{i}\right\} \subset \mathscr {D}(\Omega )$ satisfies

for every multi-index $\alpha$. Prove that then

The preceding statement becomes false if $V$ is replaced by $K$ in the hypothesis (a). Show this by means of the following example, in which $\Omega = R$. Choose $c_{1} > c_{2} > \cdots > 0$, such that $\sum c_{j} < \infty$; define

and consider functions $\phi_{i} \in \mathscr {D}(R)$ such that $\phi_{i}(x) = 0$ if $x \leq c_{i+1}, \phi_{i}(x) = 1 / i$ if $c_{i} \leq x \leq c_{1}$. Show also that this $\Lambda$ is a distribution of order 1.

However, for certain $K, V$ can be replaced by $K$ in the hypothesis (a) of Exercise 15. Show that this is so when $K$ is the closed unit ball of $\mathbb {R}^{n}$. Find other sets $K$ for which this is true.

If $\Lambda \in \mathscr {D}^{\prime }(R)$ has order $N$, show that $\Lambda = D^{N+2} f$, for some continuous function $f$. If $\Lambda = \delta$, what are the possibilities for $f$?.

Express $\delta \in \mathscr {D}^{\prime }\left(\mathbb {R}^{2}\right)$ in the form given by Theorem 6.27, as explicitly as you can.

Suppose $\Lambda \in \mathscr {D}^{\prime }(\Omega ), \phi \in \mathscr {D}(\Omega )$, and $\left(D^{\alpha } \phi \right)(x) = 0$ for every $x$ in the support of $\Lambda$ and for every multi-index $\alpha$. Prove that $\Lambda \phi = 0$. Suggestion: Do it first for distributions with compact support, by the method used in Theorem 6.25.

Prove that every continuous linear functional on $C^{\infty }(\Omega )$ is of the form $f \rightarrow \Lambda f$, where $\Lambda$ is a distribution with compact support in $\Omega$; this is a converse to $(d)$ of Theorem 6.24.

Let $C^{\infty }(T)$ be the space of all infinitely differentiable complex functions on the unit circle $T$ in $\mathscr {C}$. One may regard $C^{\infty }(T)$ as the subspace of $C^{\infty }(R)$ consisting of those functions that have period $2 \pi$. Suppose

converges in the open unit disc $U$ in $\mathcal{C}$. Prove that each of the following three properties of $f$ implies the other two:

For $u \in \mathscr {D}^{\prime }(R)$, show that

as $x \rightarrow 0$. (The derivative of $u$ may thus still be regarded as a limit of quotients).

Suppose $\left\{ f_{i}\right\}$ is a sequence of locally integrable functions in $\mathbb {R}^{n}$, such that

exists, for each $\phi \in \mathscr {D}\left(\mathbb {R}^{n}\right)$ and each $x \in \mathbb {R}^{n}$. Prove that then $\left\{ D^{a}\left(f_{i} * \phi \right)\right\}$ converges uniformly on compact sets, for each multi-index $\alpha$.

Let $H$ be the Heaviside function on $R$, defined by

and let $\delta$ be the Dirac measure.

Here is another characterization of convolutions analogous to Theorem 6.33. Suppose $L$ is a continuous linear mapping of $\mathscr {D}$ into $C^{\infty }$ which commutes with every $D^{\alpha }$, that is, (a)

Then there is a $u \in \mathscr {D}^{\prime }$ such that

Suggestion: Fix $\phi \in \mathscr {D}$, put

let $D_{e}$ be the directional derivative used in the proof of Theorem 6.30, and show that

which is 0 if (a) holds. Thus $h(x) = h(0)$, which implies that $\tau_{x} L = L \tau_{x}$. Can the assumption that the range of $L$ is in $C^{\infty }$ be weakened?

If $f \in L^{1}((-\infty ,-\delta ) \cup (\delta , \infty ))$ for every $\delta > 0$, define its principal value integral to be

if the limit exists. For $\phi \in \mathscr {D}(R)$, put

Show that

Find all distributions $u \in \mathscr {D}^{\prime }\left(\mathbb {R}^{n}\right)$ that satisfy at least one of the following two conditions: