Distributions and Fourier Transforms

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:

Suppose $A$ is an invertible linear operator on $\mathbb {R}^{n}, f \in L^{1}\left(\mathbb {R}^{n}\right)$, and $g(x) = f(A x)$. Express $\hat{g}$ in terms of $\hat{f}$. This generalizes $(d)$ of Theorem 7.2.

Is the topology of $\mathscr {S}_{n}$ induced by some invariant metric which turns the Fourier transform into an isometry of $\mathscr {S}_{n}$ onto $\mathscr {S}_{n}$?

Suppose $f(x) = e^{x}, g(x) = e^{x} \cos \left(e^{x}\right)$, on the real line. Show that $g$ is a tempered distribution but that $f$ is not.

By Exercise 3 there exist distributions in $\mathbb {R}^{n}$ which are not tempered. Such distributions are continuous linear functionals on $\mathscr {D}\left(\mathbb {R}^{n}\right)$ which have no continuous linear extension to $\mathscr {S}_{n}$. Explain why this does not contradict the Hahn-Banach theorem.

Prove that the operations listed in Theorem 7.13 are continuous mappings of $\mathscr {S}_{n}^{\prime }$ into $\mathscr {S}_{n}^{\prime }$.

If $u \in \mathscr {S}_{n}^{\prime }$, prove that

for every $x \in \mathbb {R}^{n}$.

Suppose $f \in L^{1}\left(\mathbb {R}^{n}\right), f \neq 0, \lambda$ is a complex number, and $\hat{f} = \lambda f$. What can you say about $\lambda$?

Prove (a) of Theorem 7.8 directly (without using Fourier transforms).

The Fourier transform of a complex Borel measure $\mu$ on $\mathbb {R}^{n}$ is customarily defined to be the function $\hat{\mu }$ given by

Of course, $\mu$ is also a tempered distribution, and as such its Fourier transform was defined in Section 7.14. Show that these two definitions are consistent. Prove that each $\hat{\mu }$ is bounded and uniformly continuous.

Suppose $\Lambda : \mathscr {S}_{n} \rightarrow C\left(\mathbb {R}^{n}\right)$ is continuous, linear, and $\tau_{x} \Lambda = \Lambda \tau_{x}$ for every $x \in \mathbb {R}^{n}$. Does it follow that there exists $u \in \mathscr {S}_{n}^{\prime }$ such that

for every $\phi \in \mathscr {S}_{n}$?

If $\left\{ h_{j}\right\}$ is an approximate identity, as in Definition 6.31, and $u \in \mathscr {S}_{n}^{\prime }$, does it follow that $u * h_{j} \rightarrow u$ as $j \rightarrow \infty$, in the weak*-topology of $\mathscr {S}_{n}^{\prime }$?

Suppose $X$ and $Y$ are complete metric spaces, $A$ is dense in $X, f: A \rightarrow Y$ is uniformly continuous.

Suppose $F$ is an entire function in $\mathbb {C}^{n}$, and suppose that to each $\varepsilon > 0$ there correspond an integer $N(\varepsilon )$ and a constant $\gamma (\varepsilon ) < \infty$ such that

Prove that $F$ is a polynomial.

Suppose $f$ is an entire function in $\mathbb {C}^{n}, N$ is a positive integer, $r \geq 0$, and

Prove that then

Suggestion: Fix $z = x+i y \in \mathbb {C}^{n}$; define

for $\lambda \in \mathcal{C}, s > 0$, and apply the maximum modulus theorem to a large semicircular region in the upper half-plane to deduce that $\left|g_{s}(i)\right| < 1$. Let $s \rightarrow 0$.

In (b) of Theorem 7.23 it is not asserted that $u$ has order $N$. The following example shows that this is not always true.

Let $\mu$ be the Borel probability measure on $\mathbb {R}^{3}$ which is concentrated on the unit sphere $S^{2}$ and which is invariant under all rotations of $S^{2}$. Compute (by using spherical coordinates) that

Put $u = D_{1} \mu$. Then

Deduce from Exercise 15 that

although $u$ is not a distribution of order 0. (Its order is 1). Find an explicit formula for the entire function $u\left(e_{-z}\right), z \in \mathbb {C}^{3}$.

Suppose $u$ is a distribution in $\mathbb {R}^{n}$, with compact support $K$, whose Fourier transform $\hat{u}$ is a bounded function on $\mathbb {R}^{n}$.

Was it necessary to introduce the function $\psi$ into the proof of Theorem 7.25? Could the proof have been simplified by setting $F(x) = f(x)$ on $K, F(x) = 0$ off K?

Show that the hypotheses of Theorem 7.25 imply that $D^{\alpha } f$ is locally $L^{2}$ for every multi-index $\alpha$ with $\left|\alpha \right| \leq r$.

Let $f \in L^{2}\left(\mathbb {R}^{2}\right)$ be the continuous function whose Fourier transform is

Since $\left|y\right|^{3} \hat{f}(y)$ is in $L^{2}\left(\mathbb {R}^{2}\right)$, Theorem 7.25 implies that $f \in C^{(1)}\left(\mathbb {R}^{2}\right)$. Show that the stronger conclusion $f \in C^{(2)}\left(\mathbb {R}^{2}\right)$ is false, by proving that

This shows that $>$ cannot be replaced by $\geq$ in (1) of Theorem 7.25.

Suppose $u$ is a distribution in $\mathbb {R}^{n}$ whose first derivatives $D_{1} u, \ldots , D_{n} u$ are functions in $L^{2}\left(\mathbb {R}^{n}\right)$. Prove that $u$ is also a function and that $u$ is locally $L^{2}$. (Show that "locally" cannot be omitted in the conclusion). Hint: $u$ is in fact the sum of an $L^{2}$-function and an entire function.

When $n = 1$, show that $u$ is actually a continuous function. Show that this stronger conclusion is false when $n = 2$. For example, the gradient of the function

is in $L^{2}\left(\mathbb {R}^{2}\right)$. See Exercise 11, Chapter 8, for the same result under weaker hypotheses.

Periodic distributions, or distributions on a torus $T^{n}$, have Fourier series whose theory is somewhat simpler than that of Fourier transforms. This is mainly due to the compactness of $T^{n}$ : Every distribution on $T^{n}$ has compact support. In particular, tempered distributions are nothing special.

Prove the various assertions made in the following basic outline:

Functions $\phi$ on $T^{n}$ can be identified with functions $\tilde{\phi }$ on $\mathbb {R}^{n}$ that are $2 \pi$-periodic in each variable, by setting

$\mathbb {Z}^{n}$ is the set (or additive group) of $n$-tuples $k = \left(k_{1}, \ldots , k_{n}\right)$ of integers $k_{j}$. For $k \in \mathbb {Z}^{n}$, the function $e_{k}$ is defined on $T^{n}$ by

$\sigma_{n}$ is the Haar measure of $T^{n}$. If $\phi \in L^{1}\left(\sigma_{n}\right)$, the Fourier coefficients of $\phi$ are

$\mathscr {D}\left(T^{n}\right)$ is the space of all functions $\phi$ on $T^{n}$ such that $\tilde{\phi } \in C^{\infty }\left(\mathbb {R}^{n}\right)$. If $\phi \in \mathscr {D}\left(T^{n}\right)$ then

for $N = 0,1,2, \ldots$ These norms define a Fréchet space topology on $\mathscr {D}\left(T^{n}\right)$, which coincides with the one given by the norms

$\mathscr {D}^{\prime }\left(T^{n}\right)$ is the space of all continuous linear functionals on $\mathscr {D}\left(T^{n}\right)$; its members are the distributions on $T^{n}$. The Fourier coefficients of any $u \in \mathscr {D}^{\prime }\left(T^{n}\right)$ are defined by

To each $u \in \mathscr {D}^{\prime }\left(T^{n}\right)$ correspond an $N$ and a $C$ such that

Conversely, if $g$ is a complex function on $Z^{\mathbf{n}}$ that satisfies $\left|g(k)\right| \leq C(1+\left|k\right|)^{N}$ for some $C$ and $N$, then $g = \hat{u}$ for some $u \in \mathscr {D}^{\prime }\left(T^{n}\right)$.

There is thus a linear one-to-one correspondence between distributions on $T^{n}$, on one hand, and functions of polynomial growth on $\mathbb {Z}^{n}$, on the other.

If $E_{1} \subset E_{2} \subset E_{3} \subset \cdots$ are finite sets whose union is $\mathbb {Z}^{n}$, and if $u \in \mathscr {D}^{\prime }\left(T^{n}\right)$, the "partial sums"

converge to $u$ as $j \rightarrow \infty$, in the weak*-topology of $\mathscr {D}^{\prime }\left(T^{n}\right)$. The convolution $u * v$ of $u \in \mathscr {D}^{\prime }\left(T^{n}\right)$ and $v \in \mathscr {D}^{\prime }\left(T^{n}\right)$ is most easily defined as having Fourier coefficients $\hat{u}(k) \hat{v}(k)$. The analogues of Theorems 6.30 and 6.37 are true; the proofs are much simpler.

Modify the proof of Theorem 7.25 so that Fourier series are used in place of Fourier transforms, by replacing $F$ by a suitable periodic function.

Put $c = (2 / \pi )^{1 / 2}$. For $j = 1,2,3, \ldots$, define $g_{j}$ on the real line by

Prove that $\left\{ \hat{g}_{j}\right\}$ is a uniformly bounded sequence of functions which converges pointwise, as $j \rightarrow \infty$. If $f \in L^{2}\left(\mathbb {R}^{1}\right)$, it follows that $f * g_{j}$ converges, in the $L^{2}$ metric, to a function $H f \in L^{2}$. This is the Hilbert transform of $f$; formally,

(The integral exists, in the principal value sense, for almost every $x$, but this is not so easy to prove; if $f$ satisfies a Lipschitz condition of order 1, for instance, the proof is trivial). Prove that

for every $f \in L^{2}\left(\mathbb {R}^{1}\right)$. Thus $H$ is an $L^{2}$-isometry, of period 4. Is it true that $H f \in \mathscr {S}_{1}$ if $f \in \mathscr {S}_{1}$?

The following simple properties of holomorphic functions of several variables were tacitly used in this chapter. Prove them.

Prove the statement about convex hulls made in the last sentence of the proof of Theorem 8.4.

Find a fundamental solution for the operator $\partial^{2} / \partial x_{1} \partial x_{2}$ in $\mathbb {R}^{2}$. (There is one that is the characteristic function of a certain subset of $\mathbb {R}^{2}$).

Show that the equation

is satisfied (in the distribution sense) by every locally integrable function $u$ of the form

and that even classical solutions (i.e., twice continuously differentiable functions) need not be in $C^{\infty }$. Note the contrast between this and the Laplace equation.

For $x \in \mathbb {R}^{3}$, define $f(x) = \left(1+\left|x\right|^{2}\right)^{-1}$. Show that $f \in L^{2}\left(\mathbb {R}^{3}\right)$ and that $\hat{f}$ is a fundamental solution of the operator $I-\Delta$ in $\mathbb {R}^{3}$. Find $\hat{f}$, by direct computation and also by the following reasoning: