Fourier Transforms

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 $f \in \mathscr {S}_{1}$?