Fourier series of periodic distributions

Using the expansion

prove that in $\mathcal{D}^{\prime }(\mathbb {R})$ the equality $\frac{1}{2 \pi } \sum_{k \in \mathbb {Z}} e^{i k x} = \sum_{k \in \mathbb {Z}} \delta_{2 \pi k}$ holds.

Prove in $\mathcal{D}^{\prime }(\mathbb {R})$ the equality $\frac{2}{\pi } \sum_{k = 0}^{\infty } \cos (2 k+1) x = \sum_{k \in \mathbb {Z}}(-1)^{k} \delta_{\pi k}$.

Prove the equality

(the sums of the series on both sides are taken in $\mathcal{D}^{\prime }$).

Prove the equality $\sum_{n = 1}^{\infty } \sin n x = \frac{1}{2} \operatorname {ctg} \frac{x}{2}$ (the series converges in $\mathcal{D}^{\prime }$; $\left\langle \operatorname {ctg} \frac{x}{2}, \varphi \right\rangle : =$ v.p. $\int \operatorname {ctg} \frac{x}{2} \varphi (x) d x$).

Prove that the series $\sum_{n = 1}^{\infty } a_{n} \cos n x$ converges in $\mathcal{D}^{\prime }$ if and only if there exist $C > 0$ and $k \in \mathbb {N}$ such that $\left|a_{n}\right| < C n^{k}, n = 1,2, \ldots$

Define generalized functions on $T^n$ as continuous linear functionals on $\mathcal D(T^n)$ (see Problem 423). Prove that the series $\sum_{k\in \mathbb {Z}^n}e^{2\pi ikt}$, where $t\in \mathbb {R}^{n}$, $kt=k_1t_1+\cdots +k_nt_n$, converges in $\mathcal D'(T^n)$ to the generalized function $\delta (t)$ defined by $\int_{T^n}\delta \varphi \, dt=\varphi (0)$.

Under what conditions on the coefficients $\left\{ c_n\right\}$ does the series $\sum_{n\in \mathbb {Z}}c_ne^{2\pi inx}$ converge in $\mathcal D'(\mathbb {R})$?

Prove that a generalized function on $\mathrm{T}$ is uniquely determined by its Fourier coefficients.

Prove the stated assertion. In the space $D^{\prime }$ the equalities

hold.

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.