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\centerline{\textbf{Problem Set 5: Math 201B}}
\centerline{Due: Friday, February 4}

\bigskip\noindent
\bigskip\noindent

\bigskip
\noindent
\textbf{1.} Let $\Ts^d = \Ts\times \Ts \times \dots \times \Ts$ denote the $d$-dimensional Torus.

\smallskip\noindent
(a) Show that $\mathcal{B} = \left\{e^{i\vec{n}\cdot\vec{x}} : \vec{n}\in \Ir^d\right\}$ is an orthogonal set
in $L^2(\Ts^d)$ and give an expression for the Fourier coefficients $\hat{f}(\vec{n})$
of a function
\[
f(\vec{x}) = \sum_{\vec{n}\in \Ir^d} \hat{f}(\vec{n}) e^{i\vec{n}\cdot\vec{x}} \in L^2(\Ts^d).
\]
(You can assume that $\mathcal{B}$ is complete --- the proof is similar to the one-dimensional case \textit{e.g.} use an approximate
identity
\[
\Phi_n(\vec{x}) = \phi_n(x_1) \phi_n(x_2) \dots \phi_n(x_d)\qquad n\in \Nl
\]
that is a product of one-dimensional approximate identities $\{\phi_n\}$
consisting of trigonometric polynomials.)

\smallskip\noindent
(b) For $s > 0$, let $H^s(\Ts^d)$ denote the space of functions $f\in L^2(\Ts^d)$ such that
\[
\sum_{\vec{n}\in \Ir^d} \left(1 + |\vec{n}|^{2s}\right) \left|\hat{f}(\vec{n})\right|^2 < \infty.
\]
Prove that if $s > d/2$ and $f\in H^s(\Ts^d)$, then $f\in C(\Ts^d)$.

\bigskip\noindent
\textbf{2.} (a) Show that any test function $\phi \in C^\infty(\Ts)$ can be written as
$\phi = c + \psi^\prime$
where
\[
c = \frac{1}{2\pi}\int \phi \, dx, \qquad\psi \in C^\infty(\Ts).
\]
(b) Suppose that $f\in L^1(\Ts)$ is weakly differentiable and its weak derivative $f^\prime = 0$ is zero.
Prove that $f = \mathrm{constant}$ (up to pointwise a.e.\ equivalence).


\newpage\noindent
\textbf{3.} Define the principal-value functional $T : \mathcal{D}(\Ts) \to \Cx$ by
\begin{align*}
\left\langle T,\phi\right\rangle &= \mathrm{p.v.} \int_{\Ts}\cot\left(\frac{x}{2}\right) \phi(x)\, dx
\\
&=\lim_{\epsilon \to 0^+} \left(\int_{-\pi}^{-\epsilon} + \int_{\epsilon}^\pi\right)
\cot\left(\frac{x}{2}\right) \phi(x)\, dx.
\end{align*}
(a) Show that $T \in \mathcal{D}^\prime(\Ts)$ is a well-defined periodic distribution.

\smallskip\noindent
(b) Compute the Fourier coefficients $\hat{T}(n)$ of $T$.

\bigskip\noindent
\textbf{4.} (a) If $T\in \mathcal{D}^\prime(\Ts)$ is a periodic distribution, show that
there exists an integer $k \ge 0$ and a constant $C$ such that
\begin{equation}
\left|\langle T,\phi\rangle\right| \le C \left\|\phi\right\|_{C^k}\qquad
\mbox{for all $\phi\in \mathcal{D}(\Ts)$}
\label{deford}
\end{equation}
where
\[
\left\|\phi\right\|_{C^k} = \sum_{j=0}^k \sup_{x\in \Ts} \left|\phi^{(j)}(x)\right|
\]
denotes the $C^k$-norm of $\phi$.

\smallskip\noindent
(b) The order of a distribution $T$ is the minimal integer $k\ge 0$ such that \eq{deford} holds. What it the order of:
(i) a regular distribution; (ii) the delta-function; (iii) the principal value distribution in the previous
question? Give an example of a distribution of order $100$.

\end{document}