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\centerline{\textbf{Problem Set 1: Math 201B}}

\bigskip\noindent
You can look up and use any standard results from integration theory \textit{e.g.} Fubini's theorem, H\"older's inequality, and the density
of $C(\Ts)$ in $L^p(\Ts)$.

\bigskip\bigskip\noindent
\textbf{1.} If $1\le p < q < \infty$, show that $L^p(\Ts) \supset L^q(\Ts)$.

\bigskip\noindent
\textbf{2.} (a) If $f,g\in L^1(\Ts)$, show that $f\ast g\in L^1(\Ts)$ and
\[
\|f\ast g\|_{L^1} \le \|f\|_{L^1} \|g\|_{L^1}.
\]
(b) If $f,g \in L^2(\Ts)$ show that
\[
\|f\ast g\|_{\infty} \le \|f\|_{L^2} \|g\|_{L^2}
\]
and deduce that $f\ast g \in C(\Ts)$.

\bigskip\noindent
\textbf{3.} (a) For $f\in L^p(\Ts)$ and $h\in\Rl$, let
\[
f_h(x) = f(x+h)
\]
denote the translation of $f$ by $h$.
If $1\le p < \infty$, show that $f_h \to f$ in $L^p$ as $h\to 0$.
\textsc{Hint.} Approximate $f$ by a continuous function.

\smallskip\noindent
(b) Give an example to show that this result is not true when $p=\infty$.


\bigskip\noindent
\textbf{4.} (a) Compute the Fourier series expansion of
\[
f(x) = |x|\qquad \mbox{for $|x| \le \pi$}.
\]
(b) Use Parseval's theorem to show that
\[
\sum_{n=1}^\infty \frac{1}{n^4} = \frac{\pi^4}{90}.
\]
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