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{ \large \textbf{\textsc{Math 5587 -- Homework 6 (Due Thursday Oct 20)}}}
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\begin{enumerate}
\item For each of the following functions, state whether it is even, odd, or neither, and whether it is periodic. If periodic, what is the smallest period?
\begin{enumerate}
\item $\sin(ax)$ for $a>0$
\item $e^{ax}$ for $a>0$
\item $x^m$ for an integer $m$
\item $\tan(x^2)$
\item $|\sin(x/b)|$ for $b>0$
\item $x\cos(ax)$ for $a>0$
\end{enumerate}
\item Let $f(x)$ be $2\pi$ periodic. Show that
\[\int_{-\pi}^\pi f(x) \, dx = \int_{a-\pi}^{a+\pi} f(x) \, dx\]
for all real numbers $a$. That is, the integral of a $2\pi$-periodic function is the same over any interval of length $2\pi$. [Hint: Show that there exists an integer $n$ such that
   \[a-\pi \leq 2\pi n - \pi < a+\pi.\]
Write the integral as
\[\int_{a-\pi}^{a+\pi}f(x) \, dx = \int_{a-\pi}^{2\pi n-\pi} f(x) \, dx + \int_{2\pi n-\pi}^{a+\pi} f(x) \, dx,\]
and use the periodicity of $f$ to complete the proof from here.]
\item Recall a function $f$ is Lipschitz if there exists $L>0$ such that 
\[|f(x)-f(y)|\leq L|x-y| \ \ \ \text{for all } x,y\]
\begin{enumerate}
\item Show that every Lipschitz function $f$ is continuous. [Hint: A function is continuous if $\lim_{y\to x} f(y) = f(x)$ for all $x$.]
\item Show that if $f$ is continuously differentiable and $f'$ is bounded, then $f$ is Lipschitz. [Hint: First assume $x > y$ and recall the fundamental theorem of calculus:
\[f(x) - f(y) = \int_y^x f'(s) \, ds.\]
Take absolute values of both sides, and use the inequality $|\int_a^b g(s) \, ds | \leq \int_a^b |g(s)|\, ds$, which holds for $a < b$. Then repeat a similar argument when $y > x$. Recall $f'$ is bounded means there exists $L>0$ such that $|f'(x)| \leq L$ for all $x$. ]
\end{enumerate}
\item Consider the geometric series $\sum_{n=0}^\infty (-1)^n x^{2n}$.
\begin{enumerate}
\item Does it converge pointwise in the interval $-1 < x < 1$? 
\item Does it converge uniformly in the interval $-1 < x < 1$?
\item Does it converge in the $L^2$ sense (i.e., in norm) in the interval $-1 < x < 1$? 
[Hint: You can compute the partial sums explicitly.]
\end{enumerate}
\item Prove the Cauchy-Schwarz inequality
\[|( f,g )| \leq \|f\| \|g\|,\]
for any pair of functions $f$ and $g$ on an interval $(a,b)$. 
[Hint: Consider the expression $h(t) := \|f + tg\|^2$ where $t \in \R$, and find the value of $t$ that minimizes $h$.]
\item Prove the Cauchy-Schwarz inequality for infinite series
\[\sum_{n=1}^\infty a_n b_n \leq \left(\sum_{n=1}^\infty a_n^2 \right)^\frac{1}{2}\left(\sum_{n=1}^\infty b_n^2 \right)^\frac{1}{2}.\]
[Hint: Write $(a,b) = \sum_{n=1}^N a_n b_n$ and $\|a\|^2 = (a,a)$, and use an argument similar to that for the previous question. Prove it first for finite sums, and then pass to the limit. ]
\item Show that if $f$ is a continuously differentiable $2\pi$-periodic function satisfying
\begin{equation}\label{eq:meanzero}
\int_{-\pi}^\pi f(x) \,dx = 0,
\end{equation}
then we have
\begin{equation}\label{eq:poincare}
\int_{-\pi}^\pi f(x)^2\, dx \leq \int_{-\pi}^\pi f'(x)^2 \, dx.
\end{equation}
The inequality above is called a Poincar\'e inequality. Give an example showing that \eqref{eq:poincare} may not hold when the zero mean condition \eqref{eq:meanzero} fails.
[Hint: Recall that since $f$ is continuously differentiable and $2\pi$-periodic, the Fourier series
\[f(x) = \sum_{n=-\infty}^\infty c_n e^{inx},  \ \ \text{ where } c_n = \frac{1}{2\pi} \int_{-\pi}^\pi f(x) e^{-inx} \, dx\]
converges uniformly, and hence it also converges in norm. Therefore Plancheral's identity
\[\frac{1}{2\pi}\int_{-\pi}^\pi f(x)^2 \, dx = \sum_{n=-\infty}^\infty |c_n|^2\]
holds. Use integration by parts to find a relationship between the Fourier coefficients $c_n$ of $f$, and the coefficients 
\[d_n = \frac{1}{2\pi} \int_{-\pi}^\pi f'(x) e^{-inx} \, dx\]
of $f'$. Then apply Bessel's inequality. Make sure to point out where you use the zero mean condition \eqref{eq:meanzero}.]
\end{enumerate}


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