\documentclass[11pt]{article}
\usepackage{amsfonts}
\usepackage{amsmath}
\usepackage{amssymb}

\newcommand{\be}{\begin{equation}}
\newcommand{\ee}{\end{equation}}
\newcommand{\eq}[1]{(\ref{#1})}
\newcommand{\Rl}{\mathbb{R}}
\newcommand{\Cx}{\mathbb{C}}
\newcommand{\Ts}{\mathbb{T}}
\newcommand{\Nl}{\mathbb{N}}
\newcommand{\Ir}{\mathbb{Z}}
\newcommand{\sgn}{\operatorname{sgn}}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{document}


\centerline{\textbf{Problem Set 2: Math 201B}}
\centerline{Due: Friday, January 14}

\bigskip\noindent
\textbf{1.} If $1\le p < \infty$, show that the trigonometric polynomials are dense in $L^p(\Ts)$.


\bigskip\noindent
\textbf{2.} For fixed $z\in \Cx$, let $J_n(z)$ denote the $n$th Fourier coefficient
of the function $e^{i z\sin x}$, meaning that
\[
J_n(z) = \frac{1}{2\pi}\int_0^{2\pi} e^{iz\sin x} e^{-in x}\, dx\qquad \mbox{for $n\in \Ir$}.
\]
(a) What is $J_n(0)$? Show that $J_{-n}(z) = (-1)^n J_n(z)$.

\smallskip\noindent
(b) Derive the recurrence relations
\[
\frac{2n}{z} J_n(z) = J_{n-1}(z) + J_{n+1}(z),
\qquad
2 J_n^\prime(z) = J_{n-1}(z) - J_{n+1}(z)
\]
where the prime denotes a derivative with respect to $z$.

\smallskip\noindent
(c) Deduce from (b) that $J_n(z)$ is a solution of Bessel's equation
\[
z^2 J_n^{\prime\prime} + z J_n^\prime + \left(z^2 - n^2\right) J_n = 0.
\]


\bigskip\noindent
\textbf{3.} A family of (not necessarily positive)
functions $\{\phi_n\in L^1(\Ts) : n\in \Nl\}$
is an approximate identity if:
\begin{align*}
\int \phi_n \,dx = 1\qquad &\mbox{for every $n\in \Nl$};
\\
\int |\phi_n|\, dx \le M\qquad
&\mbox{for some constant $M$ and all $n\in \Nl$};
\\
\lim_{n\to\infty} \int_{\delta<|x|<\pi} |\phi_n|\, dx = 0
\qquad
&\mbox{for every $\delta > 0$}.
\end{align*}
If $f\in L^1(\Ts)$, show that $\phi_n\ast f \to f$ in $L^1(\Ts)$ as $n\to \infty$.



\bigskip\noindent
\textbf{4.} (a) Let $\{a_n : n\ge 0\}$ be a sequence of non-negative real numbers such that $a_n \to 0$ as $n\to \infty$
and
\[
a_{n+1} - 2 a_n + a_{n-1} \ge 0.
\]
Show that the series
\[
\sum_1^\infty n\left(a_{n+1} - 2 a_n + a_{n-1}\right)
\]
converges to $a_0$. \textsc{Hint.} $\sum (a_{n+1} - a_n)$ is a convergent, decreasing telescoping series.

\smallskip\noindent
(b) For $N \ge 0$, let $K_N\ge 0$ denote the Fej\'er kernel
\[
K_N(x) = \sum_{n=-N}^N \left(1 - \frac{|n|}{N+1}\right) e^{inx}.
\]
Show that the series
\[
f(x) = \sum_{n=1}^\infty n\left(a_{n+1} - 2 a_n + a_{n-1}\right) K_{n-1}(x)
\]
converges in $L^1(\Ts)$ to a non-negative function $f\in L^1(\Ts)$ whose
Fourier coefficients are $a_{|n|}$ \textit{i.e.}
\[
f(x)\sim \sum_{n\in \Ir} a_{|n|} e^{inx}.
\]

\smallskip\noindent
(c) Show that there is a function $f\in L^1(\Ts)$ such that
\[
f(x) \sim \sum_{|n|\ge 2} \frac{1}{\log |n|}e^{inx}.
\]

\smallskip\noindent
(d) Suppose that $f\in L^1(\Ts)$ has imaginary Fourier coefficients $\{i b_n: n\in \Ir\}$
such that $b_n \ge 0$ for $n \ge 0$ and $b_{-n} = - b_{n}$.  Show that
\[
\sum_{n=1}^\infty \frac{b_n}{n}
\]
converges. \textsc{Hint.} The integral
\[
F(x) = \int_0^x f(t)\, dt
\]
is a continuous function (in fact, absolutely continuous) with Fourier coefficients
\[
\frac{1}{2\pi} \int F(x) e^{-inx}\, dx = \frac{b_n}{in}\qquad \mbox{for $n\ne 0$}.
\]
Use the fact that $K_N\ast F(0)$ converges to $F(0)$ since $\{K_N\}$ is an approximate identity.


\smallskip\noindent
(e) Show that there is no function $f\in L^1(\Ts)$ such that
\[
f(x) \sim \sum_{|n| \ge 2} \frac{i\sgn n}{\log |n|}e^{inx}.
\]
(Here, $\sgn n$ is equal to $1$ if $n>0$ and $-1$ if $n<0$.)




\end{document}
