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\centerline{\bf  Math 240C: Real Analysis, Spring 2020}

\vspace{3 mm}

\centerline{\bf \large  Homework Assignment 6 } 

\vspace{1 mm}

\centerline{\bf Due 12:00 noon, Monday, May 18, 2020}

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%{\bf Name} \rule{2.5 in}{0.005 in}  $ \quad $ {\bf ID number} \rule{2.1 in}{0.005 in}

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\begin{itemize}

\item[1.]
Let $f \in C^1(\R) $ be such that $f' \in L^1(\R)\cap L^2(\R)$ and $\lim_{|x| \to \infty} x [f(x)]^2= 0.$
Prove the following:
\begin{compactenum}
\item[(1)]
\[
\left[ \int |f(x)|^2 dx \right]^2 \le 4 \int | x f(x) |^2 dx
\int |f'(x)|^2 dx;
\]
\item[(2)]
{\bf (Heisenberg's Inequality)} For any $b,\, \beta \in \R$,
\[
\int (x-b)^2 |f(x)|^2 dx \int (\xi - \beta)^2 |\hat{f}(\xi)|^2 d\xi
\ge \frac{\| f \|_2^4 }{16 \pi^2}.
\]
\end{compactenum}
(See Exercise 18 on page 255.)

\item[2.]
Let $f \in L^1(\R^2)$ be radial, i.e., there exists $g: [0,\infty) \to \R$ such that
$f(x) = g(|x|)$ for all $x \in \R^2.$ Prove that $\hat{f}$ is also radial.
(Note that this result is true for $\R^n$ for a general $n \ge 1.$
See Exercise 22 on page 256. Here, for $n = 2$, you can use the polar coordinates and change of variables.)

\item[3.]
Let $0 < r < 1.$ Consider the Poisson kernel on $\mathbb T$:
$
P_r(x) = \sum_{k=-\infty}^\infty r^{|k|} e^{2 \pi i k x}.
$
\begin{compactenum}
\item[(1)]
Prove that
\[
P_r(x) = \frac{1 -r^2}{ 1 + r^2 - 2 r \cos 2 \pi x}.
\]
\item[(2)]
Let $f \in L^1({\mathbb T})$ and define
$
A_r f(x) = \sum_{k=-\infty}^\infty r^{|k|} \hat{f}(k) e^{2 \pi i k x}.
$
Prove that $A_r f = f * P_r.$
\end{compactenum}

\item[4.]
Let $a_k \in \C$ $(k = 0, 1, \dots)$, $S_n  = \sum_{k=0}^n a_k$ $(n = 0, 1, \dots),$
and $\sigma_m = (m+1)^{-1} \sum_{n=0}^m S_n$ $(m = 0, 1, \dots).$  
\begin{compactenum}
\item[(1)]
Show that $\sigma_m = (m+1)^{-1} \sum_{k=0}^m (m + 1 - k) a_k. $ 
\item[(2)]
Assume $\lim_{n\to \infty} S_n$ exists. Show $\lim_{m\to \infty} \sigma_m $ exists, 
and the two limits are equal. 
\item[(3)]
Show that the series $\sum_{k=0}^\infty (-1)^k$ diverges but is 
Abel and Ces\`aro summable to $1/2.$ 
\end{compactenum}

\item[5.]
Let $\sigma_m f$ denote the Ces\`aro means of the Fourier series of $f$ given by (8.39)
on page 261.
\begin{compactenum}
\item[(1)]
Denote by $F_m = (m+1)^{-1} \sum_{k=0}^m D_k, $ where $D_k$ is the $k$th Dirichlet 
kernel. Prove that $\sigma_m f =  f \ast F_m.$ 
($F_m$ is called the $m$th Fej\'er kernel.)
\item[(2)]
Prove that 
\[
F_m(x) = \frac{ \sin^2 (m+1)\pi x }{(m+1) \sin^2 \pi x}.
\]
\end{compactenum}


\item[6.]
\begin{compactenum}
\item[(1)]
Let $D_n$ denote the $n$th Dirichlet kernel. Show that  $\| D_n \|_1 > (4/\pi^2)  \sum_{j=1}^n 1/j.$ 
%% \to \infty $ as $n \to \infty.$ 
\item[(2)]
Denote by $S_m f$ the $m$th symmetric partial sum of the Fourier series of $f \in L^1({\mathbb T}).$ 
Prove that the set of all $f \in C({\mathbb T}) $ such that the sequence
$\{ S_m f (0) \} $ converges is meager in $C ({ \mathbb T}). $  
\end{compactenum}
(See Exercises 34 and 35 on page 269.) 



\end{itemize}

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