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\title{ Math 3220-3 Homework 2}
\author{Due February  11 }
\maketitle

\begin{enumerate}  \item (This problem and the next are part of problems 12,13 of Rudin,Chapter 8).  Find the Fourier coefficients $\{c_n\}$ and Fourier series $\sum_{-\infty}^\infty c_n e^{inx}$ of the following two functions: 
    \begin{enumerate}
  \item For fixed $\delta >0$ let 
  $$
  f(x) = \begin{cases}
      1 & \text{if  } |x|<\delta, \\
      0 & \text{if } \delta< |x|<\pi.
\end{cases}
$$
  on $[-\pi,\pi]$ and extend to a periodic function  of period $2\pi$ to all of $\R$.
  
    \item Let $f(x) = x$ on $[0,2\pi)$ and extend to a function periodic of period $2\pi$ to all of $\R$.
\end{enumerate}
\vskip10pt
  \item 
  \begin{enumerate}
  \item Write down Parseval's identity (Rudin, Theorem 8.16)
  $$
  \sum_{-\infty}^\infty |c_n|^2 =\frac{1}{2\pi}\int_{-\pi}^\pi |f(x)|^2 dx  
  $$
 for each function. 
\vskip3pt
\item Check that for the first function it gives
$$
\sum_{n=1}^\infty \frac{\sin^2(n\delta)}{n^2\delta} = \frac{\pi - \delta}{2},
$$
and for the second
$$
\sum_{n=1}^\infty \frac{1}{n^2} = \frac{\pi^2}{6}.
$$
  \vskip3pt
  \item Put $\delta = \frac{\pi}{2}$ in the identity for the first function.  Write down explicitly what it gives, and check that it is compatible with the answer for the second function.  
\end{enumerate}
\vskip 10pt
  \item   (Rudin, Chapter 8, Problem 16)  A variation on Fejer's theorem:
  Prove that if $f$ is Riemann integrable and there is an $x$ with the property that the one-sided limits $f(x^+), f(x^-)$ exist, then
  $$
  \lim_{N\to \infty} \sigma_N(f;x) = \frac{1}{2}(f(x^+) + f(x^-) ),
  $$
  
 where 
  $$
  \sigma_N(f;x) = \frac{s_0(f,x) +s_1f(;x) + \dots + s_N(f;x)}{N+1}
  $$
  is the sequence of Cesaro means of the Fourier sums $s_N(f,x)$.
  
  \emph{Suggestion}: Accept all the details of the proof of Fejer's theorem in Rudin, Chap 8, Exercise 15.  In particular,  accept that 
  $$
  \sigma_N(f,x) = \frac{1}{2\pi}  \int_{-\pi}^\pi f(x-t) K_N(t) dt
  $$
  where
  $$
  K_N(t) = \frac{1}{N+1}\frac{1-\cos((N+1)t)}{1-\cos(t)}
  $$
  has properties (a),,(b),(c). The proof of $\sigma_N(f,x)\to f(x)$ uniformly is based on the identity 
  $$
  f(x) - \sigma_N(f,x) =\frac{1}{2\pi} \int_{-\pi}^\pi (f(x) - f(x-t)) K_N(t) dt
  $$
 and the continuity of $f$.  Find a  suitable substitute for this identity that will give the result we want when we only know the existence of one-sided limits.
 \vskip 10pt
  \item (Rudin, Chp 8, ex 19): Suppose $f:\R\to \R$ is continuous, $f(x+2\pi) = f(x)$, and suppose $\alpha\in\R$ and $\frac{\alpha}{\pi}$ is irrational.  Prove that
  $$
  \lim_{N\to\infty} \frac{1}{N} \sum_{n=1}^N f(x + n\alpha) = \frac{1}{2\pi} \int_{-\pi}^\pi 
 f(t)dt
 $$
 \noindent\emph{Suggestion}:   Do it  first for $e^{imx}, m\in\Z$.
\vskip 10pt 
 \item Recall the $\Gamma$-function
 $$
 \Gamma(x) =  \int_0^\infty t^{x-1}e^{-t} dt
 $$ 
 One way to compute $\Gamma(\frac{1}{2})  = \int_0^\infty t^\frac{1}{2} e^{-t} dt$ is to make the substitution $ t = s^2$, which gives $\Gamma(\frac{1}{2}) = \int_{-\infty}^\infty e^{-s^2} ds$.  This can be computed by the trick of relating it to an integral over $\R^2$ and using polar coordinates:
 $$
 (\int_{-\infty}^\infty e^{-x^2}dx)^2 = \int_{-\infty}^\infty e^-(x^2 + y^2) dxdy =\int_0^{2\pi}\int_0^\infty e^{-r^2}r dr d\theta = \pi
 $$
 Use this same idea to compute 
 
 \begin{itemize}
  \item $s_{n-1} = $ the volume of $S^{n-1} = \{\vx\in\R^n : ||\vx||= 1\}$ 
  \item $b_n = $ the volume  of the unit ball $B^n = \{ \vx\in\R^n : ||\vx||\le 1\}$ 
   
\end{itemize}

in terms of the Gamma function.  
\vskip3pt
\noindent\emph{Suggestion}: Start from
 $$
 \pi^\frac{n}{2} = \int_{-\infty}^\infty \dots\int_{-\infty}^\infty e^{-(x_1^2 + \dots + x_n^2)}dx_1 \dots dx_n = \int_{S^{n-1}}\int_0^\infty e^{-r^2}r^{n-1}drd\Theta
 $$
\end{enumerate}


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