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Math 436


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{\bf Take-home problem for midterm }
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\noindent
Due at 12:50 p.m., October 13, 1999

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\noindent
You may consult your course notes, homework, and the textbook.
You may use Maple, Matlab or Mathematica as an experimental
tool  on any part of the problem.  If you do other than on
part g), please indicate how on your exam.
You may not consult any other books or notes.  You may not discuss
the exam with anyone except me.

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\noindent
The in-class part of the exam will be on October 13.  It will be
closed book, but you may bring either a sheet of paper ( $8\frac12 ''\times
11''\/$) with notes on one side or 
a  $3''\times 5''\/$
card with notes on both sides to the exam.

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\noindent
You must justify your answers.
If you cannot do one part, you may assume the result of that 
part for the remaining parts.

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\noindent
Let    
$$
K_N(x) = \frac 1{N+1}\sum _{n=0}^N D_n(x).
$$

a)  Show that  for  $0<x\le \pi\/$
$$
K_N(x) = \frac 1{2\pi(N+1)}\cdot\frac {1-\cos(N+1)x}{1-\cos x} .
$$
({\bf Hint:} Multiply the numerator and denominator of $D_n\/$ by
 $2\sin \frac x2\/$ and convert products of sines to differences
of cosines.)

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b) Show that  $K_N \ge 0\/$.

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c) Show that  $\int _{-\pi }^{\pi} K_N(x)\,dx=1\/$.

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d) Show that 
$$
K_n(x) \le \frac 1 {\pi(N+1)}\cdot\frac 1{1-\cos \delta}
$$
if  $0<\delta \le |x|\le\pi\/$.

\medskip

e)  Suppose  $f\/$ 
is piecewise smooth and  $2\pi\/$-periodic.
Let   $s_n(f;x)\/$
denote the  $n\/$th partial sum of the Fourier series of  $f\/$
and let  
$$
\sigma _N = \frac  {s_0+s_1+\dots s_N} {N+1},
$$
so  $\sigma _N \/$
is the arithmetic mean of  the first  $N+1\/$
partial sums.
Show that 
$$
\sigma _N(f;x) = \int _{-\pi }^{\pi} f(x-t)K_N(t) \,dt.
$$

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f) Prove Fej\'er's theorem:

\begin{thm}
If  $f \/$
is continuous on  $\mathbf R\/$ and periodic with period $2\pi\/$
then  
$$
\sigma _N(f;x) \to f(x)
$$
uniformly on  $[-\pi,\pi]\/$.
\end{thm}

({\bf Hints:}  Your proof will require the results of b)-d).
You will also need to use the fact that a continuous function
 $g\/$
on a closed interval  $I\/$ is {\it uniformly continuous}:  given 
 $\epsilon >0\/$
there is  $\delta >0\/$
such that  $|g(u)-g(v)|<\epsilon\/$
if  $u,v\in I\/$ and  $|u-v|<\delta\/$.)

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g)  Graph  $K_N\/$ (on a computer) for  $N=5, \ 10,\ 50 \/$. (Be
sure to have the plot command compute enough points if you 
use Maple.) Compare
with the graph of  $D_N\/$.  

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