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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 0 "" }{TEXT 256 0 "" }{TEXT 
257 0 "" }{TEXT 258 10 "Lecture 6:" }{TEXT 259 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 0 "" }{TEXT 260 0 "" }
{TEXT 261 33 "The Convergence of Fourier Series" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 0 "" }{TEXT 262 11 "Jorge A
arao" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 0 "
" }{TEXT 263 17 "PCMI, Summer 2003" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 123 "
This worksheet contains a procedure (found in the Maple Manual) that c
omputes and displays the n-th Fourier approximant for" }}{PARA 0 "" 0 
"" {TEXT -1 127 "a function f defined on a given interval. Also to be \+
found are pictures of the Dirichlet kernel, and a visual demonstration
 of " }}{PARA 0 "" 0 "" {TEXT -1 115 "Gibbs' phenomenon. Finally, at t
he end we have a procedure that computes the L^2 norm of the differenc
e f - S(f,N)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 71 "Whenever appropriate the code will be given in the form o
f a procedure." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 
0 "" {TEXT -1 30 "Computing Fourier Coefficients" }}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 124 "The procedure An takes as its input a function (fun
c), its range (xrange), and an integer n, and returns the coefficient \+
An." }}{PARA 0 "" 0 "" {TEXT -1 54 "Likewise, the procedure Bn returns
 the coefficient Bn." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "An:
=proc(func, xrange::name=range, n)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
21 "   local l; global A;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "   l:=
 rhs( rhs(xrange) ) - lhs( rhs(xrange) );" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 42 "   A[n]:= (2/l)*int(func*cos(n*x),xrange);" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 9 "end proc:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 19 "An(x,x=-Pi..Pi, 3);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 22 "An(x^2, x=-Pi..Pi, 0);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 45 "Bn:=proc(func, xrange::name=range, n::posint)" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "   local l; global B;" }}{PARA 0 ">
 " 0 "" {MPLTEXT 1 0 47 "   l:= rhs( rhs(xrange) ) - lhs( rhs(xrange) \+
);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "   B[n]:= (2/l)*int(func*sin(
n*x),xrange);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "end proc:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "Bn(x,x=-Pi..Pi, 3);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "Bn(x,x=-Pi..Pi, 2);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 31 "Displaying
 a Single Approximant" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 60 "The proce
dure below displays the n-th Fourier approximant to" }{TEXT 264 2 " f
" }{TEXT -1 29 ", together with the graph of " }{TEXT 265 1 "f" }
{TEXT -1 8 " itself." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 529 "Ap
proximantn:=proc(func, xrange::name=range, n::posint)\n local x, a, b,
 l, k, p, partsum;\n global q;  \n a:= lhs( rhs(xrange) );\n b:= rhs( \+
rhs(xrange) );\n l:= b-a;\n x:= 2*Pi*lhs(xrange)/l;\n\n partsum := (1/
l) * evalf(Int(func, xrange));\nfor k from 1 to n do\n   partsum:= par
tsum+(2/l)*evalf(Int(func*sin(k*x),xrange))*sin(k*x)+\n             (2
/l)*evalf(Int(func*cos(k*x),xrange))*cos(k*x);\nod;\nq[n]:=plot(partsu
m, xrange, color=blue, args[4..nargs]);\np:=plot(func, xrange, color=r
ed, args[4..nargs]);\nplots[display]([q[n],p]);\nend:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "Approximantn(x,x=0..Pi, 4);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 26 "Animating \+
the approximants" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 100 "This procedur
e shows an animation of the Fourier approximants, and can be found in \+
the Maple manual." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 581 "Fouri
erPicture:=proc(func, xrange::name=range, n::posint)\n local x, a, b, \+
l, k, j, p, q, partsum;\n \n a:= lhs( rhs(xrange) );\n b:= rhs( rhs(xr
ange) );\n l:= b-a;\n x:= 2*Pi*lhs(xrange)/l;\n\n partsum := (1/l) * e
valf(Int(func, xrange));\nfor k from 1 to n do\n   partsum:= partsum+(
2/l)*evalf(Int(func*sin(k*x),xrange))*sin(k*x)+\n             (2/l)*ev
alf(Int(func*cos(k*x),xrange))*cos(k*x);\n   q[k]:=plot(partsum, xrang
e, color=blue, args[4..nargs]);\nod;\nq:=plots[display]([seq(q[k],k=1.
.n)],insequence=true);\np:=plot(func, xrange, color=red, args[4..nargs
]);\nplots[display]([q,p]);\nend:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 35 "FourierPicture(x^2, x=-Pi..Pi, 20);" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 52 "#f:=x->piecewise(-Pi<x and x<0, 0, 0<x an
d x<Pi, 1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "#FourierPict
ure(f(x), x=-Pi..Pi, 40);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 
"" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 3 "
" 0 "" {TEXT -1 20 "The Dirichlet Kernel" }}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 92 "The following procedure plots the graph of the Dirichlet \+
kernel over the interval -Pi to Pi." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 11 "DN:=proc(N)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "  l
ocal j; global Dir;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "  Dir[N]:=1;
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "  for j from 1 to N do" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "    Dir[N]:=Dir[N]+2*cos(j*x);" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "  end do;" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 26 "  plot(Dir[N], x=-Pi..Pi);" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 9 "end proc:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 
"DN(5);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "Int(Dir[5], x=-P
i..Pi);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 17 "Gibbs' Phe
nomenon" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 112 "Whenever we compute th
e Fourier approximants of a discontinuous function f, there is a littl
e bump that shows up" }}{PARA 0 "" 0 "" {TEXT -1 106 "near every disco
ntinuity of f. This is known as Gibbs' phenomenon, and can be easily v
erified graphically." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "f:=
x->piecewise(-Pi<x and x<0, 0, 0<x and x<Pi, 1);" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 36 "FourierPicture(f(x), x=-Pi..Pi, 40);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 23 "Mean Squar
e Convergence" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 85 "This procedure co
mputes the L^2 norm of  f - S(f,N) over the interval from -Pi to Pi." 
}}{PARA 0 "" 0 "" {TEXT -1 53 "It calls the procedures An and Bn defin
ed previously." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "MeanSquar
e:=proc(func, N)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "  local g, j; g
lobal M;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "  g:=An(func, x=-Pi..Pi
, 0)/2;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "  for j from 1 to N do" 
}}{PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "    g:=g+An(func, x=-Pi..Pi, j)*c
os(j*x)+Bn(func, x=-Pi..Pi, j)*sin(j*x);" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 9 "  end do;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "  M:=e
valf(sqrt(int((func -g)^2, x=-Pi..Pi)));" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 9 "end proc:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
17 "MeanSquare(x, 4);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "Me
anSquare(x, 16);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "MeanSqu
are(x, 64);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "MeanSquare(x
^2, 4);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "MeanSquare(x^2, \+
16);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "MeanSquare(x^2, 64)
;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{MARK "12" 0 }
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