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{SECT 0 {EXCHG {PARA 203 "" 0 "" {TEXT 203 46 "In-class Demonstration:
 Fourier Approximations" }}{PARA 206 "" 0 "" {TEXT 201 0 "" }}{PARA 
206 "" 0 "" {TEXT 201 35 "By Russell Blyth and Mike May, S.J." }}
{PARA 206 "" 0 "" {TEXT 201 0 "" }}}{EXCHG {PARA 202 "> " 0 "" 
{MPLTEXT 1 205 20 "restart:with(plots):" }}}{EXCHG {PARA 206 "" 0 "" 
{TEXT 201 65 "We start by using Maple to perform some tedious integral
s for us." }}{PARA 206 "" 0 "" {TEXT 201 0 "" }}}{EXCHG {PARA 206 "" 0
 "" {TEXT 201 95 " Verify that the functions used as basis vectors in \+
the approximating subspace are orthogonal. " }}}{EXCHG {PARA 202 "> " 
0 "" {MPLTEXT 1 205 78 "Int(sin(i*Pi*x)*sin(j*Pi*x),x=-1..1) = int(sin
(i*Pi*x)*sin(j*Pi*x),x=-1..1);\n" }{MPLTEXT 1 205 78 "Int(sin(i*Pi*x)*
sin(i*Pi*x),x=-1..1) = int(sin(i*Pi*x)*sin(i*Pi*x),x=-1..1);\n" }
{MPLTEXT 1 205 78 "Int(sin(i*Pi*x)*cos(j*Pi*x),x=-1..1) = int(sin(i*Pi
*x)*cos(j*Pi*x),x=-1..1);\n" }{MPLTEXT 1 205 78 "Int(cos(i*Pi*x)*cos(j
*Pi*x),x=-1..1) = int(cos(i*Pi*x)*cos(j*Pi*x),x=-1..1);\n" }{MPLTEXT 
1 205 76 "Int(cos(i*Pi*x)*cos(i*Pi*x),x=-1..1) = int(cos(i*Pi*x)*cos(i
*Pi*x),x=-1..1);" }}}{EXCHG {PARA 206 "" 0 "" {TEXT 201 144 "That left
 us with some work still to do. Try again. This time we first tell Map
le that i and j are integers (Maple assumes i and j are nonzero):" }}}
{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 205 39 "assume(i,integer): assum
e(j,integer):\n" }{MPLTEXT 1 205 25 "interface(showassumed=0):" }}}
{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 205 78 "Int(sin(i*Pi*x)*sin(j*Pi
*x),x=-1..1) = int(sin(i*Pi*x)*sin(j*Pi*x),x=-1..1);\n" }{MPLTEXT 1 
205 78 "Int(sin(i*Pi*x)*sin(i*Pi*x),x=-1..1) = int(sin(i*Pi*x)*sin(i*P
i*x),x=-1..1);\n" }{MPLTEXT 1 205 78 "Int(sin(i*Pi*x)*cos(j*Pi*x),x=-1
..1) = int(sin(i*Pi*x)*cos(j*Pi*x),x=-1..1);\n" }{MPLTEXT 1 205 78 "In
t(cos(i*Pi*x)*cos(j*Pi*x),x=-1..1) = int(cos(i*Pi*x)*cos(j*Pi*x),x=-1.
.1);\n" }{MPLTEXT 1 205 76 "Int(cos(i*Pi*x)*cos(i*Pi*x),x=-1..1) = int
(cos(i*Pi*x)*cos(i*Pi*x),x=-1..1);" }}}{EXCHG {PARA 206 "" 0 "" {TEXT 
201 36 "We need to also integrate against 1:" }}}{EXCHG {PARA 202 "> "
 0 "" {MPLTEXT 1 205 58 "Int(sin(i*Pi*x)*1,x=-1..1) = int(sin(i*Pi*x)*
1,x=-1..1);\n" }{MPLTEXT 1 205 58 "Int(cos(i*Pi*x)*1,x=-1..1) = int(co
s(i*Pi*x)*1,x=-1..1);\n" }{MPLTEXT 1 205 36 "Int(1*1,x=-1..1) = int(1*
1,x=-1..1);" }}}{EXCHG {PARA 206 "" 0 "" {TEXT 201 125 "To compensate \+
for the value of this final integral, we must remember to half the coe
fficient of this constant function later." }}{PARA 206 "" 0 "" {TEXT 
201 0 "" }}{PARA 206 "" 0 "" {TEXT 201 162 "Graph several of these int
egrands as a reality check - note that in each case it appears that th
e signed area from x=-1 to x=1 has the appropriate value (0 or 1):" }}
}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 205 38 "plot(sin(2*Pi*x)*sin(3*
Pi*x),x=-1..1);" }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 205 38 "plot
(sin(3*Pi*x)*cos(4*Pi*x),x=-1..1);" }}}{EXCHG {PARA 202 "> " 0 "" 
{MPLTEXT 1 205 38 "plot(cos(5*Pi*x)*cos(3*Pi*x),x=-1..1);" }}}{EXCHG 
{PARA 202 "> " 0 "" {MPLTEXT 1 205 38 "plot(sin(3*Pi*x)*sin(3*Pi*x),x=
-1..1);" }}}{EXCHG {PARA 208 "" 0 "" {TEXT 202 0 "" }}{PARA 208 "" 0 "
" {TEXT 202 7 "Example" }}{PARA 206 "" 0 "" {TEXT 201 0 "" }}{PARA 206
 "" 0 "" {TEXT 201 41 "Find Fourier approximations for f(x) = x." }}
{PARA 206 "" 0 "" {TEXT 201 0 "" }}{PARA 206 "" 0 "" {TEXT 201 48 "We \+
compute the Fourier coefficients for f(x) = x" }}}{EXCHG {PARA 202 "> \+
" 0 "" {MPLTEXT 1 205 58 "Int(x*sin(j*Pi*x),x=-1..1) = int(x*sin(j*Pi*
x),x=-1..1);\n" }{MPLTEXT 1 205 58 "Int(x*cos(j*Pi*x),x=-1..1) = int(x
*cos(j*Pi*x),x=-1..1);\n" }{MPLTEXT 1 205 32 "Int(x,x=-1..1) = int(x,x
=-1..1);" }}}{EXCHG {PARA 206 "" 0 "" {TEXT 201 78 "Note there are onl
y sine terms (not surprisingly, since f is an odd function)." }}{PARA 
206 "" 0 "" {TEXT 201 74 "Next use Maple to write a general expression
 for the order n approximation" }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT
 1 205 22 "Fapprox := (x,n) -> \n" }{MPLTEXT 1 205 45 "    sum(-2*(-1)
^j/(j*Pi)*sin(j*Pi*x),j=1..n);" }}}{EXCHG {PARA 206 "" 0 "" {TEXT 201 
59 "We can now plot (say) a 10 term approximation against f(x)." }}}
{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 205 22 "eval(Fapprox(x,10));\n" 
}{MPLTEXT 1 205 43 "plot([Fapprox(x,10),x],x=-2..2,y = -2..2,\n" }
{MPLTEXT 1 205 114 "    title = \" 10th Order  Fourier approximation t
o y =x\",     titlefont = [HELVETICA,12], color = [green,blue]);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT 200 30 "We can also animate this plot." }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "mindeg := 2: degsteps := 2
0: bydeg :=4:\n" }{MPLTEXT 1 0 56 "A := display(seq(plot([Fapprox(x,mi
ndeg + bydeg*i),x],\n" }{MPLTEXT 1 0 21 "   x=-2..2,y=-2..2,\n" }
{MPLTEXT 1 0 74 "   title=\"degree = \"||(mindeg + bydeg*i), titlefont
=[HELVETICA, 14]), \n" }{MPLTEXT 1 0 38 "   i=0..degsteps), insequence
=true):\n" }{MPLTEXT 1 0 55 "B := animate(func,x=-2..2,y=-2..2,frames=
degsteps+1):\n" }{MPLTEXT 1 0 45 "print(\" x vs its Fourier approximat
ion\");\n" }{MPLTEXT 1 0 32 "display(A,B,view=[-2..2,-2..2]);" }}}
{EXCHG {PARA 206 "" 0 "" {TEXT 201 79 "It is also useful to plot the e
rror between the function and the approximation." }}}{EXCHG {PARA 202 
"> " 0 "" {MPLTEXT 1 205 40 "plot(Fapprox(x,10)-x,x=-1..1,y=-1..1, \n"
 }{MPLTEXT 1 205 112 "   title = \"error in 10th Order  Fourier approx
imation to y = x\",    titlefont = [HELVETICA,12], color = red);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT 200 57 "Similarly, we can animate the plo
t of the error function." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 
"mindeg := 2: degsteps := 20: bydeg :=4:\n" }{MPLTEXT 1 0 61 "print(\"
Error in nth order Fourier approximation to y=x\"):\n" }{MPLTEXT 1 0 
14 "display(seq(\n" }{MPLTEXT 1 0 39 "  plot(Fapprox(x,mindeg + bydeg*
i)-x,\n" }{MPLTEXT 1 0 34 "   x=-1..1, y=-1..1, color=blue,\n" }
{MPLTEXT 1 0 73 "   title=\"order = \"||(mindeg + bydeg*i), titlefont=
[HELVETICA, 14]), \n" }{MPLTEXT 1 0 37 "  i=0..degsteps), insequence=t
rue);\n" }}{PARA 210 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 206 "
" 0 "" {TEXT 201 67 "Higher order approximations match the function f \+
even more closely!" }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 0 "" }}
}{PARA 211 "" 0 "" {TEXT -1 0 "" }}}
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