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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT 256 11 "Math 2280-2" }}{PARA 
257 "" 0 "" {TEXT -1 21 "Tuesday April 9, 2001" }}{PARA 258 "" 0 "" 
{TEXT -1 0 "" }{TEXT 259 27 "Playing with Fourier Series" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart:\nwith(plots):" }}}{PARA 0 
"" 0 "" {TEXT -1 146 "On the interval [-Pi..Pi] the functions \{1,cos(
t),cos(2t),...sin(t),sin(2t),sin(3t),...\} are mutually orthogonal wit
h respect to the inner product" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 37 "Dot:=(f,g)->int(f(t)*g(t),t=-Pi..Pi);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%$DotGR6$%\"fG%\"gG6\"6$%)operatorG%&arrowGF)-%$intG6$
*&-9$6#%\"tG\"\"\"-9%F3F5/F4;,$%#PiG!\"\"F;F)F)F)" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 242 "Mag:=f->sqrt(Dot(f,f)); #computes the magni
tude of\n     #a vector\nDist:=(f,g)->Mag(f-g);  #computes the ``dista
nce'' between\n     #two vectors.  \nAngle:=(f,g)->arccos(Dot(f,g)/(Ma
g(f)*Mag(g)));\n     #computes the ``angle'' between two vectors" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$MagGR6#%\"fG6\"6$%)operatorG%&arrow
GF(-%%sqrtG6#-%$DotG6$9$F2F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
%DistGR6$%\"fG%\"gG6\"6$%)operatorG%&arrowGF)-%$MagG6#,&9$\"\"\"9%!\"
\"F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&AngleGR6$%\"fG%\"gG6\"6
$%)operatorG%&arrowGF)-%'arccosG6#*&-%$DotG6$9$9%\"\"\"*&-%$MagG6#F4F6
-F96#F5F6!\"\"F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 
0 "" {TEXT -1 271 "When we do the projection of a function f onto Vn, \+
the coefficients for each  basis vector are just the inner product of \+
that basis vector with f, divided by the square of the magnitude of th
e vector  Let's illustrate by projecting the absolute value function o
nto W10:  " }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 210 "f:=t->abs(t);  #we'll do Fourier for the absolute \+
value function\nn:=3;   #order 3 expansion\na:=vector(n); #cos coeffic
ients\nb:=vector(n);  #sin coefficients\na0:=(1/Pi)*Dot(f,1);  #order \+
zero Fourier coefficient" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG%$absG" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"nG\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG-%
&arrayG6$;\"\"\"\"\"$7\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bG-%&a
rrayG6$;\"\"\"\"\"$7\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#a0G%#PiG
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 129 "for i from 1 to n do  \+
#compute the projection coefficients\nb[i]:=(1/Pi)*Dot(f,t->sin(i*t));
\na[i]:=(1/Pi)*Dot(f,t->cos(i*t));\nod:\n" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 88 "evalm(b);#why will an even function have Fourier si
ne\n  #coefficients all equal to zero?" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#-%'vectorG6#7%\"\"!F'F'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }
{TEXT 257 267 "The answer to the rhetorical question above is that the
 product of an even function with an odd function is an odd function. \+
 We already asserted that the integral of an odd function over an inte
rval [-L,L] is always zero.  By the way, could you prove that assertio
n?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 31 "evalm(a); #We do these in class" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%'vectorG6#7%,$*&\"\"\"F)%#PiG!\"\"!\"%\"\"!,$F(#F,\"
\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 258 146 "The answer to th
is rhetorical question is yes:  the i=even cos coefficients are zero, \+
the i=odd ones are -4 divided by the product of Pi with i^2." }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "approx:=x->sum(b[k]*sin(k*x)
,k=1..n)\n           +sum(a[m]*cos(m*x),m=1..n)\n           +a0/2;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'approxGR6#%\"xG6\"6$%)operatorG%&ar
rowGF(,(-%$sumG6$*&&%\"bG6#%\"kG\"\"\"-%$sinG6#*&F4F59$F5F5/F4;F5%\"nG
F5-F.6$*&&%\"aG6#%\"mGF5-%$cosG6#*&FDF5F:F5F5/FDF<F5*&#F5\"\"#F5%#a0GF
5F5F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "approx(x); #or
der 3 Fourier expansion for abs(x):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#,(*&-%$cosG6#%\"xG\"\"\"%#PiG!\"\"!\"%*&#\"\"%\"\"*F)*&-F&6#,$F(\"\"$
F)F*F+F)F+*&#F)\"\"#F)F*F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 69 "almost:=plot(approx(x),x=-3*Pi..3*Pi,color=black):\ndisplay(\{al
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"" 0 "" {TEXT -1 37 "Now for the square wave, of period 2:" }}}{EXCHG 
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{MPLTEXT 1 0 31 "plot(f(t),t=-1..1,color=black);" }}{PARA 13 "" 1 "" 
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r 10 expansion\na:=vector(n); #cos coefficients\nb:=vector(n);  #sin c
oefficients\na0:=(1/1)*int(f(t),t=-1..1);  #order zero Fourier coeffic
ient" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 149 "for i from 1 to n do  #compute the projection coeffi
cients\nb[i]:=(1/1)*int(f(t)*sin(i*t*Pi),t=-1..1);\na[i]:=(1/1)*int(f(
t)*cos(i*t*Pi),t=-1..1);\nod:\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
\"nG\"#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG-%&arrayG6$;\"\"\"\"
#57\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bG-%&arrayG6$;\"\"\"\"#57
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