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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "Unit 2: Infinite Series" }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "Chapter \+
10: Fourier Series" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 52 "Section 10.6:  optimizing property of Fourier series" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 9 "C
opyright" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
48 "Copyright * 2001 by Addison Wesley Longman, Inc." }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 302 "All rights reserved.  \+
No part of this publication may be reproduced, stored in a retrieval s
ystem, or transmitted, in any form or by any means, electronic, mechan
ical, photocopying, recording, or otherwise, without the prior written
 permission of the publisher. Printed in the United States of America.
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 
15 "Initializations" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 44 "with(plots):\nwith(plottools):\nwith(student):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "
" {TEXT -1 14 "Approximating " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG
" }{TEXT -1 30 " via Trigonometric Polynomials" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 139 "A partial sum of \+
a Fourier series is a trigonometric polynomial approximating the funct
ion to which the series converges.  For the function" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "f := x;" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 24 "defined on the interval " }{XPPEDIT 18 0 "[0,Pi]" "6#7$
\"\"!%#PiG" }{TEXT -1 67 ", a two-term trigonometric polynomial which \+
might approximate it is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "g := s1*sin(x) + s2*sin(2*x);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 44 "Judging the quality of the approximation of " }{XPPEDIT 
18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 4 " by " }{XPPEDIT 18 0 "g(x)
" "6#-%\"gG6#%\"xG" }{TEXT -1 50 " requires a measure of the goodness \+
of the fit of " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG6#%\"xG" }{TEXT -1 4 " \+
to " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 187 ".  In Sect
ion 8.4 we used the notion of convergence in the mean to measure conve
rgence of a sequence of functions.  The measure of performance for our
 trigonometric approximation will be " }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 258 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "abs(abs(f(x)-g(x)
))[2]^2=Int((f(x)-g(x))^2,x=0..Pi)" "6#/*$&-%$absG6#-F'6#,&-%\"fG6#%\"
xG\"\"\"-%\"gG6#F/!\"\"6#\"\"#F6-%$IntG6$*$,&-F-6#F/F0-F26#F/F4F6/F/;
\"\"!%#PiG" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 65 "the square of the measure used for convergence in the mean.  Th
e " }{TEXT 256 9 "subscript" }{TEXT -1 41 " 2 indicates that the diffe
rence between " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 5 " \+
and " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG6#%\"xG" }{TEXT -1 26 " in the in
tegral has been " }{TEXT 257 7 "squared" }{TEXT -1 1 "." }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "We illustrate this m
easure in the following example." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 4 "" 0 "" 
{TEXT -1 13 "Example 10.12" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 10 "Assigning " }{XPPEDIT 18 0 "s[1]" "6#&%\"
sG6#\"\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "s[2]" "6#&%\"sG6#\"\"#
" }{TEXT -1 17 " the values 1 in " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG6#%
\"xG" }{TEXT -1 25 " yields the approximating" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "g1 := subs(
s1=1, s2=1, g);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 116 "appearing in the following figure
, Figure 10.27, as the solid (red) curve.  The dashed (black) curve is
, of course, " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 1 ".
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 140 "plot([f,g1],x=0..Pi,color=black,linestyle=[2,1], col
or=[black,red], xtickmarks=3, ytickmarks=3, labels=[x,``], labelfont=[
TIMES,ITALIC,12]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 15 "The calibre of " }{XPPEDIT 18 0 "g
[1](x)" "6#-&%\"gG6#\"\"\"6#%\"xG" }{TEXT -1 24 " as an approximation \+
to " }{XPPEDIT 18 0 "f(x)=x" "6#/-%\"fG6#%\"xGF'" }{TEXT -1 40 " is ca
ptured in the following animation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 385 "F := plot([f,g1], x=0..Pi,
 color=[black,red], thickness=3):\nfx := unapply(f,x):\ngx := unapply(
g1,x):\nF1 := t -> display(polygon([[t,gx(t)],[ t,fx(t)], [t+.1,fx(t+.
1)], [t+.1,gx(t+.1)]], color=green)):\nF2 := N -> display([seq(F1(k/10
),k=0..N)]):\nF3 := display([seq(F2(k),k=0..31)],insequence=true):\ndi
splay([F,F3], labels=[x,y], labelfont=[TIMES,ITALIC,12], xtickmarks=3,
 ytickmarks=3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "The integral of the square of the \+
difference between " }{XPPEDIT 18 0 "f(x)=x" "6#/-%\"fG6#%\"xGF'" }
{TEXT -1 4 " and" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 264 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "g[1](x)=sin(x)+sin(2*x)" "6#/-&%\"gG6#
\"\"\"6#%\"xG,&-%$sinG6#F*F(-F-6#*&\"\"#F(F*F(F(" }{TEXT -1 1 " " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 365 "varies w
ith the total area between the two curves.  The green area swept in th
e animation above quantifies the fit.  The integral \"measures\" the a
mount of green.  The less \"green,\" the better the fit between the tw
o functions.  Hence, minimizing the amount of green area yields the \"
best\" fit according to the constraints of this particular method of m
easuring fit." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 19 "A calculation shows" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 265 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "abs(abs(f(x)-g[1](x))
)[2]^2=Int((f(x)-g[1](x))^2,x=0..Pi)" "6#/*$&-%$absG6#-F'6#,&-%\"fG6#%
\"xG\"\"\"-&%\"gG6#F06#F/!\"\"6#\"\"#F8-%$IntG6$*$,&-F-6#F/F0-&F36#F06
#F/F6F8/F/;\"\"!%#PiG" }{TEXT -1 5 "  =  " }{XPPEDIT 18 0 "Pi^3/3=10.3
3542556" "6#/*&%#PiG\"\"$F&!\"\"$\"+cDaL5!\")" }{TEXT -1 1 " " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "since the
 value of" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 27 "q := Int((f-g1)^2,x=0..Pi);" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "evalua
tes to" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 27 "value(q) = evalf(value(q));" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 
{PARA 4 "" 0 "" {TEXT -1 32 "Optimizing the Fit Heuristically" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "Fo
r our example, the measure of fit between " }{XPPEDIT 18 0 "f(x)" "6#-
%\"fG6#%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG6#%\"
xG" }{TEXT -1 9 " would be" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "q := Int((f-g)^2,x=0..Pi);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 10 "with value" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "Q := value(q);" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "a fu
nction of the two parameters " }{XPPEDIT 18 0 "s[1]" "6#&%\"sG6#\"\"\"
" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "s[2]" "6#&%\"sG6#\"\"#" }{TEXT 
-1 33 ", graphed in Figure 10.28, below." }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 279 "p1:=plot3d(Q,s1=1..
3, s2=-2..0, axes=boxed, style=patchcontour, labels=[`s1   `,`   s2`,`
Q `], labelfont=[TIMES,ITALIC,12], tickmarks=[3,3,3], scaling=constrai
ned, orientation=[-45,65]):\np2:=spacecurve([[1,-2,2.5],[1,0,2.5],[3,0
,2.5]],color=black, linestyle=2):\ndisplay([p1,p2]);" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "T
he surface determined by " }{XPPEDIT 18 0 "Q(s[1],s[2])" "6#-%\"QG6$&%
\"sG6#\"\"\"&F'6#\"\"#" }{TEXT -1 27 " has a minimum point near (" }
{XPPEDIT 18 0 "s[1],s[2]" "6$&%\"sG6#\"\"\"&F$6#\"\"#" }{TEXT -1 4 ") \+
= " }{XPPEDIT 18 0 "``(2,-1)" "6#-%!G6$\"\"#,$\"\"\"!\"\"" }{TEXT -1 
71 " which we will momentarily find by the analytic techniques of calc
ulus." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 105 
"First, however, we build the following Maple procedure which accepts \+
a guess for the pair of parameters (" }{XPPEDIT 18 0 "s[1],s[2]" "6$&%
\"sG6#\"\"\"&F$6#\"\"#" }{TEXT -1 17 ") and plots both " }{XPPEDIT 18 
0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 39 " and the resulting approximat
ing curve " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG6#%\"xG" }{TEXT -1 78 ".  I
n addition, the value of the measuring integral is computed and displa
yed." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 196 "a := proc(u,v)\nlocal G,h,p1,p2;\nG := subs(s1=u,s2=
v,g);\nh := evalf(subs(s1=u,s2=v,Q));\np1 := plot([f,G],x=0..Pi, color
=[black,red]):\np2 := textplot([1,3,convert(h,string)]):\ndisplay([p1,
p2]);\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 39 "For example, the approximating function" 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 3 "g1;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 4 "has " }{XPPEDIT 18 0 "s[1]=1" "6#/&%\"sG6#
\"\"\"F'" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "s[2]=1" "6#/&%\"sG6#\"\"
#\"\"\"" }{TEXT -1 11 ", so we try" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "a(1,1);" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "and" 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 8 "a(1,.5);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 97 "The reader is encourages to contin
ue experimenting until a minimizing point is found empirically." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" 
{TEXT -1 31 "Optimizing the Fit Analytically" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "Equating to zero the par
tial derivatives of " }{XPPEDIT 18 0 "Q(s[1],s[2])" "6#-%\"QG6$&%\"sG6
#\"\"\"&F'6#\"\"#" }{TEXT -1 23 " taken with respect to " }{XPPEDIT 
18 0 "s[1]" "6#&%\"sG6#\"\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "s[2
]" "6#&%\"sG6#\"\"#" }{TEXT -1 21 " yields the equations" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "eq1 \+
:= diff(Q,s1) = 0;\neq2 := diff(Q,s2) = 0;" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "whose solut
ion is (" }{XPPEDIT 18 0 "s[1],s[2]" "6$&%\"sG6#\"\"\"&F$6#\"\"#" }
{TEXT -1 4 ") = " }{XPPEDIT 18 0 "``(2,-1)" "6#-%!G6$\"\"#,$\"\"\"!\"
\"" }{TEXT -1 24 ", determined in Maple by" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "qq := solve(\{eq1
,eq2\},\{s1,s2\});" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "Hence, the best fit to " }
{XPPEDIT 18 0 "f(x)=x" "6#/-%\"fG6#%\"xGF'" }{TEXT -1 68 " which we ca
n obtain under this measure, using the fitting function " }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 266 "" 0 "" {XPPEDIT 18 0 "g(x)=s[1]*sin(
x)+s[2]*sin(2*x)" "6#/-%\"gG6#%\"xG,&*&&%\"sG6#\"\"\"F--%$sinG6#F'F-F-
*&&F+6#\"\"#F--F/6#*&F4F-F'F-F-F-" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "is the function" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "G
 := subs(qq,g);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "graphed (as the red curve) in Figu
re 10.29 (below)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 127 "plot([x,G],x=0..Pi, color=[black,red], lines
tyle=[2,1], xtickmarks=3, ytickmarks=3, labels=[x,``],labelfont=[TIMES
,ITALIC,12]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "for which we have " }{XPPEDIT 18 
0 "Q(2,-1)=2.418443925" "6#/-%\"QG6$\"\"#,$\"\"\"!\"\"$\"+DRW=C!\"*" }
{TEXT -1 18 ", as determined by" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "a(2,-1);" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 21 "B
est Fit, Generalized" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 37 "We next seek, for a general function " }
{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 17 " on the interval \+
" }{XPPEDIT 18 0 "[0,Pi]" "6#7$\"\"!%#PiG" }{TEXT -1 61 ", the best ap
proximating trigonometric polynomial of the form" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 267 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "g(x)=
Sum(s[k]*sin(k*x),k=1..infinity)" "6#/-%\"gG6#%\"xG-%$SumG6$*&&%\"sG6#
%\"kG\"\"\"-%$sinG6#*&F/F0F'F0F0/F/;F0%)infinityG" }{TEXT -1 1 " " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "To minimi
ze the measure of performance " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 268 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Q=Int((f(x)-sum(s[k]*
sin(k*x),k=1..infinity))^2,x=0..Pi)" "6#/%\"QG-%$IntG6$*$,&-%\"fG6#%\"
xG\"\"\"-%$sumG6$*&&%\"sG6#%\"kGF.-%$sinG6#*&F6F.F-F.F./F6;F.%)infinit
yG!\"\"\"\"#/F-;\"\"!%#PiG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "solve the " }{TEXT 258 16 "norm
al equations" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 269 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "1/2" "6#*&\"\"\"F$\"\"#!\"\"" }{TEXT 
-1 1 " " }{XPPEDIT 18 0 "Q[s[j]]=Int((f(x)-sum(s[k]*sin(k*x),k=1..infi
nity))*sin(j*x),x=0..Pi)" "6#/&%\"QG6#&%\"sG6#%\"jG-%$IntG6$*&,&-%\"fG
6#%\"xG\"\"\"-%$sumG6$*&&F(6#%\"kGF4-%$sinG6#*&F;F4F3F4F4/F;;F4%)infin
ityG!\"\"F4-F=6#*&F*F4F3F4F4/F3;\"\"!%#PiG" }{TEXT -1 8 "  =  0, " }
{XPPEDIT 18 0 "j=1,`...`" "6$/%\"jG\"\"\"%$...G" }{TEXT -1 1 " " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "for " }
{XPPEDIT 18 0 "s[k],k=1,`...`" "6%&%\"sG6#%\"kG/F&\"\"\"%$...G" }
{TEXT -1 70 ".  Because the integrals of the \"cross terms\" vanish, t
hat is, because" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 270 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(sin(n*x)*sin(m*x),x=0..Pi)=0" "6#/-
%$IntG6$*&-%$sinG6#*&%\"nG\"\"\"%\"xGF-F--F)6#*&%\"mGF-F.F-F-/F.;\"\"!
%#PiGF5" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "n<>m" "6#0%\"nG%\"mG" }
{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 29 "these equations take the form" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 271 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(f(x)*si
n(k*x),x=0..Pi)=s[k]*Int(sin^2*``(k*x),x=0..Pi)" "6#/-%$IntG6$*&-%\"fG
6#%\"xG\"\"\"-%$sinG6#*&%\"kGF,F+F,F,/F+;\"\"!%#PiG*&&%\"sG6#F1F,-F%6$
*&F.\"\"#-%!G6#*&F1F,F+F,F,/F+;F4F5F," }{TEXT -1 5 "  =  " }{XPPEDIT 
18 0 "Pi/2" "6#*&%#PiG\"\"\"\"\"#!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 
0 "s[k]" "6#&%\"sG6#%\"kG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "from which we determine" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 272 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 
18 0 "s[k]=2/Pi" "6#/&%\"sG6#%\"kG*&\"\"#\"\"\"%#PiG!\"\"" }{TEXT -1 
1 " " }{XPPEDIT 18 0 "Int(f(x)*sin(k*x),x=0..Pi)" "6#-%$IntG6$*&-%\"fG
6#%\"xG\"\"\"-%$sinG6#*&%\"kGF+F*F+F+/F*;\"\"!%#PiG" }{TEXT -1 1 " " }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "These va
lues for the " }{XPPEDIT 18 0 "s[k]" "6#&%\"sG6#%\"kG" }{TEXT -1 63 " \+
are precisely the coefficients of the Fourier sine series for " }
{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 17 " on the interval \+
" }{XPPEDIT 18 0 "[0,Pi]" "6#7$\"\"!%#PiG" }{TEXT -1 65 ".  The Fourie
r series is the best trigonometric approximation to " }{XPPEDIT 18 0 "
f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 75 ", provided the measure used to d
etermine \"best\" is convergence in the mean." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 124 "Insight into the calcula
tions on which this derivation is based can be obtained by working wit
h the finite approximating sum" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "g := add(s[k]*sin(k*x),k=1..
5);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 10 "Restoring " }{XPPEDIT 18 0 "f" "6#%\"fG" }{TEXT 
-1 37 " to the status of a free variable via" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "f:='f';" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 38 "we write the measure of performance as" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Q =
 Int((f(x)-Sum(s[k]*sin(k*x),k=1..5))^2,x=0..Pi)" "6#/%\"QG-%$IntG6$*$
,&-%\"fG6#%\"xG\"\"\"-%$SumG6$*&&%\"sG6#%\"kGF.-%$sinG6#*&F6F.F-F.F./F
6;F.\"\"&!\"\"\"\"#/F-;\"\"!%#PiG" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 26 "and enter into Maple as it" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "Q := I
nt((f(x)-g)^2,x=0..Pi);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 76 "To minimize this measure of p
erformance, differentiate with respect to each " }{XPPEDIT 18 0 "s[k]
" "6#&%\"sG6#%\"kG" }{TEXT -1 93 ", setting each derivative to zero.  \+
The general term for such derivatives will be of the form" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 
0 "Int(-2*(f(x)-s[1]*sin(x)-s[2]*sin(2*x)-s[3]*sin(3*x)-s[4]*sin(4*x)-
s[5]*sin(5*x))*sin(k*x),x = 0 .. Pi)=0" "6#/-%$IntG6$,$*(\"\"#\"\"\",.
-%\"fG6#%\"xGF**&&%\"sG6#F*F*-%$sinG6#F/F*!\"\"*&&F26#F)F*-F56#*&F)F*F
/F*F*F7*&&F26#\"\"$F*-F56#*&FAF*F/F*F*F7*&&F26#\"\"%F*-F56#*&FHF*F/F*F
*F7*&&F26#\"\"&F*-F56#*&FOF*F/F*F*F7F*-F56#*&%\"kGF*F/F*F*F7/F/;\"\"!%
#PiGFY" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 76 
"If the integrations are carried out, the surviving terms will be of t
he form" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 261 "" 0 "" {TEXT -1 
1 " " }{XPPEDIT 18 0 "-2*Int(f(x)*sin(k*x),x=0..Pi)+2*s[k]*Int(sin^2*k
*x,x=0..Pi)=0" "6#/,&*&\"\"#\"\"\"-%$IntG6$*&-%\"fG6#%\"xGF'-%$sinG6#*
&%\"kGF'F/F'F'/F/;\"\"!%#PiGF'!\"\"*(F&F'&%\"sG6#F4F'-F)6$*(F1F&F4F'F/
F'/F/;F7F8F'F'F7" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 13 "because, for " }{XPPEDIT 18 0 "n<>m" "6#0%\"nG%\"mG" }
{TEXT -1 23 ", integrals of the form" }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{PARA 262 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(sin(n*x)*sin(m*
x),x=0..Pi)" "6#-%$IntG6$*&-%$sinG6#*&%\"nG\"\"\"%\"xGF,F,-F(6#*&%\"mG
F,F-F,F,/F-;\"\"!%#PiG" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 37 "all vanish.  Indeed, in Maple we have" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 122 "for n
 from 1 to 5 do\nfor m from 1 to n-1 do\nprint(Int(sin(n*x)*sin(m*x),x
=0..Pi) = int(sin(n*x)*sin(m*x),x=0..Pi));\nod;od;" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "On \+
the other hand, integrals of the form  " }{XPPEDIT 18 0 "Int(sin^2*k*x
,x=0..Pi)" "6#-%$IntG6$*(%$sinG\"\"#%\"kG\"\"\"%\"xGF*/F+;\"\"!%#PiG" 
}{TEXT -1 45 " are not zero.  In fact, using Maple, we find" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "f
or k from 1 to 5 do\nInt(sin(k*x)^2,x=0..Pi) = int(sin(k*x)^2,x=0..Pi)
;\nod;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 177 "Incidentally, more than thirty years ago, Tom \+
Hack, a fellow graduate student at Purdue University, pointed out that
 one could remember the value of such integrals by observing " }
{XPPEDIT 18 0 "Int(sin^2*x+cos^2*x,x=0..Pi)=Int(1,x=0..Pi)" "6#/-%$Int
G6$,&*&%$sinG\"\"#%\"xG\"\"\"F,*&%$cosGF*F+F,F,/F+;\"\"!%#PiG-F%6$F,/F
+;F1F2" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "Pi" "6#%#PiG" }{TEXT -1 166 
", and apportioning \"half\" the contribution to the sine term and \"h
alf\" to the cosine term.  (Recounting this to students since, yields \+
laughter, but correct answers!)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 84 "Thus, the equations obtained by different
iating can be evaluated to the forms below." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "for k from 1 to 5
 do\nq||k := value(expand(diff(Q,s[k]),sin)) = 0;\nod;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 174 "Solving each equation yields one coefficient each because the \+
integrations eliminate all \"cross terms.\"  The vanishing of such cro
ss terms is special to the set of functions " }{XPPEDIT 18 0 "sin(k*x)
" "6#-%$sinG6#*&%\"kG\"\"\"%\"xGF(" }{TEXT -1 71 ", and is the reason \+
why such functions are used to make Fourier series." }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "for k fro
m 1 to 5 do\nisolate(q||k,s[k]);\nod;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "These values \+
for the " }{XPPEDIT 18 0 "s[k]" "6#&%\"sG6#%\"kG" }{TEXT -1 69 " are p
recisely the Fourier sine series coefficients for the interval " }
{XPPEDIT 18 0 "[0,Pi]" "6#7$\"\"!%#PiG" }{TEXT -1 65 ".  The Fourier s
eries is the best trigonometric approximation to " }{XPPEDIT 18 0 "f(x
)" "6#-%\"fG6#%\"xG" }{TEXT -1 75 ", provided the measure used to dete
rmine \"best\" is convergence in the mean." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 528 "There are other functions with
 the same minimizing property, the essence of which is the vanishing o
f the integrals of the \"cross terms.\"  This property is called \"ort
hogonality\" in the literature, and is the basis for generalizations o
f the Fourier series.  In the next lesson, we will obtain the Fourier-
Legendre series, and similar techniques could be used to obtain series
 of Bessel functions, Tchebychev polynomials, Laguerre polynomials, He
rmite polynomials, and many other of the special functions of applied \+
mathematics." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 
{PARA 3 "" 0 "" {TEXT -1 26 "Orthogonality of Functions" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 84 "A key step in
 obtaining the Fourier coefficients by minimizing the mean-square norm
 " }{XPPEDIT 18 0 "Q" "6#%\"QG" }{TEXT -1 53 " is the vanishing of the
 \"cross-terms,\" the integrals" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 273 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(sin(n*x)*sin(m*x)
,x=0..Pi),n<>m" "6$-%$IntG6$*&-%$sinG6#*&%\"nG\"\"\"%\"xGF,F,-F(6#*&%
\"mGF,F-F,F,/F-;\"\"!%#PiG0F+F1" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 69 "This leads to a general d
efinition of the orthogonality of functions." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 
{PARA 4 "" 0 "" {TEXT -1 15 "Definition 10.1" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "Two functions " }
{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 
18 0 "g(x)" "6#-%\"gG6#%\"xG" }{TEXT -1 43 " are said to be orthogonal
 on the interval " }{XPPEDIT 18 0 "[a,b]" "6#7$%\"aG%\"bG" }{TEXT -1 
4 " if " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 263 "" 0 "" {TEXT -1 
1 " " }{XPPEDIT 18 0 "Int(f(x)*g(x),x=a..b)=0" "6#/-%$IntG6$*&-%\"fG6#
%\"xG\"\"\"-%\"gG6#F+F,/F+;%\"aG%\"bG\"\"!" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 511 "There are functions other than
 just sines and cosines with the same orthogonal, \"minimizing\" prope
rty, the basis for generalizations of the Fourier series.  In Section \+
10.7, we will use orthogonality to obtain the Fourier-Legendre series,
 and in Section 16.2 we will again use orthogonality to obtain the Fou
rier-Bessel series.  Similar techniques could be used to obtain series
 of Chebyshev polynomials, Laguerre polynomials, Hermite polynomials, \+
and many other of the special functions of applied mathematics." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 37 "Additional Remark abou
t Orthogonality" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 351 "The student is cautioned against a search for a \"g
eometric\" orthogonality in this definition.  Functions orthogonal und
er this definition will not \"look\" perpendicular in any way.  The de
finition is merely a generalization of the notion of orthogonality of \+
vectors under the usual dot product, as learned in the standard multiv
ariable calculus course." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 13 "The functions" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "f := sin(x);\ng := sin(3*x);
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 73 "have been shown, in the previous lesson, to be ortho
gonal on the interval" }{XPPEDIT 18 0 "[0,Pi]" "6#7$\"\"!%#PiG" }
{TEXT -1 10 ".  Indeed," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "Int(f*g,x=0..Pi) = int(f*g,x=0..Pi)
;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 70 "The functions don't \"look\" orthogonal, as seen in \+
the following graph." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 71 "plot([f,g],x=0..Pi, color=black, linestyl
e=[1,2], scaling=constrained);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 326 "Lest the student misi
nterpret the figure and mistakenly conclude that the two \"orthogonal
\" functions cross at right angles, we compute the points of intersect
ion, and compute the slopes of the curves at one of the intersections.
  If the curves were geometrically orthogonal at this point, the produ
ct of their slopes would be " }{XPPEDIT 18 0 "-1" "6#,$\"\"\"!\"\"" }
{TEXT -1 9 ".  Hence," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "solve(f=g,x);" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "and" }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 43 "simplify(subs(x=Pi/4,diff(f,x)*diff(g,x)));" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 267 "Cle
arly, these orthogonal curves do not cross at right angles.  Moreover,
 orthogonality of the curves in an interval property.  If the interval
 were changed, the curves might no longer be orthogonal under this def
inition.  For instance, if the interval be changed to " }{XPPEDIT 18 
0 "[0,Pi/4]" "6#7$\"\"!*&%#PiG\"\"\"\"\"%!\"\"" }{TEXT -1 15 ", we wou
ld have" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 40 "Int(f*g,x=0..Pi/4) = int(f*g,x=0..Pi/4);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 66 "and the same functions would no longer be considered orthogonal
.  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}}{MARK "1" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }
{PAGENUMBERS 0 1 2 33 1 1 }