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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 38 "Section 10.7:  Fourier-Legendre series" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 9 "Copyright" }}
{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 th
is publication may be reproduced, stored in a retrieval system, or tra
nsmitted, in any form or by any means, electronic, mechanical, photoco
pying, recording, or otherwise, without the prior written permission o
f the publisher. Printed in the United States of America." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 15 "Initializ
ations" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 
"with(student):\nwith(orthopoly):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 30 "Orthogonali
ty and Minimization" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 232 "We wish to show that, in general, orthogonal fun
ctions lead to series with the minimization property of Fourier series
.  To do this, we carry out the following experiment in a arena less f
amiliar that that of the trig functions [1]." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "On the interval " }{XPPEDIT 18 
0 "[-1,1]" "6#7$,$\"\"\"!\"\"F%" }{TEXT -1 32 ", consider the set of f
unctions " }{XPPEDIT 18 0 "p[k](x),k=0,1,`...`" "6&-&%\"pG6#%\"kG6#%\"
xG/F'\"\"!\"\"\"%$...G" }{TEXT -1 26 ", with the two properties:" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "1)       \+
" }{XPPEDIT 18 0 "Int(p[n](x)*p[m](x),x=-1..1)=0,n<>m" "6$/-%$IntG6$*&
-&%\"pG6#%\"nG6#%\"xG\"\"\"-&F*6#%\"mG6#F.F//F.;,$F/!\"\"F/\"\"!0F,F3
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "2)    \+
  " }{XPPEDIT 18 0 "Int(p[k](x)^2,x=-1..1)=2/(2*k+1)" "6#/-%$IntG6$*$-
&%\"pG6#%\"kG6#%\"xG\"\"#/F.;,$\"\"\"!\"\"F3*&F/F3,&*&F/F3F,F3F3F3F3F4
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 69 "The f
irst property is orthogonality of the functions on the interval " }
{XPPEDIT 18 0 "[-1,1]" "6#7$,$\"\"\"!\"\"F%" }{TEXT -1 34 ", and the s
econd is the analog of " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 261 "
" 0 "" {XPPEDIT 18 0 "Int(sin^2*k*x,x=0..Pi)=Pi/2" "6#/-%$IntG6$*(%$si
nG\"\"#%\"kG\"\"\"%\"xGF+/F,;\"\"!%#PiG*&F0F+F)!\"\"" }{TEXT -1 1 " " 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 143 "alread
y used when developing the Fourier series.  Are these two properties e
nough to reproduce the minimization property of the Fourier series?" }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "To tell \+
Maple that the functions " }{XPPEDIT 18 0 "p[0](x),p[1](x),p[2](x)" "6
%-&%\"pG6#\"\"!6#%\"xG-&F%6#\"\"\"6#F)-&F%6#\"\"#6#F)" }{TEXT -1 70 " \+
have these properties, embed Property (1) in the following equations.
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 104 "q01 := Int(p0(x)*p1(x),x=-1..1)=0;\nq02 := Int(p0(x)
*p2(x),x=-1..1)=0;\nq12 := Int(p1(x)*p2(x),x=-1..1)=0;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 40 "Then embed Property (2) in the equations" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "q00 := Int(
p0(x)^2,x=-1..1)=2;\nq11 := Int(p1(x)^2,x=-1..1)=2/3;\nq22 := Int(p2(x
)^2,x=-1..1)=2/5;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "The measure of performance," }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 262 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "Q=Int((f(x)-Sum(s[k]*p[k](x),k=0..N))^2,x=-1..1)" "6#/%
\"QG-%$IntG6$*$,&-%\"fG6#%\"xG\"\"\"-%$SumG6$*&&%\"sG6#%\"kGF.-&%\"pG6
#F66#F-F./F6;\"\"!%\"NG!\"\"\"\"#/F-;,$F.F@F." }{TEXT -1 1 " " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "is the sa
me as we used for the Fourier series." }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 26 "In Maple, we write this as" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "Q
 := Int((f(x)-s0*p0(x)-s1*p1(x)-s2*p2(x))^2,x=-1..1);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 34 "Again forming the normal equations" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 263 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "0=1/2" "6#/\"
\"!*&\"\"\"F&\"\"#!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Q[s[j]]=Int((
f(x)-Sum(s[k]*p[k](x),k=1..N))*p[j](x),x=-1..1)" "6#/&%\"QG6#&%\"sG6#%
\"jG-%$IntG6$*&,&-%\"fG6#%\"xG\"\"\"-%$SumG6$*&&F(6#%\"kGF4-&%\"pG6#F;
6#F3F4/F;;F4%\"NG!\"\"F4-&F>6#F*6#F3F4/F3;,$F4FDF4" }{TEXT -1 1 " " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 "we find, \+
using Properties (1) and (2), that" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 264 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(f(x)*p[j](x),x=-1
..1)=s[j]*Int(p[j](x)^2,x=-1..1)" "6#/-%$IntG6$*&-%\"fG6#%\"xG\"\"\"-&
%\"pG6#%\"jG6#F+F,/F+;,$F,!\"\"F,*&&%\"sG6#F1F,-F%6$*$-&F/6#F16#F+\"\"
#/F+;,$F,F6F,F," }{TEXT -1 5 "  =  " }{XPPEDIT 18 0 "s[j]" "6#&%\"sG6#
%\"jG" }{TEXT -1 1 " " }{XPPEDIT 18 0 "2/(2*j+1)" "6#*&\"\"#\"\"\",&*&
F$F%%\"jGF%F%F%F%!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 6 "Hence," }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 265 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "s[j]=(2*j+1)/2" "
6#/&%\"sG6#%\"jG*&,&*&\"\"#\"\"\"F'F,F,F,F,F,F+!\"\"" }{TEXT -1 1 " " 
}{XPPEDIT 18 0 "Int(f(x)*p[j](x),x=-1..1)" "6#-%$IntG6$*&-%\"fG6#%\"xG
\"\"\"-&%\"pG6#%\"jG6#F*F+/F*;,$F+!\"\"F+" }{TEXT -1 1 " " }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "These calculations
 are implemented in Maple for the case " }{XPPEDIT 18 0 "N=2" "6#/%\"N
G\"\"#" }{TEXT -1 12 " as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 222 "Differentiate the measure of performance
 with respect to each of the three coefficients s0, s1, s2.  Set the d
erivatives equal to zero to determine the values of the coefficients w
hich minimize Q, the measure of deviation." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "eq1 := diff(Q,s0)
 = 0;\neq2 := diff(Q,s1) = 0;\neq3 := diff(Q,s2) = 0;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 146 "Progress solving these equations depends on simplifying them. \+
 The parentheses need to be multiplied out, and any possible integrati
ons performed." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 59 "eq4 := expand(eq1);\neq5 := expand(eq2);\neq6 \+
:= expand(eq3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 160 "No integrations have been done be
cause everything is symbolic.  Maple knows only properties (1) and (2)
 apply, but can't apply them until told.  Do this with a " }{TEXT 257 
8 "simplify" }{TEXT -1 76 " command containing the equations that defi
ne Property (1) and Property (2)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 140 "q0 := simplify(eq4,\{q00,q1
1,q22,q01,q02,q12\});\nq1 := simplify(eq5,\{q00,q11,q22,q01,q02,q12\})
;\nq2 := simplify(eq6,\{q00,q11,q22,q01,q02,q12\});" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 58 "S
olve each equation for the one coefficient it contains.  " }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "isol
ate(q0,s0);\nisolate(q1,s1);\nisolate(q2,s2);" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "Generali
ze these definitions to a formula for the " }{TEXT 256 1 "n" }{TEXT 
-1 15 "th coefficient " }{XPPEDIT 18 0 "s[n]" "6#&%\"sG6#%\"nG" }
{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "s[n]=[(2*n+1)/2]*Int(f(x)*p[n](x),x=-1.
.1)" "6#/&%\"sG6#%\"nG*&7#*&,&*&\"\"#\"\"\"F'F.F.F.F.F.F-!\"\"F.-%$Int
G6$*&-%\"fG6#%\"xGF.-&%\"pG6#F'6#F7F./F7;,$F.F/F.F." }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 20 
"Legendre Polynomials" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 121 "Do such functions exist?  Are there actually f
unctions which satisfy Properties (1) and (2)?  Yes, Legendre polynomi
als  " }{XPPEDIT 18 0 "p[n](x)" "6#-&%\"pG6#%\"nG6#%\"xG" }{TEXT -1 
55 ", the first five of which, graphed in Figure 10.30, are" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "f
or k from 0 to 4 do\np||k := P(k,x);\nod;" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "The Legendr
e polynomials are found in Maple's " }{TEXT 258 9 "orthopoly" }{TEXT 
-1 9 " package." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 55 "Figure 10.30 is therefore the following graph in which " 
}{XPPEDIT 18 0 "p[k](x),k=0,`...`,4" "6&-&%\"pG6#%\"kG6#%\"xG/F'\"\"!%
$...G\"\"%" }{TEXT -1 77 ", appear with the colors red, black, green, \+
blue, and magenta, in that order." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 142 "plot([p||(0..4)], x=-1..1, \+
color=[red,black,green,blue,magenta], xtickmarks=3, ytickmarks=3, labe
ls=[`    x`,``],labelfont=[TIMES,ITALIC,12]);" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 67 "As seen \+
from the graph, the normalization scheme used by Maple has " }
{XPPEDIT 18 0 "p[n](1)=1" "6#/-&%\"pG6#%\"nG6#\"\"\"F*" }{TEXT -1 30 "
 for each Legendre polynomial " }{XPPEDIT 18 0 "p[n](x)" "6#-&%\"pG6#%
\"nG6#%\"xG" }{TEXT -1 120 ".  This is not the only normalization foun
d in the literature, and the reader is advised to read any new text ca
refully." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
27 "Next, consider the function" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 266 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x)=sin(Pi*x)" "6#/-
%\"fG6#%\"xG-%$sinG6#*&%#PiG\"\"\"F'F-" }{TEXT -1 1 " " }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "on the interval [-1,
1], and compute the coefficients" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 267 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "s[k]=(2*k+1)/2" "6#/&
%\"sG6#%\"kG*&,&*&\"\"#\"\"\"F'F,F,F,F,F,F+!\"\"" }{TEXT -1 1 " " }
{XPPEDIT 18 0 "Int(sin(k*x)*p[k](x),x=-1..1),k=0,`...`,4" "6&-%$IntG6$
*&-%$sinG6#*&%\"kG\"\"\"%\"xGF,F,-&%\"pG6#F+6#F-F,/F-;,$F,!\"\"F,/F+\"
\"!%$...G\"\"%" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 9 "obtaining" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "for k from 0 to 4 do\nS
[k]:=(k+1/2)*int(sin(Pi*x)*p||k,x=-1..1);\nod;" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "so that
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 27 "g := add(S[n]*p||n,n=0..4);" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 96 "plotted \+
as the solid (red) curve in Figure 10.31, below, is a Fourier-Legendre
 approximation to " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 
38 ", plotted as the dashed (black) curve." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 133 "plot([sin(Pi*x),
g],x=-1..1,linestyle=[2,1],color=[black,red], xtickmarks=3, ytickmarks
=3, labels=[x,``],labelfont=[TIMES,ITALIC,12]);" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "The ap
proximation " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG6#%\"xG" }{TEXT -1 53 " i
s a partial sum of the full Fourier-Legendre series" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x
)=Sum(s[k]*p[k](x),k=0..infinity)" "6#/-%\"fG6#%\"xG-%$SumG6$*&&%\"sG6
#%\"kG\"\"\"-&%\"pG6#F/6#F'F0/F/;\"\"!%)infinityG" }}{PARA 0 "" 0 "" 
{TEXT -1 5 "where" }}{PARA 260 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "
s[k]=(2*k+1)/2" "6#/&%\"sG6#%\"kG*&,&*&\"\"#\"\"\"F'F,F,F,F,F,F+!\"\"
" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(f(x)*p[k](x),x=-1..1)" "6#-%$Int
G6$*&-%\"fG6#%\"xG\"\"\"-&%\"pG6#%\"kG6#F*F+/F*;,$F+!\"\"F+" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "Likewise, " }
{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 125 " can be represen
ted as infinite series in other prthogonal families such as the Hermit
e, Laguerre, or Chebyshev  polynomials." }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "
" 0 "" {TEXT -1 10 "References" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 80 "1.  Robert J. Lopez, Tips for Mapl
e Instructors, MapleTech, VOL. 4, NO. 3, 1997." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 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 }