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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 40 "Partial Differential Equa
tions PowerTool" }}{PARA 19 "" 0 "" {TEXT -1 16 "by Dr. Jim Herod" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 24 "Section \+
1.5: Projections" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 3 "
" 0 "" {TEXT 259 30 "Maple Packages for Section 1.5" }}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{PARA 0 "" 0 "" {TEXT -1 282 "By now, we have become familiar with t
he tools needed to discuss Fourier Series. They are the tools of Linea
r Algebra. We want a linear space E of vectors. These vectors may them
selves be functions on an interval. We want there to be defined on thi
s linear space E a dot product  < " }{TEXT 258 1 "x" }{TEXT -1 2 ", " 
}{TEXT 257 1 "y" }{TEXT -1 277 ">. With these tools - a linear space a
nd a dot product - we can discuss orthogonality and best approximation
s. There are two more topics that are useful concepts in thinking of F
ourier Series. We deal with these last two topics from Linear Algebra \+
in this section and the next." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 41 "In this section, we consider projections \+
" }{TEXT 281 1 "P" }{TEXT -1 78 ". A projection on a linear space is a
 linear function having the property that" }}{PARA 0 "" 0 "" {TEXT -1 
32 "                                " }{XPPEDIT 18 0 "P^2;" "6#*$%\"PG
\"\"#" }{TEXT -1 3 " = " }{TEXT 260 2 "P." }}{PARA 0 "" 0 "" {TEXT -1 
123 "We make closest point projections in this Section. To do this, su
ppose we have an orthogonal sequence of elements. We take " }{TEXT 
261 1 "O" }{TEXT -1 52 " to be a finite subset of these orthogonal ele
ments:" }}{PARA 0 "" 0 "" {TEXT -1 30 "                              \+
" }{TEXT 262 2 "O " }{TEXT -1 4 "= \{ " }{XPPEDIT 18 0 "theta[1],theta
[2],theta[3];" "6%&%&thetaG6#\"\"\"&F$6#\"\"#&F$6#\"\"$" }{TEXT -1 8 "
, ... , " }{XPPEDIT 18 0 "theta[n];" "6#&%&thetaG6#%\"nG" }{TEXT -1 3 
" \}." }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{TEXT 263 1 "S" }{TEXT -1 
20 " be the span of the " }{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT 
-1 12 "'s. That is," }}{PARA 0 "" 0 "" {TEXT -1 19 "                  \+
 " }{TEXT 264 1 "S" }{TEXT -1 13 " = \{ u : u = " }{XPPEDIT 18 0 "sum(
a[p]*theta[p],p);" "6#-%$sumG6$*&&%\"aG6#%\"pG\"\"\"&%&thetaG6#F*F+F*
" }{TEXT -1 3 " , " }{XPPEDIT 18 0 "a[p];" "6#&%\"aG6#%\"pG" }{TEXT 
-1 16 " a real number\}." }}{PARA 0 "" 0 "" {TEXT -1 21 "We have seen \+
that if " }{TEXT 268 1 "f" }{TEXT -1 14 " is in E, then" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "                    \+
    " }{XPPEDIT 18 0 "sum(`<`*f*`,`*theta[p]*`>`/(abs(theta[p])^2)*the
ta[p],p = 1 .. n);" "6#-%$sumG6$*0%\"<G\"\"\"%\"fGF(%\",GF(&%&thetaG6#
%\"pGF(%\">GF(*$-%$absG6#&F,6#F.\"\"#!\"\"&F,6#F.F(/F.;F(%\"nG" }
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 276 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 6 "is in " }{TEXT 265 1 "S" }{TEXT -1 18 " and is c
loser to " }{TEXT 267 1 "f" }{TEXT -1 27 " than any other element in \+
" }{TEXT 266 1 "S" }{TEXT -1 58 ". Choosing this closest point defines
 a projection, for if" }}{PARA 0 "" 0 "" {TEXT -1 17 "                \+
 " }{TEXT 269 4 "P(f)" }{TEXT -1 5 " =   " }{XPPEDIT 18 0 "sum(`<`*f*`
,`*theta[p]*`>`/(abs(theta[p])^2)*theta[p],p = 1 .. n);" "6#-%$sumG6$*
0%\"<G\"\"\"%\"fGF(%\",GF(&%&thetaG6#%\"pGF(%\">GF(*$-%$absG6#&F,6#F.
\"\"#!\"\"&F,6#F.F(/F.;F(%\"nG" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 6 "then  " }{XPPEDIT 18 0 "P^2 = P;" "6#/*$%\"PG\"\"#F%" }
{TEXT -1 16 ". That is, take " }{TEXT 270 1 "f" }{TEXT -1 32 ". Form t
he Fourier Expansion of " }{TEXT 271 2 "f " }{TEXT -1 32 "using the or
thogonal vectors in " }{TEXT 272 1 "O" }{TEXT -1 12 ". Call this " }
{TEXT 273 4 "P(f)" }{TEXT -1 32 ". Form the Fourier expansion of " }
{TEXT 274 4 "P(f)" }{TEXT -1 18 ". What you get is " }{TEXT 275 4 "P(f
)" }{TEXT -1 8 ", again." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 37 "It is interesting to take a plane in " }{XPPEDIT 
18 0 "R^3;" "6#*$%\"RG\"\"$" }{TEXT -1 386 " and to form the closest p
oint projection onto the plane. This is properly a problem for a linea
r algebra set of lecture notes, and we resist the temptation to do thi
s here. What we would do to do that problem is to find two linearly in
dependent vectors in the plane, perform the Gramm-Schmidt process on t
hose two vectors, and construct the closest point projection as sugges
ted above." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 89 "Instead of doing that interesting problem, we take the space C(
[0, 1]), and the subspace " }{TEXT 277 1 "S" }{TEXT -1 19 " to be the \+
span of " }}{PARA 0 "" 0 "" {TEXT -1 19 "             \{ sin(" }
{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 11 " x), sin(2 " }{XPPEDIT 18 
0 "Pi;" "6#%#PiG" }{TEXT -1 11 " x), sin(3 " }{XPPEDIT 18 0 "Pi;" "6#%
#PiG" }{TEXT -1 11 " x), sin(4 " }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }
{TEXT -1 11 " x), sin(5 " }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 11 
" x), sin(6 " }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 5 " x)\}." }}
{PARA 0 "" 0 "" {TEXT -1 16 "The vectors sin(" }{XPPEDIT 18 0 "Pi;" "6
#%#PiG" }{TEXT -1 11 " x), sin(2 " }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }
{TEXT -1 11 " x), sin(3 " }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 11 
" x), sin(4 " }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 11 " x), sin(5 \+
" }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 14 " x),and sin(6 " }
{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 95 " x)  are already orthogona
l in that subspace. Now, we can make the closest point projection on \+
" }{TEXT 278 1 "S" }{TEXT -1 73 ". We illustrate by getting the closes
t point projection onto the function" }}{PARA 0 "" 0 "" {TEXT -1 48 " \+
                       f(x) = Heaviside(x-1/2)." }}{PARA 0 "" 0 "" 
{TEXT -1 40 "First, we draw a graph of this function." }}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 23 "f:=x->Heaviside(x-1/2);" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 18 "plot(f(x),x=0..1);" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 42 "Next, we comp
ute the Fourier Coefficients." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 89 "for n from 1 to 6 do\n   a[n]:=int(f(x)*sin(n*Pi*x),x=0..1)/int(
sin(n*Pi*x)^2,x=0..1);\nod;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
0 "" }}}{PARA 0 "" 0 "" {TEXT -1 56 "Finally, we form the Fourier Sum \+
as the projection onto " }{TEXT 280 1 "S" }{TEXT -1 15 " and plot both
 " }{TEXT 279 1 "f" }{TEXT -1 41 " and its approximation on the same g
raph." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "approx:=x->sum(a[p]
*sin(p*Pi*x),p=1..6);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "pl
ot([f(x),approx(x)],x=0..1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 148 "Not so bad. Should it be that \+
you think the approximation is not so good, go back and take more term
s. Change the 6 terms to 10, or 20, or whatever." }}{PARA 0 "" 0 "" 
{TEXT -1 69 "Before ending this discussion, there is one check we shou
ld make. Is " }{XPPEDIT 18 0 "P^2 = P;" "6#/*$%\"PG\"\"#F%" }{TEXT -1 
88 "? That is, are the Fourier coefficients for the approximation the \+
same as they were for " }{TEXT 282 1 "f" }{TEXT -1 1 "?" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "for n from 1 to 6 do\n   b[n]:=int
(approx(x)*sin(n*Pi*x),x=0..1)/\n                 int(sin(n*Pi*x)^2,x=
0..1);\nod;" }}}{PARA 0 "" 0 "" {TEXT -1 42 "We compare to see if the \+
coefficients for " }{TEXT 283 1 "f" }{TEXT -1 40 " and for its approxi
mation are the same." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "for \+
n from 1 to 6 do\n   is(b[n]=a[n]);\nod;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{PARA 0 "" 0 "" {TEXT -1 24 "The implication is that " }{XPPEDIT 18 0 
"P^2 = P;" "6#/*$%\"PG\"\"#F%" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 150 "In this Section, we have
 shown that the closest point approximation is a projection of the ent
ire space onto the span of the orthogonal elements used." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "EMAIL: herod@math.
gatech.edu   or    jherod@tds.net" }}{PARA 0 "" 0 "" {TEXT -1 38 "URL:
 http://www.math.gatech.edu/~herod" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 256 "" 0 "" {TEXT -1 36 "Copyright \251  2003  by James V. Herod
" }}{PARA 256 "" 0 "" {TEXT -1 19 "All rights reserved" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 
1 }{PAGENUMBERS 1 1 2 33 1 1 }