{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 261 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "Script MT Bold" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "Script MT Bold" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 268 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 271 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 272 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 273 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 275 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 276 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 277 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 278 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 279 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 280 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 281 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 282 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 283 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 284 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 285 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 286 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 287 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 288 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 289 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 290 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 291 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 292 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 295 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 296 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 297 "Script MT B old" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 298 "Script MT Bo ld" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 299 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 300 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 301 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE " " -1 302 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times " 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 } {PSTYLE "Title" -1 18 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 1 2 2 2 1 1 1 1 }3 1 0 0 12 12 1 0 1 0 2 2 19 1 }{PSTYLE "Author" -1 19 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 8 8 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {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 278 32 "Section \+ 1.3: Orthogonal Families" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 3 "" 0 "" {TEXT 261 30 "Maple Packages for Section 1.3" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 20 "with(LinearAlgebra):" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 17 "with(RealDomain):" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 500 "The notion that vectors, or functions, can be orthogon al is one of the important ideas in this set of notes. We recalled wha t orthogonality means in the previous Section 1.2. There, we connected the idea with perpendicularity through the dot product. This was the \+ connection: two vectors are perpendicular if and only if the cosine of the angle between them is zero. Relating this to the dot product, we \+ have that vectors are orthogonal if and only if their dot product is z ero. We make this precise." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 258 11 "Definition:" }{TEXT -1 29 " The collection of ve ctors \{ " }{XPPEDIT 18 0 "theta[1];" "6#&%&thetaG6#\"\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "theta[2];" "6#&%&thetaG6#\"\"#" }{TEXT -1 2 ", \+ " }{XPPEDIT 18 0 "theta[3];" "6#&%&thetaG6#\"\"$" }{TEXT -1 21 ", ... \+ \} is called an " }{TEXT 259 17 "orthogonal family" }{TEXT -1 4 " if \+ " }{TEXT 256 1 "<" }{TEXT -1 1 " " }{XPPEDIT 18 0 "theta[i];" "6#&%&th etaG6#%\"iG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "theta[j];" "6#&%&thetaG6 #%\"jG" }{TEXT -1 1 " " }{TEXT 257 2 "> " }{TEXT -1 13 "= 0 whenever \+ " }{XPPEDIT 18 0 "i <> j;" "6#0%\"iG%\"jG" }{TEXT -1 17 ". The family \+ is o" }{TEXT 260 10 "rthonormal" }{TEXT -1 11 " if each | " }{XPPEDIT 18 0 "theta[n];" "6#&%&thetaG6#%\"nG" }{TEXT -1 7 " | = 1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 124 "A glimmer into w hy this is such an important idea is suggested by the following exampl e. Consider the collection of vectors " }}{PARA 0 "" 0 "" {TEXT -1 24 " " }{TEXT 262 1 "B" }{TEXT -1 42 " = \{ [1, 2, 1], [-2, 1, 0], [-1, -2, 5] \}." }}{PARA 0 "" 0 "" {TEXT -1 15 "We ve rify that " }{TEXT 297 1 "B" }{TEXT -1 131 " is an orthogonal family. \+ To do this, three dot products need to be computed. Do them in your he ad, or let Maple calculate for you." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "u:=<1,2,1>;\nv:=<-2,1,0>;\nw:=<-1,-2,5>;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "DotProduct(u,v);\nDotProduct(v,w); \nDotProduct(w,u);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 32 "Suppose we want to find numbers " }{TEXT 267 4 "a, b" }{TEXT -1 6 ", and " }{TEXT 268 1 "c" }{TEXT -1 10 " such that" }}{PARA 0 "" 0 "" {TEXT -1 29 " [1, 2, 3] = " } {TEXT 264 1 "a" }{TEXT -1 13 " [1, 2, 1] + " }{TEXT 265 1 "b" }{TEXT -1 14 " [-2, 1, 0] + " }{TEXT 266 1 "c" }{TEXT -1 13 " [-1, -2, 5]." } }{PARA 0 "" 0 "" {TEXT -1 143 "There are many ways to do this. Linear \+ algebra techniques come to mind. On the other hand, here is a way that takes advantage of the fact that " }{TEXT 298 1 "B" }{TEXT 263 1 " " }{TEXT -1 136 "is in orthogonal family. Take the dot product of both s ides of the equation with the first vector [1, 2, 1]. This produces th e equation " }}{PARA 0 "" 0 "" {TEXT -1 28 " 8 \+ = " }{TEXT 269 1 "a" }{TEXT -1 2 " 6" }}{PARA 0 "" 0 "" {TEXT -1 11 "f rom which " }{TEXT 271 1 "a" }{TEXT -1 1 " " }{TEXT 270 0 "" }{TEXT -1 101 "is easily found. We use Maple to repeat this calculation, as w ell as the other two needed to compute " }{TEXT 272 1 "b" }{TEXT -1 5 " and " }{TEXT 273 2 "c." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 " xvec:=<1,2,3>;\nort_bas:=(a*u+b*v+c*w);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "DotProduct(xvec,u)=DotProduct(ort_bas,u);\nexpand(%); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "DotProduct(xvec,v)=DotP roduct(ort_bas,v);\nexpand(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "DotProduct(xvec,w)=DotProduct(ort_bas,w);\nexpand(%);" }}} {PARA 0 "" 0 "" {TEXT -1 28 "The result seems to be that " }{TEXT 274 1 "a" }{TEXT -1 8 " = 4/3, " }{TEXT 275 1 "b" }{TEXT -1 10 " = 0, and \+ " }{TEXT 276 1 "c" }{TEXT -1 51 " = 1/3. We check that this is correct by computing " }{TEXT 277 15 "a u + b v + c w" }{TEXT -1 1 "." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "4/3*u+0*v+1/3*w;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 381 "Now imagine a large collection of orth ogonal vectors, say 10,000 or an infinity of them. Solving for the cor rect coefficients in a problem such as the one above, where we had onl y three vectors, would be nearly impossible with the old linear algebr a matrix methods. The method above used the fact that we had an orthog onal family. Having an orthogonal family reduced the work load." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 82 "We presen t some examples of orthogonal families containing an infinity of vecto rs." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 279 31 "E xamples of orthogonal families" }}{PARA 0 "" 0 "" {TEXT -1 49 "(1) The infinite collection of functions on C([0," }{XPPEDIT 18 0 "Pi;" "6#%# PiG" }{TEXT -1 99 "]) containing \{ sin(x), sin(2 x), sin(3 x), ...\}, \n(2) The infinite collection of functions on C([0," }{XPPEDIT 18 0 "P i;" "6#%#PiG" }{TEXT -1 52 "]) containing \{ 1, cos(x), cos(2 x), cos( 3 x), ...\}," }}{PARA 0 "" 0 "" {TEXT -1 47 "(3) The infinite collecti on of functions on C([" }{XPPEDIT 18 0 "-Pi;" "6#,$%#PiG!\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 47 "]) containing t he union of the above two sets.\n" }}{PARA 0 "" 0 "" {TEXT -1 326 "We \+ verify that the first collection is an orthogonal family and leave it \+ to the student to verify that the second collection is an orthogonal f amily. We recall that the usual dot product in this space is an integr al. What remains is to compute the following integral and determine if its value is zero. The assumption is that " }{TEXT 299 1 "m" }{TEXT -1 5 " and " }{TEXT 300 1 "n" }{TEXT -1 24 " are integers is needed:" }}{PARA 0 "" 0 "" {TEXT -1 38 " \+ " }{XPPEDIT 18 0 "int(sin(n*x)*sin(m*x),x = 0 .. Pi);" "6#-%$intG6$*&- %$sinG6#*&%\"nG\"\"\"%\"xGF,F,-F(6#*&%\"mGF,F-F,F,/F-;\"\"!%#PiG" } {TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 26 "Here are the calculatio ns." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "assume(n,integer);\na ssume(m,integer);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "Int(si n(n*x)*sin(m*x),x=0..Pi)=int(sin(n*x)*sin(m*x),x=0..Pi);" }}}{PARA 0 " " 0 "" {TEXT -1 54 "You might be interested to know what happens if n \+ = m." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "Int(sin(n*x)*sin(n*x ),x=0..Pi) = int(sin(n*x)^2,x=0..Pi);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 87 "The third set brings up two ideas. First, the third set is an orthogonal family on C([-" } {XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "Pi;" "6 #%#PiG" }{TEXT -1 68 "]). And second, that third set is not an orthogo nal family on C([0, " }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 66 "]). To see the first of these we need to evaluate three integrals." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "Int(sin(n*x)*sin(m*x),x=-Pi. .Pi) = int(sin(n*x)*sin(m*x),x=-Pi..Pi);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "Int(cos(n*x)*cos(m*x),x=-Pi..Pi) = int(cos(n*x)*cos(m *x),x=-Pi..Pi);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "Int(sin( n*x)*cos(m*x),x=-Pi..Pi) = int(sin(n*x)*cos(m*x),x=-Pi..Pi);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 64 "To show that the union of the two sets is not orthogonal on [0, " }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 65 "] we evaluate one more integral to see that the norm is not zero." }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 142 "n:='n': m:='m':\nInt(sin(n*x)*cos(m*x),x=0..Pi)=in t(sin(n*x)*cos(m*x),x=0..Pi);\nInt(sin(n*x)*cos(n*x),x=0..Pi)=int(sin( n*x)*cos(n*x),x=0..Pi);\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 80 "We c an say more about the norm than its being non-zero. In particular, we know " }{XPPEDIT 18 0 "Int(sin(n*x)*cos(m*x),x=-Pi..0)=-Int(sin(n*x)* cos(m*x),x=0..Pi)" "6#/-%$IntG6$*&-%$sinG6#*&%\"nG\"\"\"%\"xGF-F--%$co sG6#*&%\"mGF-F.F-F-/F.;,$%#PiG!\"\"\"\"!,$-F%6$*&-F)6#*&F,F-F.F-F--F06 #*&F3F-F.F-F-/F.;F9F7F8" }{TEXT -1 34 ", because we knew from above th at " }{XPPEDIT 18 0 "Int(sin(n*x)*cos(m*x),x=-Pi..Pi)=0" "6#/-%$IntG6$ *&-%$sinG6#*&%\"nG\"\"\"%\"xGF-F--%$cosG6#*&%\"mGF-F.F-F-/F.;,$%#PiG! \"\"F7\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 129 "Int(sin(n*x )*cos(m*x),x=-Pi..0)=int(sin(n*x)*cos(m*x),x=-Pi..0);\nInt(sin(n*x)*co s(n*x),x=-Pi..0)=int(sin(n*x)*cos(n*x),x=-Pi..0);" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 84 "Why are we interested in orthogonal functions? We ask again: Why are they important?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 223 "We will find that they play a crucial r ole in some of the techniques for solving some of the classical partia l differential equations. Indeed, they are the basis for the notion of Fourier Series. Consider these three ideas." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 282 7 "Idea 1." }{TEXT -1 14 " If the \+ set \{ " }{XPPEDIT 18 0 "theta[1];" "6#&%&thetaG6#\"\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "theta[2];" "6#&%&thetaG6#\"\"#" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "theta[3];" "6#&%&thetaG6#\"\"$" }{TEXT -1 101 ", ... \+ \} is an orthogonal family, then any finite subset of the family is a \+ linearly independent set. " }}{SECT 0 {PARA 3 "" 0 "" {TEXT 281 22 "In dication of a proof." }}{PARA 0 "" 0 "" {TEXT -1 15 "Take the set \{ \+ " }{XPPEDIT 18 0 "theta[1];" "6#&%&thetaG6#\"\"\"" }{TEXT -1 2 ", " } {XPPEDIT 18 0 "theta[2];" "6#&%&thetaG6#\"\"#" }{TEXT -1 2 ", " } {XPPEDIT 18 0 "theta[3];" "6#&%&thetaG6#\"\"$" }{TEXT -1 60 "\}. We sh ow this set is linearly independent. To do this, let" }}{PARA 0 "" 0 " " {TEXT -1 18 " 0 = " }{XPPEDIT 18 0 "a[1]*theta[1];" "6# *&&%\"aG6#\"\"\"F'&%&thetaG6#F'F'" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "a [2]*theta[2];" "6#*&&%\"aG6#\"\"#\"\"\"&%&thetaG6#F'F(" }{TEXT -1 3 " \+ + " }{XPPEDIT 18 0 "a[3]*theta[3];" "6#*&&%\"aG6#\"\"$\"\"\"&%&thetaG6 #F'F(" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 25 "We need to show that the " }{TEXT 280 1 "a" }{TEXT -1 51 "'s are zero. Not so hard. F or example, to see that " }{XPPEDIT 18 0 "a[1];" "6#&%\"aG6#\"\"\"" } {TEXT -1 50 " is zero, take the dot product of both sides with " } {XPPEDIT 18 0 "theta[1];" "6#&%&thetaG6#\"\"\"" }{TEXT -1 8 ". Since \+ " }{XPPEDIT 18 0 "theta[1];" "6#&%&thetaG6#\"\"\"" }{TEXT -1 18 " is o rthogonal to " }{XPPEDIT 18 0 "theta[2];" "6#&%&thetaG6#\"\"#" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "theta[3];" "6#&%&thetaG6#\"\"$" }{TEXT -1 28 ", the equation is changed to" }}{PARA 0 "" 0 "" {TEXT -1 29 " \+ 0 = " }{XPPEDIT 18 0 "a[1];" "6#&%\"aG6#\"\"\" " }{TEXT -1 4 " || " }{XPPEDIT 18 0 "theta[1];" "6#&%&thetaG6#\"\"\"" }{TEXT -1 2 "||" }{XPPEDIT 18 0 "` `^2;" "6#*$%\"~G\"\"#" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "from which we conclude that " } {XPPEDIT 18 0 "a[1];" "6#&%\"aG6#\"\"\"" }{TEXT -1 9 " is zero." }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 284 7 "Idea 2." }{TEXT -1 77 " If you tell me that you have a function f that is a lin ear combination of \{ " }{XPPEDIT 18 0 "theta[1],theta[2],theta[3];" " 6%&%&thetaG6#\"\"\"&F$6#\"\"#&F$6#\"\"$" }{TEXT -1 61 " \}then I can t ell you what the coefficients are. If fact, if " }}{PARA 0 "" 0 "" {TEXT -1 24 " f = " }{XPPEDIT 18 0 "a[1]*theta[1]; " "6#*&&%\"aG6#\"\"\"F'&%&thetaG6#F'F'" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "a[2]*theta[2];" "6#*&&%\"aG6#\"\"#\"\"\"&%&thetaG6#F'F(" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "a[3]*theta[3];" "6#*&&%\"aG6#\"\"$\"\"\"&%& thetaG6#F'F(" }{TEXT -1 2 " ," }}{PARA 0 "" 0 "" {TEXT -1 17 "then \+ " }{XPPEDIT 18 0 "a[n];" "6#&%\"aG6#%\"nG" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "`<`*f,theta[n]*`>`;" "6$*&%\"GF%" }{TEXT -1 1 " " }{TEXT 283 1 "/" }{TEXT -1 4 " || \+ " }{XPPEDIT 18 0 "theta[n];" "6#&%&thetaG6#%\"nG" }{TEXT -1 3 " ||" } {XPPEDIT 18 0 "` `^2;" "6#*$%\"~G\"\"#" }{TEXT -1 17 ", n = 1, 2, or 3 ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 80 "The \+ student can make an argument for this by simply taking the dot product with " }{XPPEDIT 18 0 "theta[n];" "6#&%&thetaG6#%\"nG" }{TEXT -1 38 " of both sides of the equation for f ." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT 285 7 "Idea 3." }{TEXT -1 29 " If there are \+ an infinity of " }{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 88 "'s and only a finite number of them are used, then it may be that we onl y approximate f." }}{PARA 0 "" 0 "" {TEXT 286 8 "Example." }{TEXT -1 46 " If we take a function f on the interval [0, " }{XPPEDIT 18 0 "pi ;" "6#%#piG" }{TEXT -1 288 " ] and try to use the collection of sine t erms, but only a finite number of them, we may get only an approximati on for f. We illustrate this by drawing a graph of an f in black, two \+ terms of sines in red, and six terms of sines in blue. The graphs show that this is only an approximation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "f:=t->Pi/2-abs(t-Pi/2);" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 356 "a[1]:=int(f(t)*sin(t),t=0 ..Pi) / int(sin(t)^2,t=0..Pi); \na[2]:=int(f(t)*sin(2*t),t=0..Pi) / in t(sin(2*t)^2,t=0..Pi);\na[3]:=int(f(t)*sin(3*t),t=0..Pi) / int(sin(3*t )^2,t=0..Pi);\na[4]:=int(f(t)*sin(4*t),t=0..Pi) / int(sin(4*t)^2,t=0.. Pi);\na[5]:=int(f(t)*sin(5*t),t=0..Pi) / int(sin(5*t)^2,t=0..Pi);\na[6 ]:=int(f(t)*sin(6*t),t=0..Pi) / int(sin(6*t)^2,t=0..Pi);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 148 "plot([f(t),\n a[1] * sin(t), \n add( a[n] * sin(n*t), n=1..3),\n add( a[n] * sin(n*t), n= 1..5)],\n t=0..Pi,color=[black,red,green,blue]);" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 413 "We end this section with four inequalities. I pr esent no proof for the first two. The first two commonly appear in man y texts -- both linear algebra texts and partial differential equation s texts. The third is less common and I indicate a proof. The fourth f ollows from the third inequality. If it is true that all good analysis rests on an inequality, then the third inequality is the basis for Fo urier Analysis." }}{PARA 0 "" 0 "" {TEXT 287 13 "Inequalities." }} {PARA 0 "" 0 "" {TEXT 288 24 "The triangle inequality:" }{TEXT -1 13 " || x + y || " }{XPPEDIT 18 0 "` ` <= ` `;" "6#1%\"~GF$" }{TEXT -1 18 "|| x || + || y ||." }}{PARA 0 "" 0 "" {TEXT 289 30 "The Cauchy Swartz inequality: " }{TEXT -1 16 " | < x, y > | " }{XPPEDIT 18 0 "` ` <= \+ ` `" "6#1%\"~GF$" }{TEXT -1 16 "|| x || || y ||." }}{PARA 0 "" 0 "" {TEXT 290 23 "The Fourier Inequality:" }{TEXT -1 7 " if \{ " } {XPPEDIT 18 0 "phi[1];" "6#&%$phiG6#\"\"\"" }{TEXT -1 2 ", " } {XPPEDIT 18 0 "phi[2];" "6#&%$phiG6#\"\"#" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "phi[3];" "6#&%$phiG6#\"\"$" }{TEXT -1 14 ", ... \} is an " } {TEXT 301 11 "orthonormal" }{TEXT -1 35 " family and f is in the space , then" }}{PARA 0 "" 0 "" {TEXT -1 8 " 0 " }{XPPEDIT 18 0 "` ` <= ` `;" "6#1%\"~GF$" }{TEXT -1 2 " " }{XPPEDIT 18 0 "abs(f-sum(a[p]*ph i[p],p))^2" "6#*$-%$absG6#,&%\"fG\"\"\"-%$sumG6$*&&%\"aG6#%\"pGF)&%$ph iG6#F1F)F1!\"\"\"\"#" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "abs(f)^2" "6#* $-%$absG6#%\"fG\"\"#" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "sum(abs(-a[p])^2,p);" "6#-%$sumG6$*$-%$absG6#,&-%$<,>G6$%\"fG6#&%$phiG6#% \"pG\"\"\"&%\"aG6#F3!\"\"\"\"#F3" }{TEXT -1 4 " - " }{XPPEDIT 18 0 "s um(abs()^2,p);" "6#-%$sumG6$*$-%$absG6#-%$<,>G6$%\"fG6#&%$ph iG6#%\"pG\"\"#F2" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT 291 24 "The \+ Bessel's inequality:" }{TEXT -1 6 " if \{ " }{XPPEDIT 18 0 "phi[1];" " 6#&%$phiG6#\"\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "phi[2];" "6#&%$phi G6#\"\"#" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "phi[3];" "6#&%$phiG6#\"\"$ " }{TEXT -1 14 ", ... \} is an " }{TEXT 302 11 "orthonormal" }{TEXT -1 35 " family and f is in the space, then" }}{PARA 0 "" 0 "" {TEXT -1 44 " " }{XPPEDIT 18 0 "s um(abs()^2,p);" "6#-%$sumG6$*$-%$absG6#-%$<,>G6$%\"fG6#&%$ph iG6#%\"pG\"\"#F2" }{TEXT -1 1 " " }{XPPEDIT 18 0 "` ` <= ` `;" "6#1%\" ~GF$" }{TEXT -1 0 "" }{XPPEDIT 18 0 "abs(f)^2;" "6#*$-%$absG6#%\"fG\" \"#" }{TEXT -1 1 "." }}{SECT 0 {PARA 3 "" 0 "" {TEXT 292 30 "Proof for Fourier's Inequality" }}{PARA 0 "" 0 "" {TEXT -1 141 "We know the nor m squared is not negative, so the first part of Fourier's inequality i s easy. We start with the middle part, remembering that " }}{PARA 0 " " 0 "" {TEXT -1 25 " " }{XPPEDIT 18 0 "abs(x)^ 2;" "6#*$-%$absG6#%\"xG\"\"#" }{TEXT -1 12 " = < x, x >." }}{PARA 0 " " 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "abs(f-sum(a[p]*phi[p],p))^2" " 6#*$-%$absG6#,&%\"fG\"\"\"-%$sumG6$*&&%\"aG6#%\"pGF)&%$phiG6#F1F)F1!\" \"\"\"#" }{TEXT -1 9 " = < f - " }{XPPEDIT 18 0 "sum(a[p]*phi[p],p);" "6#-%$sumG6$*&&%\"aG6#%\"pG\"\"\"&%$phiG6#F*F+F*" }{TEXT -1 9 " , f \+ - " }{XPPEDIT 18 0 "sum(a[p]*phi[p],p);" "6#-%$sumG6$*&&%\"aG6#%\"pG\" \"\"&%$phiG6#F*F+F*" }{TEXT -1 3 " >" }}{PARA 0 "" 0 "" {TEXT -1 20 " = < f, f > - < " }{XPPEDIT 18 0 "sum(a[p]*phi[p],p);" "6#-%$sumG6 $*&&%\"aG6#%\"pG\"\"\"&%$phiG6#F*F+F*" }{TEXT -1 17 " , f > - < f , \+ " }{XPPEDIT 18 0 "sum(a[p]*phi[p],p);" "6#-%$sumG6$*&&%\"aG6#%\"pG\"\" \"&%$phiG6#F*F+F*" }{TEXT -1 10 " > + < " }{XPPEDIT 18 0 "sum(a[p]* phi[p],p);" "6#-%$sumG6$*&&%\"aG6#%\"pG\"\"\"&%$phiG6#F*F+F*" }{TEXT -1 3 " , " }{XPPEDIT 18 0 "sum(a[p]*phi[p],p);" "6#-%$sumG6$*&&%\"aG6# %\"pG\"\"\"&%$phiG6#F*F+F*" }{TEXT -1 3 " >" }}{PARA 0 "" 0 "" {TEXT -1 2 "= " }{XPPEDIT 18 0 "abs(f)^2;" "6#*$-%$absG6#%\"fG\"\"#" }{TEXT -1 8 " - 2 < " }{XPPEDIT 18 0 "sum(a[p]*phi[p],p);" "6#-%$sumG6$*&&% \"aG6#%\"pG\"\"\"&%$phiG6#F*F+F*" }{TEXT -1 9 " , f > + " }{XPPEDIT 18 0 "sum(abs(a[p])^2,p);" "6#-%$sumG6$*$-%$absG6#&%\"aG6#%\"pG\"\"#F- " }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "= " }{XPPEDIT 18 0 "ab s(f)^2;" "6#*$-%$absG6#%\"fG\"\"#" }{TEXT -1 4 " + " }{XPPEDIT 18 0 " sum(abs()^2,p);" "6#-%$sumG6$*$-%$absG6#-%$<,>G6$%\"fG6#&%$p hiG6#%\"pG\"\"#F2" }{TEXT -1 8 " - 2 < " }{XPPEDIT 18 0 "sum(a[p]*the ta[p],p);" "6#-%$sumG6$*&&%\"aG6#%\"pG\"\"\"&%&thetaG6#F*F+F*" }{TEXT -1 9 " , f > + " }{XPPEDIT 18 0 "sum(abs(a[p])^2,p);" "6#-%$sumG6$*$-% $absG6#&%\"aG6#%\"pG\"\"#F-" }{TEXT -1 4 " - " }{XPPEDIT 18 0 "sum(ab s()^2,p);" "6#-%$sumG6$*$-%$absG6#-%$<,>G6$%\"fG6#&%$phiG6#% \"pG\"\"#F2" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 4 "= " } {XPPEDIT 18 0 "abs(f)^2" "6#*$-%$absG6#%\"fG\"\"#" }{TEXT -1 3 " + " } {XPPEDIT 18 0 "sum(abs(-a[p])^2,p);" "6#-%$sumG6$*$-%$absG6# ,&-%$<,>G6$%\"fG6#&%$phiG6#%\"pG\"\"\"&%\"aG6#F3!\"\"\"\"#F3" }{TEXT -1 4 " - " }{XPPEDIT 18 0 "sum(abs()^2,p);" "6#-%$sumG6$*$- %$absG6#-%$<,>G6$%\"fG6#&%$phiG6#%\"pG\"\"#F2" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 41 "This is the right side of the inequality. " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 63 "Two \+ important consequences of The Fourier Inequality are given." }}{PARA 0 "" 0 "" {TEXT 295 24 "Important Consequence 1:" }{TEXT -1 13 " Suppo se the " }{XPPEDIT 18 0 "phi[p];" "6#&%$phiG6#%\"pG" }{TEXT -1 62 "'s \+ are orthonormal. The way to choose the coefficients of the " } {XPPEDIT 18 0 "phi[p];" "6#&%$phiG6#%\"pG" }{TEXT -1 37 "'s to make th e approximation for f by" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 48 " approximation for f = " }{XPPEDIT 18 0 "sum(a[p]*phi[p],p);" "6#-%$sumG6$*&&%\"aG6#%\"pG\"\"\" &%$phiG6#F*F+F*" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 38 "as close as possible is to choose the " } {XPPEDIT 18 0 "a[p];" "6#&%\"aG6#%\"pG" }{TEXT -1 15 "'s to be < f , \+ " }{XPPEDIT 18 0 "phi[p];" "6#&%$phiG6#%\"pG" }{TEXT -1 53 " >. To see this, note that no choice you make on the " }{XPPEDIT 18 0 "a[p];" "6 #&%\"aG6#%\"pG" }{TEXT -1 128 "'s changes the right most side of the F ourier Inequality except with the middle term, and that term can be ma de zero by choosing" }}{PARA 0 "" 0 "" {TEXT -1 46 " \+ " }{XPPEDIT 18 0 "a[p];" "6#&%\"aG6#%\"pG " }{TEXT -1 9 " = < f , " }{XPPEDIT 18 0 "phi[p];" "6#&%$phiG6#%\"pG" }{TEXT -1 3 " >." }}{PARA 0 "" 0 "" {TEXT 296 24 "Important Consequenc e 2:" }{TEXT -1 4 " If " }{XPPEDIT 18 0 "a[p];" "6#&%\"aG6#%\"pG" } {TEXT -1 9 " = < f , " }{XPPEDIT 18 0 "phi[p];" "6#&%$phiG6#%\"pG" } {TEXT -1 64 " > then the right most side of the Fourier Inequality giv es that" }}{PARA 0 "" 0 "" {TEXT -1 15 " 0 " }{XPPEDIT 18 0 "` ` <= ` `;" "6#1%\"~GF$" }{TEXT -1 1 " " }{XPPEDIT 18 0 "abs(f)^2; " "6#*$-%$absG6#%\"fG\"\"#" }{TEXT -1 4 " - " }{XPPEDIT 18 0 "sum(abs ()^2,p);" "6#-%$sumG6$*$-%$absG6#-%$<,>G6$%\"fG6#&%$phiG6#% \"pG\"\"#F2" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "which is Be ssel's inequality." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 262 "The question that should arise now is what kind of funct ions can we approximate with an orthogonal family and how close can we get with the approximations. These are the issues that we examine bef ore we start applying the ideas to partial differential equations." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{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 "Copyri ght \251 2003 by James V. Herod" }}{PARA 256 "" 0 "" {TEXT -1 19 "Al l 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 }