{VERSION 3 0 "APPLE_PPC_MAC" "3.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 256 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 1 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 }{CSTYLE "" -1 261 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 } {CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 0 1 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 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 271 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 272 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }1 0 0 0 8 4 0 0 0 0 0 0 -1 0 }{PSTYLE "H eading 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 8 2 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Plot" 0 13 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Title" 0 18 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 1 0 0 0 0 0 0 }3 0 0 -1 12 12 0 0 0 0 0 0 19 0 }{PSTYLE "" 4 256 1 {CSTYLE " " -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 18 "" 0 "" {TEXT -1 37 "Linear Methods of Applied Mathematics " }}{PARA 256 "" 0 "" {TEXT -1 11 "Convergence" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 56 "Copyright 2000 by Evans M. Harrell II and James V. Herod" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 34 "Convergence of series of functions" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 99 "In this module, we d iscuss three types of convergence in C([0, 1]): normed, pointwise, and uniform." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 " We suppose that we have a sequence of functions " }{XPPEDIT 18 0 "f[1](x);" "6#-&%\"fG6#\"\"\"6#%\"xG" }{TEXT -1 3 " , " } {XPPEDIT 18 0 "f[2](x);" "6#-&%\"fG6#\"\"#6#%\"xG" }{TEXT -1 3 " , " } {XPPEDIT 18 0 "f[3](x);" "6#-&%\"fG6#\"\"$6#%\"xG" }{TEXT -1 53 ", ... converging to a function g(x). We say that the " }{XPPEDIT 18 0 "f;" "6#%\"fG" }{TEXT -1 33 " 's converge to g in the sense of" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "1. " }{TEXT 256 16 "norm convergence" }{TEXT -1 6 " if " }{XPPEDIT 18 0 "int(abs(f[n](t )-g(t))^2,t = 0 .. 1);" "6#-%$intG6$*$-%$absG6#,&-&%\"fG6#%\"nG6#%\"tG \"\"\"-%\"gG6#F1!\"\"\"\"#/F1;\"\"!\"\"\"" }{TEXT -1 17 " -> 0 as n -> " }{XPPEDIT 18 0 "infinity;" "6#%)infinityG" }{TEXT -1 2 " ." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "2. " } {TEXT 257 9 "pointwise" }{TEXT -1 22 " if, for each x, " } {XPPEDIT 18 0 "f[n](x);" "6#-&%\"fG6#%\"nG6#%\"xG" }{TEXT -1 19 " -> g (x) as n -> " }{XPPEDIT 18 0 "infinity;" "6#%)infinityG" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "3 . " }{TEXT 258 9 "uniformly" }{TEXT -1 59 " if the maximum for all x \+ in [0, 1] of the difference in " }{XPPEDIT 18 0 "f[n](x);" "6#-&%\"fG 6#%\"nG6#%\"xG" }{TEXT -1 34 " and g(x) goes to zero as n -> " } {XPPEDIT 18 0 "infinity;" "6#%)infinityG" }{TEXT -1 2 " ." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "These methods of conver gence can be contrasted. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 86 "1. Uniform Convergence implies pointwise convergen ce. To see this, note only that if " }}{PARA 0 "" 0 "" {TEXT -1 2 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 30 " " } {XPPEDIT 18 0 "MAX[x]*abs(f[n](x)-g(x));" "6#*&&%$MAXG6#%\"xG\"\"\"-%$ absG6#,&-&%\"fG6#%\"nG6#F'F(-%\"gG6#F'!\"\"F(" }{TEXT -1 5 " -> 0" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "then for \+ each x, " }{XPPEDIT 18 0 "f[n](x)-g(x);" "6#,&-&%\"fG6#%\"nG6#%\" xG\"\"\"-%\"gG6#F*!\"\"" }{TEXT -1 9 " -> 0." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 74 "2. Uniform Convergence implies normed convergence. To see this, note that " }}{PARA 0 "" 0 "" {TEXT -1 27 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 26 " " } {XPPEDIT 18 0 "abs(int((f[n](x)-g(x))^2,x = 0 .. 1)) <= MAX[x]*abs(f[n ](x)-g(x))^2;" "6#1-%$absG6#-%$intG6$*$,&-&%\"fG6#%\"nG6#%\"xG\"\"\"-% \"gG6#F2!\"\"\"\"#/F2;\"\"!\"\"\"*&&%$MAXG6#F2F3*$-F%6#,&-&F.6#F06#F2F 3-F56#F2F7\"\"#F3" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 241 "3. Pointwise convergence does not imply \+ uniform convergence. To see this, just note that the following sequenc e of functions converges to zero pointwise, but the maximum of each fu nction differs from the zero function by 1/e = 0.367689... . " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "plot([seq(n*x*exp(-n*x),n=1..10)],x=0..1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 176 "4. Pointwise conv ergence does not imply normed convergence. To see this, just note that the previous sequence of functions converges to zero pointwise, but t he integral of the " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "plot( [seq(sqrt(n)*exp(-n*x),n=1..10)],x=0..1);" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 34 "int((sqrt(n)*exp(-n*x))^2,x=0..1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "limit(%,n=infinity);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 188 "5. Norm convergence does not imply poin t wise convergence. The following is nine terms of an infinite sequenc e. The sequence does not converge point wise. To see this, after execu ting the " }{TEXT 259 7 "display" }{TEXT -1 320 " command, touch the g raph with the mouse, see a new tool bar above; the tool bar looks like a CD player control. Push the ->| symbol with the mouse. The graphs w ill progress through in order. You should see that you do not have poi ntwise convergence. To see that the norm converges to zero, compute th e norm in the next " }{TEXT 260 7 "do loop" }{TEXT -1 36 ". See how th e continuation might go." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 383 "f[1]:=x->(1+signum(1/2-x))/2:\nf[2]:=x->(1+si gnum(x-1/2))/2:\nf[3]:=x->(1+signum(1/3-x))/2:\nf[4]:=x->(1+signum(2/3 -x))/2-(1+signum(1/3-x))/2:\nf[5]:=x->(1+signum(1-x))/2-(1+signum(2/3- x))/2:\nf[6]:=x->(1+signum(1/4-x))/2-(1+signum(0-x))/2:\nf[7]:=x->(1+s ignum(1/2-x))/2-(1+signum(1/4-x))/2:\nf[8]:=x->(1+signum(3/4-x))/2-(1+ signum(1/2-x))/2:\nf[9]:=x->(1+signum(1-x))/2-(1+signum(3/4-x))/2:" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "for n from 1 to 9 do\np[n]: =plot([f[n](x)],x=0..1):\nod:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "display([seq(p[n],n=1..9)],insequence=true);" }}{PARA 13 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "for n fro m 1 to 9 do\nint(f[n](x)^2,x=0..1);\nod;" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 29 "Convergence of Fourier \+ series" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 225 " In the previous sec tion we discussed three general types of convergence in C([0, 1]): nor med, pointwise, and uniform. Here, we apply these ideas to the special case that we have a function f and we have the Fourier series" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 " \+ " }{XPPEDIT 18 0 "S[n](x) = Sum(anglebracket(f,phi[p])*phi [p](x),p = 0 .. n);" "6#/-&%\"SG6#%\"nG6#%\"xG-%$SumG6$*&-%-anglebrack etG6$%\"fG&%$phiG6#%\"pG\"\"\"-&F46#F66#F*F7/F6;\"\"!F(" }{TEXT -1 1 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "phi[1],phi[2],phi[3];" "6%&%$phiG6#\"\"\"&F$6#\"\"# &F$6#\"\"$" }{TEXT -1 302 " ... is the usual orthonormal sequence. \+ These results are presented here as Theorems, without proofs. The resu lts may be found in any of the four sets of text materials referenced \+ in Module 1. In particular, these results are contained in a general t heorem found in Chapter 3 of the Electonic Text. " }}{PARA 0 "" 0 "" {TEXT -1 56 " There are several terms we will use in this module. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "A fun ction is " }{TEXT 261 22 "sectionally continuous" }{TEXT -1 135 " on a n interval [a, b] if it is continuous on that interval except for poss ibly a finite number of jumps and removable discontinuities." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "A function is \+ " }{TEXT 262 18 "sectionally smooth" }{TEXT -1 28 " on an interval [a, b] if f " }{TEXT 263 3 "and" }{TEXT -1 55 " f ' are sectionally conti nuous on the interval [a, b]." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 183 "Examples: The function f(x) = signum(x) \+ is sectionally continuous on [-1, 1], but the function g(x) = 1/x is n ot sectionally continuous is not sectionally continuous on that interv al." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "plot([signum(x),1/x+1/10],x=-1..1,y =-3..3,discont=true,\n color=[BLACK, RED]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "Example: The function " }{XPPEDIT 18 0 "sqrt(abs(x)) ;" "6#-%%sqrtG6#-%$absG6#%\"xG" }{TEXT -1 92 " is (even) continuous, b ut not sectionally smooth on [-1, 1] because the derivative goes to " }{XPPEDIT 18 0 "infinity;" "6#%)infinityG" }{TEXT -1 22 " as x approac hes zero." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "diff(sqrt(abs(x )),x);\nplot(%,x=-1..1,y=-3..3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 "Here is the first result for convergence of series. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 264 9 "THEOREM: \+ " }{TEXT -1 130 " If the function f is sectionally smooth and periodic with period 2 c, then at each point x the Fourier series for f conver ges and" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 " " }{XPPEDIT 18 0 "a[0]+sum(a[n]*cos(n*pi*x/c)+b[n]*sin(n*pi*x/ c),n = 1 .. infinity);" "6#,&&%\"aG6#\"\"!\"\"\"-%$sumG6$,&*&&F%6#%\"n GF(-%$cosG6#**F0F(%#piGF(%\"xGF(%\"cG!\"\"F(F(*&&%\"bG6#F0F(-%$sinG6#* *F0F(F5F(F6F(F7F8F(F(/F0;\"\"\"%)infinityGF(" }{TEXT -1 24 " = [ f(x+) + f(x-)] / 2." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 276 "Example: The function signum(x) is sectionally smooth. T herefore the Fourier series for this function converges to 1 for 0 < x < 1, to -1 for -1 < x < 0, and to 0 for x = -1, 0, or 1. The Fourier series for the function has period 2. Here is a plot for 5 terms of t he series." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 184 "c:=1;\nf:=x->signum(x);\na[0]:=1/(2*c)*int(f(x),x= -c..c);\nfor n from 1 to 5 do\n a[n]:=1/c*int(f(x)*cos((n*Pi*x)/c),x =-c..c);\n b[n]:=1/c*int(f(x)*sin((n*Pi*x)/c),x=-c..c);\nod;\nn:='n' :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "s:=x->a[0]+sum(a[n]*co s((n*Pi*x)/c)+b[n]*sin((n*Pi*x)/c),n=1..5);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "plot([[x,f(x),x=-c..c],[x,s(x),x=-2*c..2*c]]);" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 291 " In this previous example, the function was not continuous. We will see that a discontinuity alw ays leads to the over shoot that could be observed, no matter how many terms of the series are taken. Is it not clear that the function woul d be getting close to f(x) if more terms were used?" }}{PARA 0 "" 0 " " {TEXT -1 105 " In this next example, we see a more powerful resu lt which will insure uniform convergence for all x." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "THEOREM: If the series \+ " }{XPPEDIT 18 0 "sum(abs(a[n])+abs(b[n]),n = 1 .. infinity);" "6#-%$s umG6$,&-%$absG6#&%\"aG6#%\"nG\"\"\"-F(6#&%\"bG6#F-F./F-;\"\"\"%)infini tyG" }{TEXT -1 86 " converges, then the Fourier series for f converges uniformly in the interval [-c, c]." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 25 "Example: Take the series " }{XPPEDIT 18 0 "a[n] = 2*(-1+(-1)^n)/(Pi*n^2);" "6#/&%\"aG6#%\"nG*(\"\"#\"\"\",&\" \"\"!\"\"),$\"\"\"F-F'F*F**&%#PiGF**$F'\"\"#F*F-" }{TEXT -1 9 " , wit h " }{XPPEDIT 18 0 "b[n] = 0;" "6#/&%\"bG6#%\"nG\"\"!" }{TEXT -1 26 ". Since the number series " }{XPPEDIT 18 0 "sum(1/(n^2),n = 1 .. infini ty);" "6#-%$sumG6$*&\"\"\"\"\"\"*$%\"nG\"\"#!\"\"/F*;\"\"\"%)infinityG " }{TEXT -1 140 " converges, the series of absolute values converges, \+ and so a Fourier Series with these coefficients converges. We provide \+ an illustration.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "c:=Pi; \na[0]:=Pi/2;\nfor n from 1 to 5 do\n a[n]:=2/Pi*(-1+(-1)^n)/n^2;\n b[n]:=0;\nod;\nn:='n':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "s:=x->a[0]+sum(a[n]*cos((n*Pi*x)/c)+b[n]*sin((n*Pi*x)/c),n=1..5); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "plot(s(x),x=-2*c..2*c); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 293 "Here is the last Theorem that we state in this section. It gives a condition for uniform conve rgence based on properties of the function, and not on properties of t he series as the last theorem did. After all, it is usually the functi on we know at the outset, and not properties of the series." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 183 "THEOREM: If f( x) is periodic, continuous, and has a sectionally continuous derivativ e, then the Fourier Series corresponding to f converges uniformly to f (x) for the entire real line." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 82 " As an example of this Theorem, take \+ the function f(x) = |x| on the interval [" }{XPPEDIT 18 0 "-pi;" "6#,$ %#piG!\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "pi;" "6#%#piG" }{TEXT -1 235 " ]. It is continuous on that interval and its periodic extension \+ is also continuous. While its derivative is not continuous, it is sect ionally continuous. Further, the coefficients are the ones used in the previous example. Check that." }}{PARA 0 "" 0 "" {TEXT -1 304 " O n the other hand, the function f(x) = x on the interval [-1, 1] is con tinuous there. While the periodic extension is sectionally continuous, the derivative is not. Thus, you cannot expect the Fourier Series to \+ coverge uniformly. At these discontinuities, you should expect the ser ies to converge to" }}{PARA 0 "" 0 "" {TEXT -1 31 " ( f(c-) + f(-c+) )/2," }}{PARA 0 "" 0 "" {TEXT -1 196 "the average of the jump \+ from the left and the jump from the right at the points of discontinui ties. (Actually, this is true at the points of continuity, too, for at such places x, f(x-) = f(x+). )" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 176 "c:=1;\nf:=x->x;\na[0]:=1/(2 *c)*int(f(x),x=-c..c);\nfor n from 1 to 5 do\n a[n]:=1/c*int(f(x)*co s((n*Pi*x)/c),x=-c..c);\n b[n]:=1/c*int(f(x)*sin((n*Pi*x)/c),x=-c..c );\nod;\nn:='n':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "s:=x->a [0]+sum(a[n]*cos((n*Pi*x)/c)+b[n]*sin((n*Pi*x)/c),n=1..5);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "plot([[x,f(x),x=-c..c],[x,s(x),x=-2 *c..2*c]]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 139 " These results have implications for functio ns defined on an interval [0, c]. Simply decide what type periodic ext ension is to be made." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "ASSIGNMENT: Here is an exercise found in the electroni c text." }}{PARA 0 "" 0 "" {TEXT -1 140 "Graph the following functions . Either create or imagine the periodic extension. Predict the nature \+ of the convergence of the Fourier Series." }}{PARA 0 "" 0 "" {TEXT -1 15 "1. cosh(x), " }{XPPEDIT 18 0 "-pi <= x;" "6#1,$%#piG!\"\"%\"xG " }{TEXT -1 5 " and " }{XPPEDIT 18 0 "x <= pi;" "6#1%\"xG%#piG" } {TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 13 "2. x |x|, " } {XPPEDIT 18 0 "-pi <= x;" "6#1,$%#piG!\"\"%\"xG" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "x <= pi;" "6#1%\"xG%#piG" }{TEXT -1 3 " . " }}{PARA 0 " " 0 "" {TEXT -1 15 "3. 1 - |x|, " }{XPPEDIT 18 0 "-1 <= x;" "6#1,$ \"\"\"!\"\"%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "x <= 1;" "6#1%\" xG\"\"\"" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 18 "4. | sin(x) |, " }{XPPEDIT 18 0 "0 <= x;" "6#1\"\"!%\"xG" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "x <= 1;" "6#1%\"xG\"\"\"" }{TEXT -1 17 " .\n5. 2 - 2 c os(" }{XPPEDIT 18 0 "pi;" "6#%#piG" }{TEXT -1 6 " x), " }{XPPEDIT 18 0 "-1 <= x;" "6#1,$\"\"\"!\"\"%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "x <= 1;" "6#1%\"xG\"\"\"" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 27 "6. x Heaviside(1/2 - x), " }{XPPEDIT 18 0 "0 <= x;" "6#1\"\"! %\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "x <= 1;" "6#1%\"xG\"\"\"" } {TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 12 "Hint: for 6." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "plot(x*Heaviside(1/2-x),x=0..1,disc ont=true);" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 52 "The critical noti on: Complete orthonormal sequences" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 140 " We pause here to emphasiz e the critical notion of a complete orthonormal sequence. Recall that \+ an orthonormal sequence is one for which" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 " " }{XPPEDIT 18 0 "anglebracke t(phi[n],phi[m]);" "6#-%-anglebracketG6$&%$phiG6#%\"nG&F'6#%\"mG" } {TEXT -1 8 " = 0 if " }{XPPEDIT 18 0 "n <> m;" "6#0%\"nG%\"mG" }{TEXT -1 6 " and " }{XPPEDIT 18 0 "anglebracket(phi[n],phi[n]) = abs(phi[n] )^2;" "6#/-%-anglebracketG6$&%$phiG6#%\"nG&F(6#F**$-%$absG6#&F(6#F*\" \"#" }{TEXT -1 5 " = 1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 71 "The important inequality is that if f is any element of the space, then" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 10 " 0 ² " }{XPPEDIT 18 0 "abs(f-sum(a[p]*phi[p],p))^2 " "6#*$-%$absG6#,&%\"fG\"\"\"-%$sumG6$*&&%\"aG6#%\"pGF)&%$phiG6#F1F)F1 !\"\"\"\"#" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "abs(f)^2" "6#*$-%$absG6# %\"fG\"\"#" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "sum(abs(anglebracket(f,p hi[p])-a[p])^2,p);" "6#-%$sumG6$*$-%$absG6#,&-%-anglebracketG6$%\"fG&% $phiG6#%\"pG\"\"\"&%\"aG6#F2!\"\"\"\"#F2" }{TEXT -1 4 " - " } {XPPEDIT 18 0 "sum(abs(anglebracket(f,phi[p]))^2,p);" "6#-%$sumG6$*$-% $absG6#-%-anglebracketG6$%\"fG&%$phiG6#%\"pG\"\"#F1" }{TEXT -1 1 "." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "Several \+ important facts followed from this inequality." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 265 6 "Fact 1" }{TEXT -1 48 ". S uppose that n is a positive integer, that \{ " }{XPPEDIT 18 0 "phi[1] ;" "6#&%$phiG6#\"\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "phi[2];" "6#&% $phiG6#\"\"#" }{TEXT -1 7 ", ..., " }{XPPEDIT 18 0 "phi[n];" "6#&%$phi G6#%\"nG" }{TEXT -1 41 " \} is an orthonormal sequence, and that " } {XPPEDIT 18 0 "S[n];" "6#&%\"SG6#%\"nG" }{TEXT -1 20 " is the span of \+ the " }{XPPEDIT 18 0 "phi[i];" "6#&%$phiG6#%\"iG" }{TEXT -1 55 " 's. I f f is in the space, then the closest element in " }{XPPEDIT 18 0 "S[n ];" "6#&%\"SG6#%\"nG" }{TEXT -1 40 " to f is given by the Fourier expa nsion:" }}{PARA 0 "" 0 "" {TEXT -1 15 " " }{XPPEDIT 18 0 "sum(anglebracket(f,phi[p])*phi[p],p = 1 .. n);" "6#-%$sumG6$*&-%-an glebracketG6$%\"fG&%$phiG6#%\"pG\"\"\"&F,6#F.F//F.;\"\"\"%\"nG" } {TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "We argue this by asking how to choose the " }{XPPEDIT 18 0 "a[p];" "6#&%\"aG6#%\"pG" }{TEXT -1 84 "' s on the right side of the inequality to make the right side as small as possible." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 266 7 "Fact 2." }{TEXT -1 8 " If the " }{XPPEDIT 18 0 "phi[i];" "6#&%$phiG6#%\"iG" }{TEXT -1 54 "'s is an infinite sequence, then the infinite series " }{XPPEDIT 18 0 "sum(abs(anglebracket(f,phi[p]))^2,p);" "6#-%$sumG6$*$-%$absG6#-% -anglebracketG6$%\"fG&%$phiG6#%\"pG\"\"#F1" }{TEXT -1 11 " converges. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "We ar gue this by observing that " }{XPPEDIT 18 0 "sum(abs(anglebracket(f,p hi[p]))^2,p);" "6#-%$sumG6$*$-%$absG6#-%-anglebracketG6$%\"fG&%$phiG6# %\"pG\"\"#F1" }{TEXT -1 18 " does not exceed " }{XPPEDIT 18 0 "abs(f) ^2" "6#*$-%$absG6#%\"fG\"\"#" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 267 7 "Fact 3." }{TEXT -1 8 " If the " }{XPPEDIT 18 0 "phi[i];" "6#&%$phiG6#%\"iG" }{TEXT -1 57 "'s is an infinite sequence, then the infinite series " } {XPPEDIT 18 0 "sum(anglebracket(f,phi[p])*phi[p],p);" "6#-%$sumG6$*&-% -anglebracketG6$%\"fG&%$phiG6#%\"pG\"\"\"&F,6#F.F/F." }{TEXT -1 31 " i n the vector space converges." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 50 "We argue this by observing that if n > m \+ then " }{XPPEDIT 18 0 "abs(sum(anglebracket(f,phi[p])*phi[p],p = m .. n))^2;" "6#*$-%$absG6#-%$sumG6$*&-%-anglebracketG6$%\"fG&%$phiG6#% \"pG\"\"\"&F06#F2F3/F2;%\"mG%\"nG\"\"#" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "sum(abs(anglebracket(f,phi[p]))^2,p = m .. n);" "6#-%$sumG6$*$-% $absG6#-%-anglebracketG6$%\"fG&%$phiG6#%\"pG\"\"#/F1;%\"mG%\"nG" } {TEXT -1 30 " , and this latter converges. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 "Remark: We wish now to have tha t the series " }{XPPEDIT 18 0 "sum(anglebracket(f,phi[p])*phi[p],p = 1 .. n);" "6#-%$sumG6$*&-%-anglebracketG6$%\"fG&%$phiG6#%\"pG\"\"\"&F,6 #F.F//F.;\"\"\"%\"nG" }{TEXT -1 97 " converges to f. This is not nec essarily so. One more critical idea is needed. It is that of a " } {TEXT 268 8 "complete" }{TEXT -1 23 " orthonormal sequence. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 270 11 "Definition:" } {TEXT -1 27 " An orthogonal sequence is " }{TEXT 269 8 "complete" } {TEXT -1 104 " if the only vector in the space that is orthogonal to e very element in the sequence is the zero vector." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 271 6 "Fact 4" } {TEXT -1 17 ". Suppose that \{ " }{XPPEDIT 18 0 "phi[1];" "6#&%$phiG6# \"\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "phi[2];" "6#&%$phiG6#\"\"#" } {TEXT -1 7 ", ..., " }{XPPEDIT 18 0 "phi[n];" "6#&%$phiG6#%\"nG" } {TEXT -1 13 " , ...\} is a " }{TEXT 272 8 "complete" }{TEXT -1 29 " or thonormal sequence. Then " }{XPPEDIT 18 0 "sum(anglebracket(f,phi[p]) *phi[p],p = 1 .. infinity);" "6#-%$sumG6$*&-%-anglebracketG6$%\"fG&%$p hiG6#%\"pG\"\"\"&F,6#F.F//F.;\"\"\"%)infinityG" }{TEXT -1 24 " converg es to f in norm." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 64 "We argue this by showing that the element g defined as f - " }{XPPEDIT 18 0 "sum(anglebracket(f,phi[p])*phi[p],p = 1 .. in finity);" "6#-%$sumG6$*&-%-anglebracketG6$%\"fG&%$phiG6#%\"pG\"\"\"&F, 6#F.F//F.;\"\"\"%)infinityG" }{TEXT -1 24 " is orthogonal to each " } {XPPEDIT 18 0 "phi[p];" "6#&%$phiG6#%\"pG" }{TEXT -1 1 "." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 8 "Exercise" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 11 "Let f(x) = " }{XPPEDIT 18 0 "x^2;" "6 #*$%\"xG\"\"#" }{TEXT -1 8 " on [0, " }{XPPEDIT 18 0 "pi;" "6#%#piG" } {TEXT -1 3 " ]." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "1. Draw the graph of the projection of f onto the span of sin(x), sin(3 x), sin(5 x), ... . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 94 "2. Draw the graph of the projection of f onto the span of sin(2 x), sin(4 x), sin(6 x), ... . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "3. Compute six terms of the decreasing sequence formed by the norm of \n f \+ - " }{XPPEDIT 18 0 "sum(anglebracket(f,phi[p])*phi[p],p = 1 .. infini ty);" "6#-%$sumG6$*&-%-anglebracketG6$%\"fG&%$phiG6#%\"pG\"\"\"&F,6#F. F//F.;\"\"\"%)infinityG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 10 "where the " }{XPPEDIT 18 0 "phi[p];" "6#&%$phiG6#%\"pG" }{TEXT -1 46 " 's are formed from the sine functions on [0, " }{XPPEDIT 18 0 "pi ;" "6#%#piG" }{TEXT -1 3 " ]." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "f:=x->x^2; " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "assume(n,integer):\nint (sin(n*x)^2,x=0..Pi);\nn:='n':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "for n from 1 to 10 do\n a[n]:=2*int(f(x)*sin(n*x),x=0..Pi)/P i;\nod;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "p:='p';Sum(a[p]* sin(p*x),p=1..5);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 167 "plot( \{[x,f(x),x=0..Pi],\n [x,sum(a[2*p-1]*sin((2*p-1)*x),p=1..5),x=-Pi.. 2*Pi],\n [x,sum(a[2*p]*sin((2*p)*x),p=1..5),x=-Pi..2*Pi]\},\n \+ color=[BLACK,RED, GREEN]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 123 "plot(\{[x,f(x),x=0..Pi],\n [x,sum(a[2*p]*sin((2*p)*x),p=1..5)\n +sum(a[2*p-1]*sin((2*p-1)*x),p=1..5),x=-Pi..2*Pi]\});" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "for n from 1 to 10 do\n \+ sqrt(evalf(int((f(x)-sum(a[p]*sin(p*x),p=1..n))^2,x=0..Pi)));\nod;" }} }}}}{MARK "9" 0 }{VIEWOPTS 1 1 0 1 1 1803 }