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"_cstyle4 " -1 252 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle5" -1 253 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{PSTYLE "_pstyle5" -1 206 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{CSTYLE "_cstyle6" -1 254 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle7" -1 255 "Times" 1 12 0 0 0 1 2 2 1 2 2 2 0 0 0 1 }{CSTYLE "_cstyle8" -1 256 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{PSTYLE "_pstyle6" -1 207 1 {CSTYLE "" -1 -1 "T imes" 1 18 0 0 0 1 2 1 2 2 2 2 1 0 0 1 }1 1 0 0 8 4 2 0 2 0 2 2 -1 1 } {CSTYLE "_cstyle9" -1 257 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 0 0 0 1 } {PSTYLE "_pstyle7" -1 208 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }3 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{CSTYLE "_cstyle10" -1 258 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{PSTYLE "_pstyle8" -1 209 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 2 2 2 0 0 0 1 }0 0 0 -1 -1 -1 1 0 1 0 2 2 -1 1 }} {SECT 0 {PARA 202 "" 0 "" {TEXT 249 42 "Section 2.3: Convergence of Fo urier Series" }}{PARA 203 "" 0 "" }{SECT 1 {PARA 204 "" 0 "" {TEXT 250 30 "Maple Packages for Section 2.3" }}{EXCHG {PARA 205 "> " 0 "" {MPLTEXT 1 251 8 "restart:" }}}{EXCHG {PARA 205 "> " 0 "" }}}{PARA 203 "" 0 "" }{PARA 203 "" 0 "" {TEXT 252 190 " In the Section 2.1, we discussed three general types of convergence in C([0, 1]): normed, pointwise, and uniform. Here, we apply these ideas to the special cas e that we have a function " }{TEXT 253 1 "f" }{TEXT 252 31 " and we ha ve the Fourier series" }}{PARA 203 "" 0 "" }{PARA 206 "" 0 "" {TEXT 254 21 " " }{XPPEDIT 2 0 "S[n](x)" "6#-&I\"SG6\"6# I\"nGF&6#I\"xGF&" }{TEXT 254 3 " = " }{XPPEDIT 2 0 "Sum(`<`*f*`,`*phi[ p]*`>`*phi[p](x),p = 0 .. n)" "6#-I$SumG6\"6$*.I\"GF)F*-F-6#I\"xGF%F*/F0;\" \"!I\"nGF%" }{TEXT 254 2 " ." }}{PARA 203 "" 0 "" }{PARA 203 "" 0 "" {TEXT 252 6 "where " }{XPPEDIT 18 0 "phi[1],phi[2],phi[3];" "6%&%$phiG 6#\"\"\"&F$6#\"\"#&F$6#\"\"$" }{TEXT 252 200 " ... is the usual ort honormal sequence. These results are presented here as Theorems, witho ut proofs. The results may be found in many texts in texts on Fourier \+ Series and Boundary Value Problems. " }}{PARA 203 "" 0 "" }{PARA 203 " " 0 "" {TEXT 252 51 "There are several terms we will use in this modul e." }}{PARA 203 "" 0 "" }{PARA 203 "" 0 "" {TEXT 252 14 "A function is " }{TEXT 253 22 "sectionally continuous" }{TEXT 252 135 " on an inter val [a, b] if it is continuous on that interval except for possibly a \+ finite number of jumps and removable discontinuities." }}{PARA 203 "" 0 "" }{PARA 203 "" 0 "" {TEXT 252 14 "A function is " }{TEXT 253 18 "s ectionally smooth" }{TEXT 252 31 " on an interval [a, b] if both " } {TEXT 253 1 "f" }{TEXT 252 1 " " }{TEXT 252 3 "and" }{TEXT 252 1 " " } {TEXT 253 1 "f" }{TEXT 252 53 " ' are sectionally continuous on the in terval [a, b]." }}{PARA 203 "" 0 "" }{PARA 203 "" 0 "" {TEXT 255 9 "Ex amples:" }{TEXT 252 280 " The function f(x) = signum(x) is sectionally continuous on [-1, 1], but the function g(x) = 1/x is not sectionally continuous on that interval. Both are continuous on that interval exc ept at zero. The signum function has a jump at x = 0, but 1/x has a mo re serious discontinuity." }}{EXCHG {PARA 205 "> " 0 "" {MPLTEXT 1 251 55 "plot([signum(x),1/x+1/10],x=-1..1,y=-3..3,discont=true," } {MPLTEXT 1 251 24 "\n color=[BLACK, RED]);" }}}{EXCHG {PARA 205 "> " 0 "" }}{PARA 203 "" 0 "" }{PARA 203 "" 0 "" {TEXT 255 9 "Example: " } {TEXT 252 13 "The function " }{XPPEDIT 18 0 "sqrt(abs(x));" "6#-%%sqrt G6#-%$absG6#%\"xG" }{TEXT 252 115 " is continuous, but not sectionally smooth on [-1, 1]. It is not sectionally smooth because the derivativ e goes to " }{XPPEDIT 18 0 "infinity;" "6#%)infinityG" }{TEXT 252 22 " as x approaches zero." }}{EXCHG {PARA 205 "> " 0 "" {MPLTEXT 1 251 21 "diff(sqrt(abs(x)),x);" }{MPLTEXT 1 251 71 "\nplot([%,sqrt(abs(x))] ,x=-1..1,y=-3..3,discont=true,color=[red,black]);" }}}{EXCHG {PARA 205 "> " 0 "" }}{PARA 203 "" 0 "" }{PARA 203 "" 0 "" {TEXT 252 53 "Her e is the first result for convergence of series. " }}{PARA 203 "" 0 " " }{PARA 203 "" 0 "" {TEXT 256 9 "THEOREM: " }{TEXT 252 17 " If the fu nction " }{TEXT 253 1 "f" }{TEXT 252 49 " is sectionally smooth and pe riodic with period 2" }{TEXT 253 2 " c" }{TEXT 252 46 ", then at each \+ point x the Fourier series for " }{TEXT 253 1 "f" }{TEXT 252 14 " conv erges and" }}{PARA 203 "" 0 "" }{PARA 203 "" 0 "" {TEXT 252 7 " \+ " }{XPPEDIT 18 0 "a[0]+sum(a[n]*cos(n*pi*x/c)+b[n]*sin(n*pi*x/c),n = 1 .. infinity);" "6#,&&%\"aG6#\"\"!\"\"\"-%$sumG6$,&*&&F%6#%\"nGF(-%$co sG6#**F0F(%#piGF(%\"xGF(%\"cG!\"\"F(F(*&&%\"bGF/F(-%$sinGF3F(F(/F0;F(% )infinityGF(" }{TEXT 252 24 " = [ f(x+) + f(x-)] / 2." }}{PARA 203 "" 0 "" }{PARA 203 "" 0 "" {TEXT 255 8 "Example:" }{TEXT 252 268 " The fu nction signum(x) is sectionally smooth. Therefore the Fourier series f or this function converges to 1 for 0 < x < 1, to -1 for -1 < x < 0, \+ and to 0 for x = -1, 0, or 1. The Fourier series for the function has \+ period 2. Here is a plot for 5 terms of the series." }}{PARA 203 "" 0 "" }{EXCHG {PARA 205 "> " 0 "" {MPLTEXT 1 251 5 "c:=1;" }{MPLTEXT 1 251 17 "\nf:=x->signum(x);" }{MPLTEXT 1 251 42 "\na[0]:=int(f(x),x=-c. .c)/int(1^2,x=-c..c);" }{MPLTEXT 1 251 21 "\nfor n from 1 to 5 do" } {MPLTEXT 1 251 44 "\n a[n]:=int(f(x)*cos((n*Pi*x)/c),x=-c..c)/" } {MPLTEXT 1 251 40 "\n int(cos(n*Pi*x/c)^2,x=-c..c);" } {MPLTEXT 1 251 44 "\n b[n]:=int(f(x)*sin((n*Pi*x)/c),x=-c..c)/" } {MPLTEXT 1 251 40 "\n int(sin(n*Pi*x/c)^2,x=-c..c);" } {MPLTEXT 1 251 4 "\nod;" }{MPLTEXT 1 251 8 "\nn:='n':" }}}{EXCHG {PARA 205 "> " 0 "" {MPLTEXT 1 251 65 "s:=x->a[0]+sum(a[n]*cos((n*Pi*x )/c)+b[n]*sin((n*Pi*x)/c),n=1..5);" }}}{EXCHG {PARA 205 "> " 0 "" {MPLTEXT 1 251 46 "plot([[x,f(x),x=-c..c],[x,s(x),x=-2*c..2*c]]);" }}} {EXCHG {PARA 205 "> " 0 "" }}{PARA 203 "" 0 "" }{PARA 203 "" 0 "" {TEXT 252 255 "In this previous example, the function was not continuo us. We will see that a discontinuity always leads to the over shoot th at can be observed, no matter how many terms of the series are taken. \+ Is it not clear that the function would be getting close to " }{TEXT 253 1 "f" }{TEXT 252 1 "(" }{TEXT 253 1 "x" }{TEXT 252 26 ") if more t erms were used?" }}{PARA 203 "" 0 "" }{PARA 203 "" 0 "" {TEXT 252 98 " In this next Theorem, we see a more powerful result which will insure \+ uniform convergence for all " }{TEXT 253 1 "x" }{TEXT 252 1 "." }} {PARA 203 "" 0 "" }{PARA 203 "" 0 "" {TEXT 256 8 "THEOREM:" }{TEXT 252 15 " If the series " }{XPPEDIT 18 0 "sum(abs(a[n])+abs(b[n]),n = 1 .. infinity);" "6#-%$sumG6$,&-%$absG6#&%\"aG6#%\"nG\"\"\"-F(6#&%\"bGF ,F./F-;F.%)infinityG" }{TEXT 252 40 " converges, then the Fourier seri es for " }{TEXT 253 1 "f" }{TEXT 252 45 " converges uniformly in the i nterval [-c, c]." }}{PARA 203 "" 0 "" }{PARA 203 "" 0 "" {TEXT 255 8 " Example:" }{TEXT 252 17 " Take the series " }{XPPEDIT 18 0 "a[n] = 2*( -1+(-1)^n)/(Pi*n^2);" "6#/&%\"aG6#%\"nG*(\"\"#\"\"\",&F*!\"\"),$F*F,F' F*F**&%#PiGF**$F'F)F*F," }{TEXT 252 9 " , with " }{XPPEDIT 18 0 "b[n] = 0;" "6#/&%\"bG6#%\"nG\"\"!" }{TEXT 252 26 ". Since the number serie s " }{XPPEDIT 18 0 "sum(1/(n^2),n = 1 .. infinity);" "6#-%$sumG6$*&\" \"\"F'*$%\"nG\"\"#!\"\"/F);F'%)infinityG" }{TEXT 252 148 " converges, \+ the series of absolute values converges, and so a Fourier Series with \+ these coefficients converges. We draw graphs for this series on [-" } {XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT 252 2 ", " }{XPPEDIT 18 0 "Pi;" " 6#%#PiG" }{TEXT 252 3 " ]." }{TEXT 252 1 "\n" }}{EXCHG {PARA 205 "> " 0 "" {MPLTEXT 1 251 6 "c:=Pi;" }{MPLTEXT 1 251 12 "\na[0]:=Pi/2;" } {MPLTEXT 1 251 21 "\nfor n from 1 to 5 do" }{MPLTEXT 1 251 32 "\n a [n]:=2/Pi*(-1+(-1)^n)/n^2;" }{MPLTEXT 1 251 13 "\n b[n]:=0;" } {MPLTEXT 1 251 4 "\nod;" }{MPLTEXT 1 251 8 "\nn:='n':" }}}{EXCHG {PARA 205 "> " 0 "" {MPLTEXT 1 251 65 "s:=x->a[0]+sum(a[n]*cos((n*Pi*x )/c)+b[n]*sin((n*Pi*x)/c),n=1..5);" }}}{EXCHG {PARA 205 "> " 0 "" {MPLTEXT 1 251 23 "plot(s(x),x=-2*c..2*c);" }}}{EXCHG {PARA 205 "> " 0 "" }}{PARA 203 "" 0 "" }{PARA 203 "" 0 "" {TEXT 255 8 "Example:" } {TEXT 252 6 " If \{ " }{XPPEDIT 18 0 "phi[1],phi[2],phi[3];" "6%&%$phi G6#\"\"\"&F$6#\"\"#&F$6#\"\"$" }{TEXT 252 70 ", ... \} is an orthogona l sequence and we construct the Fourier Series " }}{PARA 203 "" 0 "" {TEXT 252 22 " " }{XPPEDIT 18 0 "sum(a[p]*phi[p], p);" "6#-%$sumG6$*&&%\"aG6#%\"pG\"\"\"&%$phiGF)F+F*" }}{PARA 203 "" 0 "" {TEXT 252 54 "then this series will converge in norm if and only if " }}{PARA 203 "" 0 "" {TEXT 252 22 " " } {XPPEDIT 18 0 "sum(abs(a[p])^2,p);" "6#-%$sumG6$*$-%$absG6#&%\"aG6#%\" pG\"\"#F-" }{TEXT 252 1 " " }}{PARA 203 "" 0 "" {TEXT 252 228 "converg es. We established this earlier with the Fourier Inequality. This resu lt and the following example should be contrasted with the previous ex ample. In this example, we have normed convergence, but not uniform co nvergence. " }}{PARA 203 "" 0 "" {TEXT 252 15 "The example is " } {XPPEDIT 18 0 "sum(sin(n*x)/n,n = 1 .. infinity);" "6#-%$sumG6$*&-%$si nG6#*&%\"nG\"\"\"%\"xGF,F,F+!\"\"/F+;F,%)infinityG" }{TEXT 252 145 ". \+ I try to convince you and me that the convergence is not uniform by se eing that the series converges in norm, but not to a continuous functi on." }}{EXCHG {PARA 205 "> " 0 "" {MPLTEXT 1 251 31 "f2:=x->sum(sin(n* x)/n,n=1..15):" }{MPLTEXT 1 251 32 "\ng2:=x->sum(sin(n*x)/n,n=1..25):" }{MPLTEXT 1 251 32 "\nh2:=x->sum(sin(n*x)/n,n=1..35):" }}}{EXCHG {PARA 205 "> " 0 "" {MPLTEXT 1 251 43 "plot(\{f2(x),g2(x),h2(x)\},x=-P i..Pi,y=-2..2," }{MPLTEXT 1 251 28 "\n color=[red,blue,green]);" }}} {EXCHG {PARA 205 "> " 0 "" }}{PARA 203 "" 0 "" }{PARA 203 "" 0 "" {TEXT 252 293 "Here is the last Theorem that we state in this section. It gives a condition for uniform convergence based on properties of t he function, and not on properties of the series as the last theorem d id. After all, it is usually the function we know at the outset, and n ot properties of the series." }}{PARA 203 "" 0 "" }{PARA 203 "" 0 "" {TEXT 256 8 "THEOREM:" }{TEXT 252 4 " If " }{TEXT 253 1 "f" }{TEXT 252 1 "(" }{TEXT 253 1 "x" }{TEXT 252 113 ") is periodic, continuous, \+ and has a sectionally continuous derivative, then the Fourier Series c orresponding to " }{TEXT 253 1 "f" }{TEXT 252 24 " converges uniformly to " }{TEXT 253 4 "f(x)" }{TEXT 252 26 " for the entire real line." } }{PARA 203 "" 0 "" }{PARA 203 "" 0 "" {TEXT 252 49 "As an example of t his Theorem, take the function " }{TEXT 253 1 "f" }{TEXT 252 1 "(" } {TEXT 253 1 "x" }{TEXT 252 5 ") = |" }{TEXT 253 1 "x" }{TEXT 252 19 "| on the interval [" }{XPPEDIT 18 0 "-pi;" "6#,$%#piG!\"\"" }{TEXT 252 2 ", " }{XPPEDIT 18 0 "pi;" "6#%#piG" }{TEXT 252 235 " ]. It is contin uous on that interval and its periodic extension is also continuous. W hile its derivative is not continuous, it is sectionally continuous. F urther, the coefficients are the ones used in the previous example. Ch eck that." }}{PARA 203 "" 0 "" }{PARA 203 "" 0 "" {TEXT 252 32 "On the other hand, the function " }{TEXT 253 1 "f" }{TEXT 252 1 "(" }{TEXT 253 1 "x" }{TEXT 252 4 ") = " }{TEXT 253 1 "x" }{TEXT 252 78 " on the \+ interval [-1, 1] is continuous there. While the periodic extension is \+ " }{TEXT 255 11 "sectionally" }{TEXT 252 17 " continuous, the " } {TEXT 255 36 "periodic extension is not continuous" }{TEXT 252 137 ". \+ Thus, you cannot expect the Fourier Series to converge uniformly. At t hese discontinuities, you should expect the series to converge to" }} {PARA 203 "" 0 "" {TEXT 252 12 " ( " }{TEXT 253 1 "f" }{TEXT 252 1 "(" }{TEXT 253 1 "c" }{TEXT 252 5 "-) + " }{TEXT 253 1 "f" } {TEXT 252 1 "(" }{TEXT 253 1 "c" }{TEXT 252 7 "+) )/2," }}{PARA 203 "" 0 "" {TEXT 252 140 "the average of the jump from the left and the jum p from the right at the points of discontinuities. Recall the first Th eorem in this Section" }}{PARA 203 "" 0 "" }{SECT 1 {PARA 207 "" 0 "" {TEXT 257 58 "Test for Failure of a Fourier Series to Converge Uniform ly" }}{PARA 203 "" 0 "" }{PARA 203 "" 0 "" {TEXT 252 170 "A commonly k nown result in undergraduate mathematics is that if a sequence of cont inuous functions converges uniformly, the the limit must be continuous . A Fourier Series" }}{PARA 203 "" 0 "" {TEXT 252 9 " " } {TEXT 253 1 "S" }{TEXT 252 1 "(" }{TEXT 253 1 "N" }{TEXT 252 4 ") = " }{XPPEDIT 18 0 "sum(A[p]*sin(p*x),p = 1 .. N);" "6#-%$sumG6$*&&%\"AG6# %\"pG\"\"\"-%$sinG6#*&F*F+%\"xGF+F+/F*;F+%\"NG" }}{PARA 203 "" 0 "" {TEXT 252 47 "is a sequence of continuous functions. Indeed, " }{TEXT 253 1 "S" }{TEXT 252 1 "(" }{TEXT 253 1 "N" }{TEXT 252 84 ") is simply a sum of trigonometric function (and, hence, infinitely differentiabl e)." }}{PARA 203 "" 0 "" {TEXT 252 1 " " }}{PARA 203 "" 0 "" {TEXT 252 355 "Thus, if the series converges uniformly, the limit must be co ntinuous. This result is invoked most often if the periodic extension \+ of a function is not continuous. The test for failure to have uniform \+ convergence goes like this: If the periodic extension of a function is not continuous, then the Fourier Series for the function cannot conve rge uniformly." }}{PARA 203 "" 0 "" }{PARA 203 "" 0 "" {TEXT 252 33 "H ere is an illustration: we take " }{TEXT 253 1 "f" }{TEXT 252 1 "(" } {TEXT 253 1 "x" }{TEXT 252 4 ") = " }{TEXT 253 1 "x" }{TEXT 252 159 " \+ over [-1,1]. Think about the periodic extension, with period 2. This p eriodic extension is NOT continuous. Thus, the Fourier series cannot c onverge uniformly." }}{EXCHG {PARA 205 "> " 0 "" {MPLTEXT 1 251 5 "c:= 1;" }{MPLTEXT 1 251 9 "\nf:=x->x;" }{MPLTEXT 1 251 42 "\na[0]:=int(f(x ),x=-c..c)/int(1^2,x=-c..c);" }{MPLTEXT 1 251 21 "\nfor n from 1 to 5 \+ do" }{MPLTEXT 1 251 41 "\na[n]:=int(f(x)*cos((n*Pi*x)/c),x=-c..c)/" } {MPLTEXT 1 251 40 "\n int(cos(n*Pi*x/c)^2,x=-c..c);" } {MPLTEXT 1 251 44 "\n b[n]:=int(f(x)*sin((n*Pi*x)/c),x=-c..c)/" } {MPLTEXT 1 251 40 "\n int(sin(n*Pi*x/c)^2,x=-c..c);" } {MPLTEXT 1 251 4 "\nod;" }{MPLTEXT 1 251 8 "\nn:='n':" }}}{EXCHG {PARA 205 "> " 0 "" {MPLTEXT 1 251 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 205 "> " 0 "" {MPLTEXT 1 251 46 "plot([[x,f(x),x=-c..c],[x,s(x),x=-2*c..2*c]]);" }}} {EXCHG {PARA 205 "> " 0 "" }}{PARA 203 "" 0 "" }}{PARA 203 "" 0 "" } {PARA 203 "" 0 "" }{PARA 203 "" 0 "" {TEXT 252 485 "We have presented \+ two type results in this Section. One type result was information abou t convergence based on the character of the coefficients. The other r esult was based on the character of the function to which the series w as to converge. The most frequent application is the last result: if t he function is periodic, piecewise smooth, and continuous, the Fourier Trigonometric Series converges uniformly. If the function is not cont inuous, the series will not converge uniformly." }}{PARA 203 "" 0 "" } {PARA 203 "" 0 "" {TEXT 252 51 "EMAIL: herod@math.gatech.edu or j herod@tds.net" }}{PARA 203 "" 0 "" {TEXT 252 38 "URL: http://www.math. gatech.edu/~herod" }}{PARA 203 "" 0 "" }{PARA 208 "" 0 "" {TEXT 258 36 "Copyright \251 2003 by James V. Herod" }}{PARA 208 "" 0 "" {TEXT 258 19 "All rights reserved" }}{PARA 209 "" 0 "" }}{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 1 1 2 33 1 1 }