{VERSION 6 0 "IBM INTEL NT" "6.0" } {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 "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "Blue Emphasis" -1 256 "Times" 0 0 0 0 255 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Green Emphasis" -1 257 "Times" 1 12 0 128 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Maroon Emphasis" -1 258 "Times" 1 12 128 0 128 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Dark Red Emphasis" -1 259 "Times" 1 12 128 0 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Red Emphasis" -1 260 "Times" 1 12 255 0 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Purple Emphasis" -1 261 " Times" 1 12 115 0 230 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" 260 262 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" 260 263 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" 260 264 "" 1 24 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" 260 265 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE " " 260 266 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" 260 267 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" 260 268 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{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 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 279 "" 0 1 0 0 0 0 1 0 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 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 282 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 283 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 284 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 285 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 286 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 287 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 288 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 289 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" 260 290 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 291 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 292 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Tim es" 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 0 128 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 "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 128 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot" -1 13 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 }{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 }{PSTYLE "Normal" -1 257 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 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 49 "Parseval's formula and the Rieman n-Lebesgue Lemma" }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nanai mo, B.C., Canada" }}{PARA 0 "" 0 "" {TEXT -1 19 "Version: 26.3.2007" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 54 "Approximation of functions by tru ncated Fourier series" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" } }}{PARA 0 "" 0 "" {TEXT -1 41 "Suppose that we have a periodic functio n " }{XPPEDIT 18 0 "phi(x);" "6#-%$phiG6#%\"xG" }{TEXT -1 11 " of peri od " }{XPPEDIT 18 0 "2*L" "6#*&\"\"#\"\"\"%\"LGF%" }{TEXT -1 32 ", whi ch has the Fourier series: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "c+``;" "6#,&%\"cG\"\"\"%!GF%" }{XPPEDIT 18 0 "Sum(a[k]* cos(k*Pi*x/L)+b[k]*sin(k*Pi*x/L),k = 1 .. infinity);" "6#-%$SumG6$,&*& &%\"aG6#%\"kG\"\"\"-%$cosG6#**F+F,%#PiGF,%\"xGF,%\"LG!\"\"F,F,*&&%\"bG 6#F+F,-%$sinG6#**F+F,F1F,F2F,F3F4F,F,/F+;F,%)infinityG" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "c = 1/(2*L)" "6#/%\"cG*&\"\"\"F&*&\"\"#F&%\"LGF&!\" \"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(phi(x),x = -L .. L);" "6#-%$In tG6$-%$phiG6#%\"xG/F);,$%\"LG!\"\"F-" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[k] = 1/L" "6#/&%\"aG6#%\"kG*&\"\"\"F)%\"LG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(phi(x)*cos(k*Pi*x/L),x = -L .. L);" "6#-%$IntG6$*& -%$phiG6#%\"xG\"\"\"-%$cosG6#**%\"kGF+%#PiGF+F*F+%\"LG!\"\"F+/F*;,$F2F 3F2" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "b[k] = 1/L" "6#/&%\"bG6#%\"kG *&\"\"\"F)%\"LG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(phi(x)*sin(k *Pi*x/L),x = -L .. L);" "6#-%$IntG6$*&-%$phiG6#%\"xG\"\"\"-%$sinG6#**% \"kGF+%#PiGF+F*F+%\"LG!\"\"F+/F*;,$F2F3F2" }{TEXT -1 1 "." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "We can consider " }{XPPEDIT 18 0 "phi(x);" "6#-%$phiG6#%\"xG" }{TEXT -1 39 " to be a per iodic function with period " }{XPPEDIT 18 0 "2*L" "6#*&\"\"#\"\"\"%\"L GF%" }{TEXT -1 61 " defined over the whole of the real line, or we can think of " }{XPPEDIT 18 0 "phi(x);" "6#-%$phiG6#%\"xG" }{TEXT -1 39 " as being defined just on the interval " }{XPPEDIT 18 0 "[-L,L]" "6#7$ ,$%\"LG!\"\"F%" }{TEXT -1 21 ", or on the interval " }{XPPEDIT 18 0 "[ 0, 2*L];" "6#7$\"\"!*&\"\"#\"\"\"%\"LGF'" }{TEXT -1 1 "." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "phi[n](x);" "6#-&%$phiG6#%\"nG6#%\"xG" }{TEXT -1 37 " denote the tr uncated Fourier series " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "phi[n](x) = c;" "6#/-&%$phiG6#%\"nG6#%\"xG%\"cG" }{TEXT -1 2 " + " }{XPPEDIT 18 0 "Sum(a[k]*cos(k*Pi*x/L)+b[k]*sin(k*Pi*x/L),k = 1 .. n );" "6#-%$SumG6$,&*&&%\"aG6#%\"kG\"\"\"-%$cosG6#**F+F,%#PiGF,%\"xGF,% \"LG!\"\"F,F,*&&%\"bG6#F+F,-%$sinG6#**F+F,F1F,F2F,F3F4F,F,/F+;F,%\"nG " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 50 "When using such a tr uncated series to approximate " }{XPPEDIT 18 0 "phi(x)" "6#-%$phiG6#% \"xG" }{TEXT -1 95 " it is useful to be able to determine how many ter ms are needed to obtain the desired accuracy." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "Sometimes we require that " }{XPPEDIT 18 0 "abs(phi(x)-phi[n](x)) < epsilon;" "6#2-%$absG6#,&- %$phiG6#%\"xG\"\"\"-&F)6#%\"nG6#F+!\"\"%(epsilonG" }{TEXT -1 10 " for all " }{TEXT 269 1 "x" }{TEXT -1 49 ", in which case we say that the \+ approximation is " }{TEXT 261 7 "uniform" }{TEXT -1 71 ". This is the \+ requirement that the maximum absolute error is less than " }{XPPEDIT 18 0 "epsilon" "6#%(epsilonG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "An alternative is to require th at " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int([phi(x)-ph i[n](x)]^2,x = -L .. L) < epsilon;" "6#2-%$IntG6$*$7#,&-%$phiG6#%\"xG \"\"\"-&F+6#%\"nG6#F-!\"\"\"\"#/F-;,$%\"LGF4F9%(epsilonG" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 12 "We say that " }{XPPEDIT 18 0 "phi[n ](x);" "6#-&%$phiG6#%\"nG6#%\"xG" }{TEXT -1 14 " approximates " } {XPPEDIT 18 0 "phi(x)" "6#-%$phiG6#%\"xG" }{TEXT -1 4 " is " }{TEXT 261 11 "in the mean" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 12 "Th e quantity" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Epsilon [n] = Int([phi(x)-phi[n](x)]^2,x = -L .. L);" "6#/&%(EpsilonG6#%\"nG-% $IntG6$*$7#,&-%$phiG6#%\"xG\"\"\"-&F/6#F'6#F1!\"\"\"\"#/F1;,$%\"LGF7F< " }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 7 "is the " }{TEXT 261 17 "mean square error" }{TEXT -1 31 " of the approximating function " }{XPPEDIT 18 0 "phi[n](x);" "6#-&%$phiG6#%\"nG6#%\"xG" }{TEXT -1 70 ", and may be regarded as providing an average error over the interval \+ " }{XPPEDIT 18 0 "[-L, L];" "6#7$,$%\"LG!\"\"F%" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 18 "Mean square error " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 46 "Given a finite Fourier series approximation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "phi[n](x) = c;" "6#/-&%$phiG6#%\"n G6#%\"xG%\"cG" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "Sum(a[k]*cos(k*Pi*x/L )+b[k]*sin(k*Pi*x/L),k = 1 .. n);" "6#-%$SumG6$,&*&&%\"aG6#%\"kG\"\"\" -%$cosG6#**F+F,%#PiGF,%\"xGF,%\"LG!\"\"F,F,*&&%\"bG6#F+F,-%$sinG6#**F+ F,F1F,F2F,F3F4F,F,/F+;F,%\"nG" }{TEXT -1 12 " ------- (i)" }}{PARA 0 " " 0 "" {TEXT -1 15 "for a function " }{XPPEDIT 18 0 "phi(x)" "6#-%$phi G6#%\"xG" }{TEXT -1 8 ", where " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "c = 1/(2*L)" "6#/%\"cG*&\"\"\"F&*&\"\"#F&%\"LGF&!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(phi(x),x = -L .. L);" "6#-%$IntG6$ -%$phiG6#%\"xG/F);,$%\"LG!\"\"F-" }{TEXT -1 4 ", " }{XPPEDIT 18 0 "a [k] = 1/L" "6#/&%\"aG6#%\"kG*&\"\"\"F)%\"LG!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "Int(phi(x)*cos(k*Pi*x/L),x = -L .. L);" "6#-%$IntG6$*&- %$phiG6#%\"xG\"\"\"-%$cosG6#**%\"kGF+%#PiGF+F*F+%\"LG!\"\"F+/F*;,$F2F3 F2" }{TEXT -1 10 " and " }{XPPEDIT 18 0 "b[k] = 1/L" "6#/&%\"bG6# %\"kG*&\"\"\"F)%\"LG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(phi(x)* sin(k*Pi*x/L),x = -L .. L);" "6#-%$IntG6$*&-%$phiG6#%\"xG\"\"\"-%$sinG 6#**%\"kGF+%#PiGF+F*F+%\"LG!\"\"F+/F*;,$F2F3F2" }{TEXT -1 2 ", " }} {PARA 0 "" 0 "" {TEXT -1 40 "we can show that the mean square error \+ " }{XPPEDIT 18 0 "Epsilon[n] = 1/(2*L)" "6#/&%(EpsilonG6#%\"nG*&\"\"\" F)*&\"\"#F)%\"LGF)!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int([phi(x)-p hi[n](x)]^2,x = -L .. L);" "6#-%$IntG6$*$7#,&-%$phiG6#%\"xG\"\"\"-&F*6 #%\"nG6#F,!\"\"\"\"#/F,;,$%\"LGF3F8" }{TEXT -1 12 " reduces to" }} {PARA 256 "" 0 "" {TEXT -1 6 " " }{XPPEDIT 18 0 "Epsilon[n] = 1/( 2*L)" "6#/&%(EpsilonG6#%\"nG*&\"\"\"F)*&\"\"#F)%\"LGF)!\"\"" }{TEXT -1 2 " " }{XPPEDIT 18 0 "Int(phi(x)^2,x = -L .. L)-c^2-1/2;" "6#,(-%$ IntG6$*$-%$phiG6#%\"xG\"\"#/F+;,$%\"LG!\"\"F0\"\"\"*$%\"cGF,F1*&F2F2F, F1F1" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(a[k]^2+b[k]^2,k=1..n)" "6#-% $SumG6$,&*$&%\"aG6#%\"kG\"\"#\"\"\"*$&%\"bG6#F+F,F-/F+;F-%\"nG" } {TEXT -1 16 " ------- (ii). " }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {TEXT 262 28 "____________________________" }{TEXT -1 14 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 52 "Justificati on of the mean square error formula (ii) " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 7 "We have" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "Epsilon[n] = 1/(2*L)" "6#/&%(EpsilonG6#%\"n G*&\"\"\"F)*&\"\"#F)%\"LGF)!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int( [phi(x)-phi[n](x)]^2,x = -L .. L);" "6#-%$IntG6$*$7#,&-%$phiG6#%\"xG\" \"\"-&F*6#%\"nG6#F,!\"\"\"\"#/F,;,$%\"LGF3F8" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/(2*L);" "6#/%!G*&\" \"\"F&*&\"\"#F&%\"LGF&!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(phi(x )^2,x = -L .. L)-1/L;" "6#,&-%$IntG6$*$-%$phiG6#%\"xG\"\"#/F+;,$%\"LG! \"\"F0\"\"\"*&F2F2F0F1F1" }{TEXT -1 2 " " }{XPPEDIT 18 0 "Int(phi(x)* phi[n](x),x = -L .. L)+1/(2*L);" "6#,&-%$IntG6$*&-%$phiG6#%\"xG\"\"\"- &F)6#%\"nG6#F+F,/F+;,$%\"LG!\"\"F5F,*&F,F,*&\"\"#F,F5F,F6F," }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(phi[n](x)^2,x = -L .. L);" "6#-%$IntG6$*$ -&%$phiG6#%\"nG6#%\"xG\"\"#/F-;,$%\"LG!\"\"F2" }{TEXT -1 14 " ----- (i ii). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 " Now, substituting for " }{XPPEDIT 18 0 "phi[n](x);" "6#-&%$phiG6#%\"nG 6#%\"xG" }{TEXT -1 35 " from (i) the middle term in (iii) " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "1/L" "6#*&\"\"\"F$%\"LG!\" \"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(phi(x)*phi[n](x),x = -L .. L); " "6#-%$IntG6$*&-%$phiG6#%\"xG\"\"\"-&F(6#%\"nG6#F*F+/F*;,$%\"LG!\"\"F 4" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 3 "is " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1/L;" "6#*&\"\"\"F$%\"LG!\"\"" } {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(c*phi(x),x = -L .. L)+1/L;" "6#,&-% $IntG6$*&%\"cG\"\"\"-%$phiG6#%\"xGF)/F-;,$%\"LG!\"\"F1F)*&F)F)F1F2F)" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(Int(a[k]*phi(x)*cos(k*Pi*x/L),x = \+ -L .. L),k = 1 .. n);" "6#-%$SumG6$-%$IntG6$*(&%\"aG6#%\"kG\"\"\"-%$ph iG6#%\"xGF.-%$cosG6#**F-F.%#PiGF.F2F.%\"LG!\"\"F./F2;,$F8F9F8/F-;F.%\" nG" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "1/L" "6#*&\"\"\"F$%\"LG!\"\"" } {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(Int(b[k]*phi(x)*sin(k*Pi*x/L),x = - L .. L),k = 1 .. n);" "6#-%$SumG6$-%$IntG6$*(&%\"bG6#%\"kG\"\"\"-%$phi G6#%\"xGF.-%$sinG6#**F-F.%#PiGF.F2F.%\"LG!\"\"F./F2;,$F8F9F8/F-;F.%\"n G" }{TEXT -1 2 " " }}{PARA 256 "" 0 "" {TEXT -1 3 " = " }{XPPEDIT 18 0 "c/L;" "6#*&%\"cG\"\"\"%\"LG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "I nt(phi(x),x = -L .. L);" "6#-%$IntG6$-%$phiG6#%\"xG/F);,$%\"LG!\"\"F- " }{TEXT -1 4 " + " }{XPPEDIT 18 0 "Sum(a[k]/L,k = 1 .. n);" "6#-%$Su mG6$*&&%\"aG6#%\"kG\"\"\"%\"LG!\"\"/F*;F+%\"nG" }{TEXT -1 1 " " } {XPPEDIT 18 0 "Int(phi(x)*cos(k*Pi*x/L),x = -L .. L);" "6#-%$IntG6$*&- %$phiG6#%\"xG\"\"\"-%$cosG6#**%\"kGF+%#PiGF+F*F+%\"LG!\"\"F+/F*;,$F2F3 F2" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "Sum(b[k]/L,k = 1 .. n);" "6#-%$S umG6$*&&%\"bG6#%\"kG\"\"\"%\"LG!\"\"/F*;F+%\"nG" }{TEXT -1 1 " " } {XPPEDIT 18 0 "Int(phi(x)*sin(k*Pi*x/L),x = -L .. L);" "6#-%$IntG6$*&- %$phiG6#%\"xG\"\"\"-%$sinG6#**%\"kGF+%#PiGF+F*F+%\"LG!\"\"F+/F*;,$F2F3 F2" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 3 " = " }{XPPEDIT 18 0 "2*c^2" "6#*&\"\"#\"\"\"*$%\"cGF$F%" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "Sum(a[k]^2+b[k]^2,k = 1 .. n)" "6#-%$SumG6$,&*$&%\"aG6#%\"kG\"\"#\" \"\"*$&%\"bG6#F+F,F-/F+;F-%\"nG" }{TEXT -1 14 " ------- (iv)." }} {PARA 0 "" 0 "" {TEXT -1 23 "The last term in (iii) " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1/(2*L)" "6#*&\"\"\"F$*&\"\"#F$%\" LGF$!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(phi[n](x)^2,x = -L .. L );" "6#-%$IntG6$*$-&%$phiG6#%\"nG6#%\"xG\"\"#/F-;,$%\"LG!\"\"F2" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 3 "is " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1/(2*L)" "6#*&\"\"\"F$*&\"\"#F$%\"LGF$! \"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int((c+Sum(``(a[k]*cos(k*Pi*x/L) +b[k]*sin(k*Pi*x/L)),k = 1 .. n))^2,x = -L .. L);" "6#-%$IntG6$*$,&%\" cG\"\"\"-%$SumG6$-%!G6#,&*&&%\"aG6#%\"kGF)-%$cosG6#**F5F)%#PiGF)%\"xGF )%\"LG!\"\"F)F)*&&%\"bG6#F5F)-%$sinG6#**F5F)F:F)F;F)Fn],[L,m=n])" "6#/-%$IntG6$*&-%$cos G6#**%\"kG\"\"\"%#PiGF-%\"xGF-%\"LG!\"\"F--F)6#**%\"mGF-F.F-F/F-F0F1F- /F/;,$F0F1F0-%*PIECEWISEG6$7$\"\"!0F5%\"nG7$F0/F5F?" }{TEXT -1 2 " ," }}{PARA 256 "" 0 "" {XPPEDIT 18 0 " Int(sin(k*Pi*x/L)*sin(m*Pi*x/L),x= -L..L)=PIECEWISE([0,m<>n],[L,m=n])" "6#/-%$IntG6$*&-%$sinG6#**%\"kG\" \"\"%#PiGF-%\"xGF-%\"LG!\"\"F--F)6#**%\"mGF-F.F-F/F-F0F1F-/F/;,$F0F1F0 -%*PIECEWISEG6$7$\"\"!0F5%\"nG7$F0/F5F?" }{TEXT -1 2 " ," }}{PARA 0 " " 0 "" {TEXT -1 9 "to give: " }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "1/(2* L)" "6#*&\"\"\"F$*&\"\"#F$%\"LGF$!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(phi[n](x)^2,x = -L .. L) = c^2+1/2;" "6#/-%$IntG6$*$-&%$phiG6#% \"nG6#%\"xG\"\"#/F.;,$%\"LG!\"\"F3,&*$%\"cGF/\"\"\"*&F8F8F/F4F8" } {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(a[k]^2+b[k]^2,k = 1 .. n);" "6#-%$S umG6$,&*$&%\"aG6#%\"kG\"\"#\"\"\"*$&%\"bG6#F+F,F-/F+;F-%\"nG" }{TEXT -1 14 " ------- (v). " }}{PARA 0 "" 0 "" {TEXT -1 42 "Substituting (iv ) and (v) in (iii) gives: " }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "Epsilon[n] = 1/(2*L)" "6#/&%(EpsilonG6#%\"nG*&\"\"\"F)* &\"\"#F)%\"LGF)!\"\"" }{TEXT -1 2 " " }{XPPEDIT 18 0 "Int(phi(x)^2,x \+ = -L .. L)-2*c^2-``;" "6#,(-%$IntG6$*$-%$phiG6#%\"xG\"\"#/F+;,$%\"LG! \"\"F0\"\"\"*&F,F2*$%\"cGF,F2F1%!GF1" }{XPPEDIT 18 0 "Sum(a[k]^2+b[k]^ 2,k = 1 .. n) " "6#-%$SumG6$,&*$&%\"aG6#%\"kG\"\"#\"\"\"*$&%\"bG6#F+F, F-/F+;F-%\"nG" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "c^2+1/2;" "6#,&*$%\"c G\"\"#\"\"\"*&F'F'F&!\"\"F'" }{XPPEDIT 18 0 "Sum(a[k]^2+b[k]^2,k=1..n) " "6#-%$SumG6$,&*$&%\"aG6#%\"kG\"\"#\"\"\"*$&%\"bG6#F+F,F-/F+;F-%\"nG " }{TEXT -1 2 " " }}{PARA 256 "" 0 "" {TEXT -1 3 " = " }{XPPEDIT 18 0 "1/(2*L);" "6#*&\"\"\"F$*&\"\"#F$%\"LGF$!\"\"" }{TEXT -1 2 " " } {XPPEDIT 18 0 "Int(phi(x)^2,x = -L .. L)-c^2-1/2;" "6#,(-%$IntG6$*$-%$ phiG6#%\"xG\"\"#/F+;,$%\"LG!\"\"F0\"\"\"*$%\"cGF,F1*&F2F2F,F1F1" } {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(a[k]^2+b[k]^2,k=1..n)" "6#-%$SumG6$ ,&*$&%\"aG6#%\"kG\"\"#\"\"\"*$&%\"bG6#F+F,F-/F+;F-%\"nG" }{TEXT -1 1 " ," }}{PARA 0 "" 0 "" {TEXT -1 27 "which is the formula (ii). " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 48 "Mean square error via the complex Fourier series" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 69 "We can derive the formula of the last section using \+ the complex form " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "phi[n](x) = Sum(c[k]*eta[k](x),k = -n .. n);" "6#/-&%$phiG6#%\"nG6#% \"xG-%$SumG6$*&&%\"cG6#%\"kG\"\"\"-&%$etaG6#F26#F*F3/F2;,$F(!\"\"F(" } {TEXT -1 13 " ------- (i) " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "of the Fourier series approximation for a functio n " }{XPPEDIT 18 0 "phi(x)" "6#-%$phiG6#%\"xG" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "eta[k] = exp(k*Pi*i*x/L)" "6#/&%$etaG6#%\"kG-%$expG6#* ,F'\"\"\"%#PiGF,%\"iGF,%\"xGF,%\"LG!\"\"" }{TEXT -1 4 " and" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[k]=1/(2*L)" "6#/&%\"cG6#% \"kG*&\"\"\"F)*&\"\"#F)%\"LGF)!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "I nt(phi(x)*conjugate(eta[k])(x),x = -L .. L) = 1/(2*L);" "6#/-%$IntG6$* &-%$phiG6#%\"xG\"\"\"--%*conjugateG6#&%$etaG6#%\"kG6#F+F,/F+;,$%\"LG! \"\"F9*&F,F,*&\"\"#F,F9F,F:" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(phi(x )*exp(-k*Pi*i*x/L),x = -L .. L);" "6#-%$IntG6$*&-%$phiG6#%\"xG\"\"\"-% $expG6#,$*,%\"kGF+%#PiGF+%\"iGF+F*F+%\"LG!\"\"F5F+/F*;,$F4F5F4" } {TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 17 "for each integer " } {TEXT 270 1 "k" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 257 "" 0 "" {TEXT -1 50 "In this more general context, where the functions " }{XPPEDIT 18 0 "phi(x)" "6#-%$phiG6#%\"xG" }{TEXT -1 5 " \+ and " }{XPPEDIT 18 0 "phi[n](x);" "6#-&%$phiG6#%\"nG6#%\"xG" }{TEXT -1 51 " may have complex values, the mean square error is " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Epsilon[n] = 1/(2*L)" "6#/& %(EpsilonG6#%\"nG*&\"\"\"F)*&\"\"#F)%\"LGF)!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "Int(abs(phi(x)-phi[n](x))^2,x = -L .. L);" "6#-%$IntG6$ *$-%$absG6#,&-%$phiG6#%\"xG\"\"\"-&F,6#%\"nG6#F.!\"\"\"\"#/F.;,$%\"LGF 5F:" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 40 "We can show that \+ the mean square error " }{XPPEDIT 18 0 "Epsilon[n];" "6#&%(EpsilonG6# %\"nG" }{TEXT -1 12 " reduces to " }}{PARA 256 "" 0 "" {TEXT -1 9 " \+ " }{XPPEDIT 18 0 "Epsilon[n] = 1/(2*L);" "6#/&%(EpsilonG6#%\"nG* &\"\"\"F)*&\"\"#F)%\"LGF)!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(ab s(f(x))^2,x = -L .. L)-``;" "6#,&-%$IntG6$*$-%$absG6#-%\"fG6#%\"xG\"\" #/F.;,$%\"LG!\"\"F3\"\"\"%!GF4" }{XPPEDIT 18 0 " Sum(abs(c[k])^2,k = - n .. n)" "6#-%$SumG6$*$-%$absG6#&%\"cG6#%\"kG\"\"#/F-;,$%\"nG!\"\"F2" }{TEXT -1 14 " ------- (i)." }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {TEXT 265 21 "_____________________" }{TEXT -1 9 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "In the case whe re " }{XPPEDIT 18 0 "phi(x)" "6#-%$phiG6#%\"xG" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "phi[n](x);" "6#-&%$phiG6#%\"nG6#%\"xG" }{TEXT -1 41 " a re real valued functions the relations " }{XPPEDIT 18 0 "c[0]=c,c[k]=( a[k]-i*b[k])/2" "6$/&%\"cG6#\"\"!F%/&F%6#%\"kG*&,&&%\"aG6#F+\"\"\"*&% \"iGF1&%\"bG6#F+F1!\"\"F1\"\"#F7" }{TEXT -1 5 " and " }{XPPEDIT 18 0 " c[-k]=(a[k]+i*b[k])/2" "6#/&%\"cG6#,$%\"kG!\"\"*&,&&%\"aG6#F(\"\"\"*&% \"iGF/&%\"bG6#F(F/F/F/\"\"#F)" }{TEXT -1 13 ", imply that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "abs(c[k])^2=c[k]*conjugate( c[k])" "6#/*$-%$absG6#&%\"cG6#%\"kG\"\"#*&&F)6#F+\"\"\"-%*conjugateG6# &F)6#F+F0" }{XPPEDIT 18 0 "`` = c[k]*c[-k];" "6#/%!G*&&%\"cG6#%\"kG\" \"\"&F'6#,$F)!\"\"F*" }{XPPEDIT 18 0 "``= (a[k]^2+b[k]^2)/4" "6#/%!G*& ,&*$&%\"aG6#%\"kG\"\"#\"\"\"*$&%\"bG6#F+F,F-F-\"\"%!\"\"" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 8 "so that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(abs(c[k])^2,k = -n .. n) = Sum(c[k] *c[-k],k = -n .. n);" "6#/-%$SumG6$*$-%$absG6#&%\"cG6#%\"kG\"\"#/F.;,$ %\"nG!\"\"F3-F%6$*&&F,6#F.\"\"\"&F,6#,$F.F4F:/F.;,$F3F4F3" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = c[0]^2+2*Sum(c[k]*c[-k],k = 0 .. n);" "6#/%!G, &*$&%\"cG6#\"\"!\"\"#\"\"\"*&F+F,-%$SumG6$*&&F(6#%\"kGF,&F(6#,$F4!\"\" F,/F4;F*%\"nGF,F," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = c^2+2*Sum((a[k]^ 2+b[k]^2)/4,k = 0 .. n);" "6#/%!G,&*$%\"cG\"\"#\"\"\"*&F(F)-%$SumG6$*& ,&*$&%\"aG6#%\"kGF(F)*$&%\"bG6#F4F(F)F)\"\"%!\"\"/F4;\"\"!%\"nGF)F)" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = c^2+1/2;" "6#/%!G,&*$%\"cG\"\"#\" \"\"*&F)F)F(!\"\"F)" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(a[k]^2+b[k]^2 ,k=1..n)" "6#-%$SumG6$,&*$&%\"aG6#%\"kG\"\"#\"\"\"*$&%\"bG6#F+F,F-/F+; F-%\"nG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 61 "Hence we obta in the formula of the previous section, namely: " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Epsilon[n] = 1/(2*L)" "6#/&%(EpsilonG6 #%\"nG*&\"\"\"F)*&\"\"#F)%\"LGF)!\"\"" }{TEXT -1 2 " " }{XPPEDIT 18 0 "Int(f(x)^2,x = -L .. L)-c^2-1/2;" "6#,(-%$IntG6$*$-%\"fG6#%\"xG\"\" #/F+;,$%\"LG!\"\"F0\"\"\"*$%\"cGF,F1*&F2F2F,F1F1" }{TEXT -1 1 " " } {XPPEDIT 18 0 "Sum(a[k]^2+b[k]^2,k=1..n)" "6#-%$SumG6$,&*$&%\"aG6#%\"k G\"\"#\"\"\"*$&%\"bG6#F+F,F-/F+;F-%\"nG" }{TEXT -1 3 ". " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 51 "Justification of the mean square \+ error formula (i) " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT -1 31 "In terms of the \"dot product\" " } {XPPEDIT 18 0 "f(x)*`.`*g(x) = Int(f(x)*conjugate(g(x)),x = -L .. L); " "6#/*(-%\"fG6#%\"xG\"\"\"%\".GF)-%\"gG6#F(F)-%$IntG6$*&-F&6#F(F)-%*c onjugateG6#-F,6#F(F)/F(;,$%\"LG!\"\"F<" }{TEXT -1 11 ", we have " } {XPPEDIT 18 0 "c[k]=1/(2*L)" "6#/&%\"cG6#%\"kG*&\"\"\"F)*&\"\"#F)%\"LG F)!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "[phi(x)*`.`*eta[k](x)];" "6#7 #*(-%$phiG6#%\"xG\"\"\"%\".GF)-&%$etaG6#%\"kG6#F(F)" }{TEXT -1 10 " fo r each " }{TEXT 271 1 "k" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 44 "The mean square error in the approximation " }{XPPEDIT 18 0 "phi[ n](x) = Sum(c[k]*eta[k](x),k = -n .. n);" "6#/-&%$phiG6#%\"nG6#%\"xG-% $SumG6$*&&%\"cG6#%\"kG\"\"\"-&%$etaG6#F26#F*F3/F2;,$F(!\"\"F(" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "phi(x)" "6#-%$phiG6#%\"xG" }{TEXT -1 5 " \+ is " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Epsilon[n](x) = 1/(2*L);" "6#/-&%(EpsilonG6#%\"nG6#%\"xG*&\"\"\"F,*&\"\"#F,%\"LGF,! \"\"" }{TEXT -1 2 " " }{XPPEDIT 18 0 "Int(abs(phi(x)-phi[n](x))^2,x = -L .. L);" "6#-%$IntG6$*$-%$absG6#,&-%$phiG6#%\"xG\"\"\"-&F,6#%\"nG6# F.!\"\"\"\"#/F.;,$%\"LGF5F:" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/(2*L);" "6#/%!G*&\"\"\"F&*&\"\"# F&%\"LGF&!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int([phi(x)-phi[n](x)] *[conjugate(phi(x)-phi[n](x))],x = -L .. L);" "6#-%$IntG6$*&7#,&-%$phi G6#%\"xG\"\"\"-&F*6#%\"nG6#F,!\"\"F-7#-%*conjugateG6#,&-F*6#F,F--&F*6# F16#F,F3F-/F,;,$%\"LGF3FB" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/(2*L);" "6#/%!G*&\"\"\"F&*&\"\"#F&%\"L GF&!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "[phi(x)-phi[n](x)]*`.`*[phi( x)-phi[n](x)];" "6#*(7#,&-%$phiG6#%\"xG\"\"\"-&F'6#%\"nG6#F)!\"\"F*%\" .GF*7#,&-F'6#F)F*-&F'6#F.6#F)F0F*" }{TEXT -1 1 " " }}{PARA 256 "" 0 " " {TEXT -1 2 "= " }{XPPEDIT 18 0 "1/(2*L)" "6#*&\"\"\"F$*&\"\"#F$%\"LG F$!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "[phi(x)*`.`*phi(x)-phi(x)*`.` *phi[n](x)-phi[n](x)*`.`*phi(x)+phi[n](x)*`.`*phi[n](x)];" "6#7#,**(-% $phiG6#%\"xG\"\"\"%\".GF*-F'6#F)F*F**(-F'6#F)F*F+F*-&F'6#%\"nG6#F)F*! \"\"*(-&F'6#F46#F)F*F+F*-F'6#F)F*F6*(-&F'6#F46#F)F*F+F*-&F'6#F46#F)F*F *" }{TEXT -1 15 " ------- (ii)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 261 4 "Note" }{TEXT -1 40 ": The expanded integr al form of (ii) is " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1/(2*L)" "6#*&\"\"\"F$*&\"\"#F$% \"LGF$!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``(Int(abs(phi(x))^2,x = \+ -L .. L)-Int(phi(x)*conjugate(phi[n](x)),x = -L .. L)-Int(phi[n](x)*co njugate(phi(x)),x = -L .. L)+Int(abs(phi[n](x))^2,x = -L .. L));" "6#- %!G6#,*-%$IntG6$*$-%$absG6#-%$phiG6#%\"xG\"\"#/F1;,$%\"LG!\"\"F6\"\"\" -F(6$*&-F/6#F1F8-%*conjugateG6#-&F/6#%\"nG6#F1F8/F1;,$F6F7F6F7-F(6$*&- &F/6#FD6#F1F8-F?6#-F/6#F1F8/F1;,$F6F7F6F7-F(6$*$-F,6#-&F/6#FD6#F1F2/F1 ;,$F6F7F6F8" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 60 "We can simplify the terms in the bracket of (ii ) as follows." }}{PARA 0 "" 0 "" {TEXT -1 2 " " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "phi[n](x)*`.`*phi(x) = Sum(c[k]*eta[k] (x),k = -n .. n)*`.`*phi(x);" "6#/*(-&%$phiG6#%\"nG6#%\"xG\"\"\"%\".GF ,-F'6#F+F,*(-%$SumG6$*&&%\"cG6#%\"kGF,-&%$etaG6#F86#F+F,/F8;,$F)!\"\"F )F,F-F,-F'6#F+F," }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`` = Sum(c[k]*[eta[k](x)*`.`*phi(x)],k = -n .. n);" "6# /%!G-%$SumG6$*&&%\"cG6#%\"kG\"\"\"7#*(-&%$etaG6#F,6#%\"xGF-%\".GF--%$p hiG6#F5F-F-/F,;,$%\"nG!\"\"F=" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = \+ 2*L;" "6#/%!G*&\"\"#\"\"\"%\"LGF'" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Sum (c[k]*conjugate(c[k]),k = -n .. n)" "6#-%$SumG6$*&&%\"cG6#%\"kG\"\"\"- %*conjugateG6#&F(6#F*F+/F*;,$%\"nG!\"\"F4" }{TEXT -1 1 " " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 2*L;" "6#/%!G*&\"\"#\"\"\"%\"LGF'" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(abs(c[k])^2,k = -n .. n)" "6#-%$SumG6$*$-%$absG6#&%\"cG6#%\" kG\"\"#/F-;,$%\"nG!\"\"F2" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Then " }{XPPEDIT 18 0 "phi(x)*`.`*phi[n](x) = conjugate(phi[n]( x)*`.`*phi(x));" "6#/*(-%$phiG6#%\"xG\"\"\"%\".GF)-&F&6#%\"nG6#F(F)-%* conjugateG6#*(-&F&6#F.6#F(F)F*F)-F&6#F(F)" }{XPPEDIT 18 0 "``=2*L " "6 #/%!G*&\"\"#\"\"\"%\"LGF'" }{XPPEDIT 18 0 "Sum(abs(c[k])^2,k = -n .. n );" "6#-%$SumG6$*$-%$absG6#&%\"cG6#%\"kG\"\"#/F-;,$%\"nG!\"\"F2" } {TEXT -1 7 ", also." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 " " {TEXT -1 39 "The last term in the bracket of (ii) is" }}{PARA 256 " " 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "phi[n](x)*`.`*phi[n](x) = Sum( c[k]*eta[k](x),k = -n .. n)*`.`*Sum(c[k]*eta[k](x),k = -n .. n);" "6#/ *(-&%$phiG6#%\"nG6#%\"xG\"\"\"%\".GF,-&F'6#F)6#F+F,*(-%$SumG6$*&&%\"cG 6#%\"kGF,-&%$etaG6#F:6#F+F,/F:;,$F)!\"\"F)F,F-F,-F46$*&&F86#F:F,-&F=6# F:6#F+F,/F:;,$F)FCF)F," }{TEXT -1 5 ". " }}{PARA 0 "" 0 "" {TEXT -1 56 "This simplifies by means of the orthogonality relations " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "eta[k](x)*`.`*eta[m]( x) = PIECEWISE([0, m <> k],[2*L, m = k])" "6#/*(-&%$etaG6#%\"kG6#%\"xG \"\"\"%\".GF,-&F'6#%\"mG6#F+F,-%*PIECEWISEG6$7$\"\"!0F1F)7$*&\"\"#F,% \"LGF,/F1F)" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 7 "to give" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "phi[n](x)*`.`*phi[n]( x) = 2*L;" "6#/*(-&%$phiG6#%\"nG6#%\"xG\"\"\"%\".GF,-&F'6#F)6#F+F,*&\" \"#F,%\"LGF," }{TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(abs(c[k])^2,k = -n . . n)" "6#-%$SumG6$*$-%$absG6#&%\"cG6#%\"kG\"\"#/F-;,$%\"nG!\"\"F2" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 22 "From (ii) we now have \+ " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Epsilon[n](x)=1/ (2*L)" "6#/-&%(EpsilonG6#%\"nG6#%\"xG*&\"\"\"F,*&\"\"#F,%\"LGF,!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(abs(phi(x))^2,x = -L .. L)-``;" "6 #,&-%$IntG6$*$-%$absG6#-%$phiG6#%\"xG\"\"#/F.;,$%\"LG!\"\"F3\"\"\"%!GF 4" }{XPPEDIT 18 0 " Sum(abs(c[k])^2,k = -n .. n)" "6#-%$SumG6$*$-%$abs G6#&%\"cG6#%\"kG\"\"#/F-;,$%\"nG!\"\"F2" }{TEXT -1 1 "." }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{TEXT 264 17 "_________________" }{TEXT -1 1 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 19 "Parseval's formula " }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 4 "Let " } {XPPEDIT 18 0 "phi(x)" "6#-%$phiG6#%\"xG" }{TEXT -1 48 " be a real val ued periodic function with period " }{XPPEDIT 18 0 "2*L" "6#*&\"\"#\" \"\"%\"LGF%" }{TEXT -1 41 " which has the Fourier series expansion: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "phi(x) = c;" "6#/-% $phiG6#%\"xG%\"cG" }{TEXT -1 2 " +" }{XPPEDIT 18 0 "Sum(a[k]*cos(k*Pi* x/L)+b[k]*sin(k*Pi*x/L),k = 1 .. infinity);" "6#-%$SumG6$,&*&&%\"aG6#% \"kG\"\"\"-%$cosG6#**F+F,%#PiGF,%\"xGF,%\"LG!\"\"F,F,*&&%\"bG6#F+F,-%$ sinG6#**F+F,F1F,F2F,F3F4F,F,/F+;F,%)infinityG" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 5 "Then " }}{PARA 256 "" 0 "" {TEXT -1 2 " \+ " }{XPPEDIT 18 0 "1/(2*L);" "6#*&\"\"\"F$*&\"\"#F$%\"LGF$!\"\"" } {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(phi(x)^2,x = -L .. L) = c^2+1/2;" " 6#/-%$IntG6$*$-%$phiG6#%\"xG\"\"#/F+;,$%\"LG!\"\"F0,&*$%\"cGF,\"\"\"*& F5F5F,F1F5" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(a[k]^2+b[k]^2,k = 1 .. infinity);" "6#-%$SumG6$,&*$&%\"aG6#%\"kG\"\"#\"\"\"*$&%\"bG6#F+F,F-/ F+;F-%)infinityG" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 263 24 "________________________" }{TEXT -1 2 " " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "We can establish t his formula as follows." }}{PARA 0 "" 0 "" {TEXT -1 43 "From the formu la for the mean square error " }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "Epsilon[n] = 1/(2*L);" "6#/&%(EpsilonG6#%\"nG*&\"\"\"F) *&\"\"#F)%\"LGF)!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int([phi(x)-phi [n](x)]^2,x = -L .. L) = 1/(2*L);" "6#/-%$IntG6$*$7#,&-%$phiG6#%\"xG\" \"\"-&F+6#%\"nG6#F-!\"\"\"\"#/F-;,$%\"LGF4F9*&F.F.*&F5F.F9F.F4" } {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(phi(x)^2,x = -L .. L)-c^2-1/2;" "6# ,(-%$IntG6$*$-%$phiG6#%\"xG\"\"#/F+;,$%\"LG!\"\"F0\"\"\"*$%\"cGF,F1*&F 2F2F,F1F1" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(a[k]^2+b[k]^2,k=1..n)" "6#-%$SumG6$,&*$&%\"aG6#%\"kG\"\"#\"\"\"*$&%\"bG6#F+F,F-/F+;F-%\"nG" } {TEXT -1 14 " ------- (i), " }}{PARA 0 "" 0 "" {TEXT -1 13 "we see tha t " }{XPPEDIT 18 0 "Epsilon[n+1] = Epsilon[n]-(a[n]^2+b[n]^2)/2;" "6# /&%(EpsilonG6#,&%\"nG\"\"\"F)F),&&F%6#F(F)*&,&*$&%\"aG6#F(\"\"#F)*$&% \"bG6#F(F3F)F)F3!\"\"F8" }{TEXT -1 11 " for each " }{TEXT 272 1 "n" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 34 "Hence the sequence of re al numbers" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Epsilo n[1],Epsilon[2],` . . `*Epsilon[n],` . . . `;" "6&&%(EpsilonG6#\"\"\"& F$6#\"\"#*&%&~.~.~GF&&F$6#%\"nGF&%(~.~.~.~G" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 42 "is decreasing, or at least non-increasing." }} {PARA 0 "" 0 "" {TEXT -1 60 "Since it is clear from the definition of \+ mean square error, " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Epsilon[n] = 1/(2*L)" "6#/&%(EpsilonG6#%\"nG*&\"\"\"F)*&\"\"#F)%\"L GF)!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int([phi(x)-phi[n](x)]^2,x = -L .. L);" "6#-%$IntG6$*$7#,&-%$phiG6#%\"xG\"\"\"-&F*6#%\"nG6#F,!\"\" \"\"#/F,;,$%\"LGF3F8" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 5 "t hat " }{XPPEDIT 18 0 "Epsilon[n];" "6#&%(EpsilonG6#%\"nG" }{TEXT -1 36 " is never negative, it follows that " }{XPPEDIT 18 0 "Limit(Epsilo n[n],n = infinity) = 0.;" "6#/-%&LimitG6$&%(EpsilonG6#%\"nG/F*%)infini tyG-%&FloatG6$\"\"!F0" }}{PARA 0 "" 0 "" {TEXT -1 37 "Parsevals formul a follows by letting " }{TEXT 273 1 "n" }{TEXT -1 25 " tend to infinit y in (i)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 49 "Parseval's formula for a complex valued function " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "phi(x)" "6#-%$phiG6#%\"xG" }{TEXT -1 51 " be a complex valued perio dic function with period " }{XPPEDIT 18 0 "2*L" "6#*&\"\"#\"\"\"%\"LGF %" }{TEXT -1 41 " which has the Fourier series expansion: " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "phi(x) = Sum(c[k]*exp(k*Pi *i*x/L),k = -infinity .. infinity);" "6#/-%$phiG6#%\"xG-%$SumG6$*&&%\" cG6#%\"kG\"\"\"-%$expG6#*,F/F0%#PiGF0%\"iGF0F'F0%\"LG!\"\"F0/F/;,$%)in finityGF8F<" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 4 "Then" }} {PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "1/(2*L);" "6#*&\"\" \"F$*&\"\"#F$%\"LGF$!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(abs(phi (x))^2,x = -L .. L) = Sum(abs(c[k])^2,k = -infinity .. infinity);" "6# /-%$IntG6$*$-%$absG6#-%$phiG6#%\"xG\"\"#/F.;,$%\"LG!\"\"F3-%$SumG6$*$- F)6#&%\"cG6#%\"kGF//F>;,$%)infinityGF4FB" }{TEXT -1 3 " . " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 266 19 "___________________" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 90 "We can establish this formula by a similar argument to that given \+ in the previous section." }}{PARA 0 "" 0 "" {TEXT -1 42 "From the form ula for the mean square error" }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "Epsilon[n] = 1/(2*L)" "6#/&%(EpsilonG6#%\"nG*&\"\"\"F)* &\"\"#F)%\"LGF)!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(abs(phi(x)-p hi[n](x))^2,x = -L .. L) = 1/(2*L);" "6#/-%$IntG6$*$-%$absG6#,&-%$phiG 6#%\"xG\"\"\"-&F-6#%\"nG6#F/!\"\"\"\"#/F/;,$%\"LGF6F;*&F0F0*&F7F0F;F0F 6" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(abs(phi(x))^2,x = -L .. L)-``; " "6#,&-%$IntG6$*$-%$absG6#-%$phiG6#%\"xG\"\"#/F.;,$%\"LG!\"\"F3\"\"\" %!GF4" }{XPPEDIT 18 0 "Sum(abs(c[k])^2,k = -n .. n)" "6#-%$SumG6$*$-%$ absG6#&%\"cG6#%\"kG\"\"#/F-;,$%\"nG!\"\"F2" }{TEXT -1 15 " ------- (i ), " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "we see that " }{XPPEDIT 18 0 "Epsilon[n+1] = Epsilon[n]+abs(c[n])^2+abs (c[-n])^2;" "6#/&%(EpsilonG6#,&%\"nG\"\"\"F)F),(&F%6#F(F)*$-%$absG6#&% \"cG6#F(\"\"#F)*$-F/6#&F26#,$F(!\"\"F4F)" }{TEXT -1 11 " for each " } {TEXT 274 1 "n" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 13 "Note th at if " }{XPPEDIT 18 0 "phi(x)" "6#-%$phiG6#%\"xG" }{TEXT -1 33 " is a real valued function then " }{XPPEDIT 18 0 "abs(c[n])^2 = abs(c[-n]) ^2;" "6#/*$-%$absG6#&%\"cG6#%\"nG\"\"#*$-F&6#&F)6#,$F+!\"\"F," }{TEXT -1 10 ", because " }{XPPEDIT 18 0 "conjugate(c[n]) = c[-n];" "6#/-%*co njugateG6#&%\"cG6#%\"nG&F(6#,$F*!\"\"" }{TEXT -1 24 ". In this case, w e have " }{XPPEDIT 18 0 "E[n+1]=E[n]+2*c[n]^2" "6#/&%\"EG6#,&%\"nG\"\" \"F)F),&&F%6#F(F)*&\"\"#F)*$&%\"cG6#F(F.F)F)" }{TEXT -1 10 " for each \+ " }{TEXT 275 1 "n" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 40 "In \+ any case the sequence of real numbers" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Epsilon[1],Epsilon[2],` . . `*Epsilon[n],` . . . \+ `;" "6&&%(EpsilonG6#\"\"\"&F$6#\"\"#*&%&~.~.~GF&&F$6#%\"nGF&%(~.~.~.~G " }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 42 "is decreasing, or at least non-increasing." }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " } {XPPEDIT 18 0 "Epsilon[n];" "6#&%(EpsilonG6#%\"nG" }{TEXT -1 36 " is n ever negative, it follows that " }{XPPEDIT 18 0 "Limit(Epsilon[n],n = \+ infinity) = 0.;" "6#/-%&LimitG6$&%(EpsilonG6#%\"nG/F*%)infinityG-%&Flo atG6$\"\"!F0" }}{PARA 0 "" 0 "" {TEXT -1 61 "The complex version of Pa rseval's formula follows by letting " }{TEXT 276 1 "n" }{TEXT -1 25 " \+ tend to infinity in (i)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 28 "The Riemann-Lebesgue Lemma " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 14 " Suppose that " } {XPPEDIT 18 0 "phi(x)" "6#-%$phiG6#%\"xG" }{TEXT -1 36 " is a periodic function with period " }{XPPEDIT 18 0 "2*L" "6#*&\"\"#\"\"\"%\"LGF%" }{TEXT -1 32 " which has the Fourier series: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c+``;" "6#,&%\"cG\"\"\"%!GF%" } {XPPEDIT 18 0 "Sum(a[k]*cos(k*Pi*x/L)+b[k]*sin(k*Pi*x/L),k = 1 .. infi nity);" "6#-%$SumG6$,&*&&%\"aG6#%\"kG\"\"\"-%$cosG6#**F+F,%#PiGF,%\"xG F,%\"LG!\"\"F,F,*&&%\"bG6#F+F,-%$sinG6#**F+F,F1F,F2F,F3F4F,F,/F+;F,%)i nfinityG" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c = 1/(2*L)" "6#/%\"c G*&\"\"\"F&*&\"\"#F&%\"LGF&!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int( phi(x),x = -L .. L);" "6#-%$IntG6$-%$phiG6#%\"xG/F);,$%\"LG!\"\"F-" } {TEXT -1 4 ", " }{XPPEDIT 18 0 "a[k] = 1/L" "6#/&%\"aG6#%\"kG*&\"\" \"F)%\"LG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(phi(x)*cos(k*Pi*x/ L),x = -L .. L);" "6#-%$IntG6$*&-%$phiG6#%\"xG\"\"\"-%$cosG6#**%\"kGF+ %#PiGF+F*F+%\"LG!\"\"F+/F*;,$F2F3F2" }{TEXT -1 10 " and " } {XPPEDIT 18 0 "b[k] = 1/L" "6#/&%\"bG6#%\"kG*&\"\"\"F)%\"LG!\"\"" } {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(phi(x)*sin(k*Pi*x/L),x = -L .. L); " "6#-%$IntG6$*&-%$phiG6#%\"xG\"\"\"-%$sinG6#**%\"kGF+%#PiGF+F*F+%\"LG !\"\"F+/F*;,$F2F3F2" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 4 "The n" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Limit(a[k],k=in finity) =0" "6#/-%&LimitG6$&%\"aG6#%\"kG/F*%)infinityG\"\"!" }{TEXT -1 8 " and " }{XPPEDIT 18 0 "Limit(b[k],k=infinity) =0" "6#/-%&Limi tG6$&%\"bG6#%\"kG/F*%)infinityG\"\"!" }{TEXT -1 1 "." }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{TEXT 290 19 "___________________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "Sinc e the mean square error " }{XPPEDIT 18 0 "Epsilon[n] = 1/(2*L)" "6#/& %(EpsilonG6#%\"nG*&\"\"\"F)*&\"\"#F)%\"LGF)!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "Int([phi(x)-phi[n](x)]^2,x = -L .. L);" "6#-%$IntG6$*$7 #,&-%$phiG6#%\"xG\"\"\"-&F*6#%\"nG6#F,!\"\"\"\"#/F,;,$%\"LGF3F8" } {TEXT -1 31 " is non-negative, the formula " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Epsilon[n] = 1/(2*L)" "6#/&%(EpsilonG6# %\"nG*&\"\"\"F)*&\"\"#F)%\"LGF)!\"\"" }{TEXT -1 2 " " }{XPPEDIT 18 0 "Int(phi(x)^2,x = -L .. L)-c^2-1/2;" "6#,(-%$IntG6$*$-%$phiG6#%\"xG\" \"#/F+;,$%\"LG!\"\"F0\"\"\"*$%\"cGF,F1*&F2F2F,F1F1" }{TEXT -1 1 " " } {XPPEDIT 18 0 "Sum(a[k]^2+b[k]^2,k=1..n)" "6#-%$SumG6$,&*$&%\"aG6#%\"k G\"\"#\"\"\"*$&%\"bG6#F+F,F-/F+;F-%\"nG" }{TEXT -1 2 " " }}{PARA 0 " " 0 "" {TEXT -1 12 "shows that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "c^2+1/2;" "6#,&*$%\"cG\"\"#\"\"\"*&F'F'F&!\"\"F'" } {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(a[k]^2+b[k]^2,k = 1 .. n) <= 1/(2*L );" "6#1-%$SumG6$,&*$&%\"aG6#%\"kG\"\"#\"\"\"*$&%\"bG6#F,F-F./F,;F.%\" nG*&F.F.*&F-F.%\"LGF.!\"\"" }{TEXT -1 2 " " }{XPPEDIT 18 0 "Int(phi(x )^2,x = -L .. L);" "6#-%$IntG6$*$-%$phiG6#%\"xG\"\"#/F*;,$%\"LG!\"\"F/ " }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 8 "for all " }{TEXT 291 1 "n" }{TEXT -1 7 ". Hence" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "c^2+1/2;" "6#,&*$%\"cG\"\"#\"\"\"*&F'F'F&!\"\"F'" } {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(a[k]^2+b[k]^2,k = 1 .. infinity) <= 1/(2*L);" "6#1-%$SumG6$,&*$&%\"aG6#%\"kG\"\"#\"\"\"*$&%\"bG6#F,F-F./F ,;F.%)infinityG*&F.F.*&F-F.%\"LGF.!\"\"" }{TEXT -1 2 " " }{XPPEDIT 18 0 "Int(phi(x)^2,x = -L .. L);" "6#-%$IntG6$*$-%$phiG6#%\"xG\"\"#/F* ;,$%\"LG!\"\"F/" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 27 "This m eans that the series " }{XPPEDIT 18 0 "Sum(a[k]^2+b[k]^2,k = 1 .. infi nity)" "6#-%$SumG6$,&*$&%\"aG6#%\"kG\"\"#\"\"\"*$&%\"bG6#F+F,F-/F+;F-% )infinityG" }{TEXT -1 16 " is convergent. " }}{PARA 0 "" 0 "" {TEXT -1 16 "It follows that " }{XPPEDIT 18 0 "Limit([a[k]^2+b[k]^2],k = inf inity) = 0;" "6#/-%&LimitG6$7#,&*$&%\"aG6#%\"kG\"\"#\"\"\"*$&%\"bG6#F- F.F//F-%)infinityG\"\"!" }{TEXT -1 21 ", which implies that " } {XPPEDIT 18 0 "Limit(a[k],k=infinity)=0" "6#/-%&LimitG6$&%\"aG6#%\"kG/ F*%)infinityG\"\"!" }{TEXT -1 6 " and " }{XPPEDIT 18 0 "Limit(b[k],k= infinity)=0" "6#/-%&LimitG6$&%\"bG6#%\"kG/F*%)infinityG\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 58 "The Rieman n-Lebesgue Lemma for a complex valued function " }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 14 " Suppose tha t " }{XPPEDIT 18 0 "phi(x)" "6#-%$phiG6#%\"xG" }{TEXT -1 51 " is a com plex valued periodic function with period " }{XPPEDIT 18 0 "2*L" "6#*& \"\"#\"\"\"%\"LGF%" }{TEXT -1 31 " which has the Fourier series: " }} {PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Sum(c[k]*exp(k*Pi*i* x/L),k = -infinity .. infinity);" "6#-%$SumG6$*&&%\"cG6#%\"kG\"\"\"-%$ expG6#*,F*F+%#PiGF+%\"iGF+%\"xGF+%\"LG!\"\"F+/F*;,$%)infinityGF4F8" } {TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[k] = 1/(2*L);" "6#/&%\"cG6#%\"kG *&\"\"\"F)*&\"\"#F)%\"LGF)!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(p hi(x)*exp(-m*Pi*i*x/L),x = -L .. L);" "6#-%$IntG6$*&-%$phiG6#%\"xG\"\" \"-%$expG6#,$*,%\"mGF+%#PiGF+%\"iGF+F*F+%\"LG!\"\"F5F+/F*;,$F4F5F4" } {TEXT -1 1 "." }{TEXT 292 1 " " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 5 "Then " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Limit(abs(c[k]),k = infinity) = 0;" "6#/-%&LimitG6$-%$absG6#&%\"cG6#% \"kG/F-%)infinityG\"\"!" }{TEXT -1 6 " and " }{XPPEDIT 18 0 "Limit(ab s(c[k]),k = -infinity) = 0" "6#/-%&LimitG6$-%$absG6#&%\"cG6#%\"kG/F-,$ %)infinityG!\"\"\"\"!" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{TEXT 268 21 "_____________________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "Since the mean \+ square error " }{XPPEDIT 18 0 "Epsilon[n] = 1/(2*L)" "6#/&%(EpsilonG6 #%\"nG*&\"\"\"F)*&\"\"#F)%\"LGF)!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int([phi(x)-phi[n](x)]^2,x = -L .. L);" "6#-%$IntG6$*$7#,&-%$phiG6#% \"xG\"\"\"-&F*6#%\"nG6#F,!\"\"\"\"#/F,;,$%\"LGF3F8" }{TEXT -1 31 " is non-negative, the formula " }}{PARA 256 "" 0 "" {TEXT -1 9 " \+ " }{XPPEDIT 18 0 "Epsilon[n] = 1/(2*L);" "6#/&%(EpsilonG6#%\"nG*&\"\" \"F)*&\"\"#F)%\"LGF)!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(abs(f(x ))^2,x = -L .. L)-``;" "6#,&-%$IntG6$*$-%$absG6#-%\"fG6#%\"xG\"\"#/F.; ,$%\"LG!\"\"F3\"\"\"%!GF4" }{XPPEDIT 18 0 " Sum(abs(c[k])^2,k = -n .. \+ n)" "6#-%$SumG6$*$-%$absG6#&%\"cG6#%\"kG\"\"#/F-;,$%\"nG!\"\"F2" } {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 12 "shows that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(abs(c[k])^2,k = -n .. n ) <= 1/(2*L);" "6#1-%$SumG6$*$-%$absG6#&%\"cG6#%\"kG\"\"#/F.;,$%\"nG! \"\"F3*&\"\"\"F6*&F/F6%\"LGF6F4" }{TEXT -1 2 " " }{XPPEDIT 18 0 "Int( phi(x)^2,x = -L .. L);" "6#-%$IntG6$*$-%$phiG6#%\"xG\"\"#/F*;,$%\"LG! \"\"F/" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 8 "for all " } {TEXT 277 1 "n" }{TEXT -1 7 ". Hence" }}{PARA 256 "" 0 "" {TEXT -1 6 " " }{XPPEDIT 18 0 "Sum(abs(c[k])^2,k = -infinity .. infinity) <= \+ 1/(2*L);" "6#1-%$SumG6$*$-%$absG6#&%\"cG6#%\"kG\"\"#/F.;,$%)infinityG! \"\"F3*&\"\"\"F6*&F/F6%\"LGF6F4" }{TEXT -1 2 " " }{XPPEDIT 18 0 "Int( phi(x)^2,x = -L .. L);" "6#-%$IntG6$*$-%$phiG6#%\"xG\"\"#/F*;,$%\"LG! \"\"F/" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 35 "This means that both of the series " }{XPPEDIT 18 0 "Sum(abs(c[k])^2,k = 1 .. infinit y);" "6#-%$SumG6$*$-%$absG6#&%\"cG6#%\"kG\"\"#/F-;\"\"\"%)infinityG" } {TEXT -1 6 " and " }{XPPEDIT 18 0 "Sum(abs(c[-k])^2,k = 1 .. infinity )" "6#-%$SumG6$*$-%$absG6#&%\"cG6#,$%\"kG!\"\"\"\"#/F.;\"\"\"%)infinit yG" }{TEXT -1 17 " are convergent. " }}{PARA 0 "" 0 "" {TEXT -1 17 "It follows that " }{XPPEDIT 18 0 "abs(c[k]) -> 0" "6#f*6#-%$absG6#&%\"c G6#%\"kG7\"6$%)operatorG%&arrowG6\"\"\"!F0F0F0" }{TEXT -1 5 " as " } {XPPEDIT 18 0 "k -> infinity" "6#f*6#%\"kG7\"6$%)operatorG%&arrowG6\"% )infinityGF*F*F*" }{TEXT -1 4 " or " }{XPPEDIT 18 0 "k -> -infinity" " 6#f*6#%\"kG7\"6$%)operatorG%&arrowG6\",$%)infinityG!\"\"F*F*F*" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 60 "The R iemann-Lebesgue Lemma for a complex valued function II " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 116 " In this section we give a different proof of a version of the Riemann \+ - Lebesgue lemma for a complex Fourier series." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "If " }{XPPEDIT 18 0 "phi(x )" "6#-%$phiG6#%\"xG" }{TEXT -1 62 " is a continuous periodic complex \+ valued function with period " }{XPPEDIT 18 0 "2*L" "6#*&\"\"#\"\"\"%\" LGF%" }{TEXT -1 5 ", and" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "c[k] =1/(2*L)" "6#/&%\"cG6#%\"kG*&\"\"\"F)*&\"\"#F)%\"L GF)!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(phi(x)*exp(-k*Pi*i*x/L), x = -L .. L);" "6#-%$IntG6$*&-%$phiG6#%\"xG\"\"\"-%$expG6#,$*,%\"kGF+% #PiGF+%\"iGF+F*F+%\"LG!\"\"F5F+/F*;,$F4F5F4" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 59 "is the complex Fourier series coefficient for e ach integer " }{TEXT 280 1 "k" }{TEXT -1 7 ", then " }{XPPEDIT 18 0 "a bs(c[k]) -> 0" "6#f*6#-%$absG6#&%\"cG6#%\"kG7\"6$%)operatorG%&arrowG6 \"\"\"!F0F0F0" }{TEXT -1 5 " as " }{XPPEDIT 18 0 "k -> infinity" "6#f *6#%\"kG7\"6$%)operatorG%&arrowG6\"%)infinityGF*F*F*" }{TEXT -1 4 " or " }{XPPEDIT 18 0 "k -> -infinity" "6#f*6#%\"kG7\"6$%)operatorG%&arrow G6\",$%)infinityG!\"\"F*F*F*" }{TEXT -1 1 "." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 267 20 "____________________" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "First we consider a " }{TEXT 261 12 "special ca se" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{TEXT 278 1 " a" }{TEXT -1 5 " and " }{TEXT 279 1 "b" }{TEXT -1 27 " be two real num bers with " }{XPPEDIT 18 0 "0 <= a;" "6#1\"\"!%\"aG" }{XPPEDIT 18 0 " `` < b;" "6#2%!G%\"bG" }{XPPEDIT 18 0 "`` OF'F9F+7$$\"3K+++]o;BuF9F+7$$\"3w++ ]7hAq!)F9F+7$$\"3B,++v`G<()F9F+7$$\"3X,+D1]\"3/*F9F+7$$\"3o,+]PYMk$*F9 F+7$$\"3N-]7`%4h_*F9F+7$$\"3#>+](oU(yo*F9F+7$$\"3C-Dcwmvo(*F9F+7$$\"3Y ,]P%3R'\\)*F9F+7$$\"3!=](=#\\@0$**F9F+7$$\"3@+++!RS6+\"F/$\"\"\"F*7$$ \"32+++?O3J6F/F[p7$$\"3#*******\\o-h7F/F[p7$$\"31+++XHL#H\"F/F[p7$$\"3 ********R!ROK\"F/F[p7$$\"38++vjbYJ8F/F[p7$$\"31++](3#HR8F/F+7$$\"3)*** *\\7h=rM\"F/F+7$$\"3!******\\8X\\N\"F/F+7$$\"3)*****\\#=)fq8F/F+7$$\"3 1+++I7D'Q\"F/F+7$$\"37+++?M')[9F/F+7$$\"3>+++5cZ6:F/F+7$$\"32++vV8>D;F /F+7$$\"3'*****\\xq!*Qam%*\\F/F+7$$\"3b\\(oHd @D+&F/F[p7$$\"3T+v$fsx.,&F/F[p7$$\"3R]i!*yQB=]F/F[p7$$\"3O+](=.!4E]F/F [p7$$\"3J+D\"yL-=/&F/F[p7$$\"3E++vVY^d]F/F[p7$$\"3;+]ib#R*)3&F/F[p7$$ \"31++]nQO?^F/F[p7$$\"3%)***\\748K=&F/F[p7$$\"3k*****\\JigC&F/F[p7$$\" 3C*\\7GCnkF&F/F[p7$$\"3s**\\iq@(oI&F/F[p7$$\"31D\"yDStWJ&F/F[p7$$\"3_ \\7`MY2A`F/F[p7$$\"3'[P%[menH`F/F[p7$$\"3?+vV)4xsL&F/F+7$$\"3+]PMi&zCN &F/F+7$$\"3!)***\\i-#on`F/F+7$$\"3()**\\(=)=\\GaF/F+7$$\"3%*****\\PA&zh*F/F+7$$\"3F++]A_ER(*F/F+7$$ \"#5F*F+-%&COLORG6&%$RGBG$\"\"%!\"\"F*$\"\"*F[al-%%TEXTG6%7$F\\al$!\"( F)Q\"a6\"-%'COLOURG6&Fh`lF*F*F*-F_al6%7$$\"#:F[alFbalQ\"bFealFfal-F_al 6%7$$\"#ZF[alFbalQ%a+2LFealFfal-F_al6%7$$\"#dF[alFbalQ%b+2LFealFfal-F_ al6%7$$\"#()F[alFbalQ%a+4LFealFfal-F_al6%7$$\"#(*F[alFbalQ%b+4LFealFfa l-%*AXESTICKSG6$F*7#/F\\p%\"1G-%+AXESLABELSG6$Q\"xFealQ!Feal-%%VIEWG6$ ;F(Fc`l%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" }}{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 9 "Then the " }{TEXT 281 1 "k" }{TEXT -1 35 " th complex Fourier coefficient of " }{XPPEDIT 18 0 "phi[[a, b]](x);" "6#-&%$phiG6#7$%\"aG%\"bG6#%\"xG" }{TEXT -1 4 " is " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[k] =1/(2*L)" "6#/&%\"cG6#%\"kG*&\" \"\"F)*&\"\"#F)%\"LGF)!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(exp(- k*Pi*i*x/L),x = a .. b);" "6#-%$IntG6$-%$expG6#,$*,%\"kG\"\"\"%#PiGF,% \"iGF,%\"xGF,%\"LG!\"\"F1/F/;%\"aG%\"bG" }{TEXT -1 1 " " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/(-2*k*Pi*i);" "6#/%!G*&\" \"\"F&,$**\"\"#F&%\"kGF&%#PiGF&%\"iGF&!\"\"F-" }{TEXT -1 2 " " } {XPPEDIT 18 0 "exp(-k*Pi*i*x/L);" "6#-%$expG6#,$*,%\"kG\"\"\"%#PiGF)% \"iGF)%\"xGF)%\"LG!\"\"F." }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([ b, ``],[``, ``],[a, ``]);" "6#-%*PIECEWISEG6%7$%\"bG%!G7$F(F(7$%\"aGF( " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/(-2*k*Pi*i);" "6#/%!G*&\"\"\"F&, $**\"\"#F&%\"kGF&%#PiGF&%\"iGF&!\"\"F-" }{XPPEDIT 18 0 "``(exp(-k*Pi*i *b/L)-exp(-k*Pi*i*a/L));" "6#-%!G6#,&-%$expG6#,$*,%\"kG\"\"\"%#PiGF-% \"iGF-%\"bGF-%\"LG!\"\"F2F--F(6#,$*,F,F-F.F-F/F-%\"aGF-F1F2F2F2" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 6 "Hence " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "abs(c[k]) <= 1/(abs(k)*Pi);" "6#1-% $absG6#&%\"cG6#%\"kG*&\"\"\"F,*&-F%6#F*F,%#PiGF,!\"\"" }{TEXT -1 1 ", " }}{PARA 0 "" 0 "" {TEXT -1 8 "so that " }{XPPEDIT 18 0 "abs(c[k]) -> 0" "6#f*6#-%$absG6#&%\"cG6#%\"kG7\"6$%)operatorG%&arrowG6\"\"\"!F0F0F 0" }{TEXT -1 4 " as " }{XPPEDIT 18 0 "abs(k) -> infinity" "6#f*6#-%$ab sG6#%\"kG7\"6$%)operatorG%&arrowG6\"%)infinityGF-F-F-" }{TEXT -1 1 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "Now c onsider a " }{TEXT 261 7 "general" }{TEXT -1 45 " continuous complex v alued periodic function " }{XPPEDIT 18 0 "phi(x)" "6#-%$phiG6#%\"xG" } {TEXT -1 12 " defined on " }{XPPEDIT 18 0 "[-L,L]" "6#7$,$%\"LG!\"\"F% " }{TEXT -1 13 " with period " }{XPPEDIT 18 0 "2*L" "6#*&\"\"#\"\"\"% \"LGF%" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 19 "The restriction of " }{XPPEDIT 18 0 "phi(x)" "6#-%$phiG6#%\"xG" }{TEXT -1 17 " to the interval " }{XPPEDIT 18 0 "[-L,L]" "6#7$,$%\"LG!\"\"F%" }{TEXT -1 19 " is continuous and " }{XPPEDIT 18 0 "Limit(phi(x),x = -L,right) = Lim it(phi(x),x = L,left);" "6#/-%&LimitG6%-%$phiG6#%\"xG/F*,$%\"LG!\"\"%& rightG-F%6%-F(6#F*/F*F-%%leftG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 88 "It is a result from basi c analysis that, on any closed interval, a continuous function " } {XPPEDIT 18 0 "phi(x)" "6#-%$phiG6#%\"xG" }{TEXT -1 13 " is actually \+ " }{TEXT 261 20 "uniformly continuous" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 58 "This means that if we are given a (small) positive num ber " }{XPPEDIT 18 0 "epsilon" "6#%(epsilonG" }{TEXT -1 25 ", we can f ind an integer " }{TEXT 284 1 "n" }{TEXT -1 47 " which is large enough so that for any numbers " }{TEXT 282 1 "x" }{TEXT -1 5 " and " } {TEXT 283 1 "y" }{TEXT -1 17 " in the interval " }{XPPEDIT 18 0 "[-L,L ]" "6#7$,$%\"LG!\"\"F%" }{TEXT -1 1 "," }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "abs(x-y)<2*L/n" "6#2-%$absG6#,&%\"xG\"\"\"%\"yG! \"\"*(\"\"#F)%\"LGF)%\"nGF+" }{TEXT -1 16 " implies that " } {XPPEDIT 18 0 "abs(phi(x)-phi(y)) < epsilon;" "6#2-%$absG6#,&-%$phiG6# %\"xG\"\"\"-F)6#%\"yG!\"\"%(epsilonG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "phi(x)" "6#-%$phiG6#%\"xG" } {TEXT -1 25 " is periodic with period " }{XPPEDIT 18 0 "2*L" "6#*&\"\" #\"\"\"%\"LGF%" }{TEXT -1 43 ", this holds for all pairs of real numbe rs " }{TEXT 285 1 "x" }{TEXT -1 5 " and " }{TEXT 286 1 "y" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "N ow consider the subdivision of " }{XPPEDIT 18 0 "[-L,L]" "6#7$,$%\"LG! \"\"F%" }{TEXT -1 46 " into intervals by the equally spaced points " }{XPPEDIT 18 0 "2*j*L/n-L;" "6#,&**\"\"#\"\"\"%\"jGF&%\"LGF&%\"nG!\"\" F&F(F*" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "j = 0,1,` . . . `,n;" "6&/% \"jG\"\"!\"\"\"%(~.~.~.~G%\"nG" }{TEXT -1 29 ", and construct the func tion " }{XPPEDIT 18 0 "s[n](x)" "6#-&%\"sG6#%\"nG6#%\"xG" }{TEXT -1 5 " by: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "s[n](x) = phi(2*j*L/n-L);" "6#/-&%\"sG6#%\"nG6#% \"xG-%$phiG6#,&**\"\"#\"\"\"%\"jGF1%\"LGF1F(!\"\"F1F3F4" }{TEXT -1 6 " if " }{XPPEDIT 18 0 "2*(j-1)*L/n-L <= x;" "6#1,&**\"\"#\"\"\",&%\"j GF'F'!\"\"F'%\"LGF'%\"nGF*F'F+F*%\"xG" }{XPPEDIT 18 0 "`` < 2*j*L/n-L; " "6#2%!G,&**\"\"#\"\"\"%\"jGF(%\"LGF(%\"nG!\"\"F(F*F," }{TEXT -1 4 ", " }{XPPEDIT 18 0 "j=1,` . . . `,n" "6%/%\"jG\"\"\"%(~.~.~.~G%\"nG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 5 "Then " }{XPPEDIT 18 0 " abs(s[n](x)-phi(x)) < epsilon;" "6#2-%$absG6#,&-&%\"sG6#%\"nG6#%\"xG\" \"\"-%$phiG6#F.!\"\"%(epsilonG" }{TEXT -1 17 " for all numbers " } {TEXT 287 1 "x" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {XPPEDIT 18 0 "s[n](x" "6#-&%\"sG6#%\"nG6#%\"xG" } {TEXT -1 62 " can be thought of as a \"stepped function\" which approx imates " }{XPPEDIT 18 0 "phi(x)" "6#-%$phiG6#%\"xG" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 7 "Example" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 4 "Let " } {XPPEDIT 18 0 "f(x);" "6#-%\"fG6#%\"xG" }{TEXT -1 28 " be defined on t he interval " }{XPPEDIT 18 0 "[-2,2]" "6#7$,$\"\"#!\"\"F%" }{TEXT -1 4 " by " }{XPPEDIT 18 0 "f(x) = PIECEWISE([4-x^2, -2 <= x and x < 0],[ 4-2*x, 0 <= x and x < 2]);" "6#/-%\"fG6#%\"xG-%*PIECEWISEG6$7$,&\"\"% \"\"\"*$F'\"\"#!\"\"31,$F0F1F'2F'\"\"!7$,&F-F.*&F0F.F'F.F131F6F'2F'F0 " }{TEXT -1 13 ", and extend " }{XPPEDIT 18 0 "f(x);" "6#-%\"fG6#%\"xG " }{TEXT -1 91 " to the whole of the real line by periodicity, so that the resulting function has period 4." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 106 "f := x -> piecewise(x<0 ,4-x^2,4-2*x):\n'f(x)'=f(x);\nf_ := x -> f(x-4*floor((x+2)/4)):\nplot( f_(x),x=-2..10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG-%*P IECEWISEG6$7$,&\"\"%\"\"\"*$)F'\"\"#F.!\"\"2F'\"\"!7$,&F-F.*&F1F.F'F.F 2%*otherwiseG" }}{PARA 13 "" 1 "" {GLPLOT2D 537 215 215 {PLOTDATA 2 "6 %-%'CURVESG6$7ar7$$!\"#\"\"!$F*F*7$$!3&*****\\P&3Y$>!#<$\"3G96Km!)*Gd# !#=7$$!3!******\\2<#p=F/$\"3[[WGl_Fg]F27$$!3%)****\\7c#Q!=F/$\"3g-+*of J@Y(F27$$!3,+++]TVQt #F/7$$!37******\\lfs**F2$\"39\"G40=ta+$F/7$$!3')****\\P\"40p)F2$\"3>!H 5$40vWKF/7$$!3i+++DiF2$\"3x$H1^`XJh $F/7$$!3&3++]U.6.&F2$\"3wK&pK)*zou$F/7$$!3\"3++]s:.!QF2$\"3+xJ!R+wb&QF 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0 "s(x)" "6#-%\"sG6#%\"xG" }{TEXT -1 37 " be a stepped function app roximating " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 35 " an d based on 20 intervals between " }{XPPEDIT 18 0 "-2" "6#,$\"\"#!\"\" " }{TEXT -1 7 " and 2." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 191 "n := 20: d := 4/n:\nf := x -> piec ewise(x<0,4-x^2,4-2*x):\nf_ := x -> f(x-4*floor((x+2)/4)):\ns := x -> \+ f_(d*ceil(x/d));\nplot([f_(x),s(x)],x=-2..10,color=[red,COLOR(RGB,.4,0 ,.9)],numpoints=100);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"sGf*6#%\" xG6\"6$%)operatorG%&arrowGF(-%#f_G6#*&%\"dG\"\"\"-%%ceilG6#*&9$F1F0!\" \"F1F(F(F(" }}{PARA 13 "" 1 "" {GLPLOT2D 658 253 253 {PLOTDATA 2 "6&-% 'CURVESG6$7\\s7$$!\"#\"\"!$F*F*7$$!3(pppp4!fO>!#<$\"3/'**)=jz='\\#!#=7 $$!3%RRRR>!=t=F/$\"3WY\"\\F27$$!3IOOOwv+==F/$\"3SpT%G_%[[pF27$$! 3myyye\\$Gw\"F/$\"3T#Q`23HT#*)F27$$!33+++!e(y++h$HF/7$$!3q4444,?%3*F2$\"31F*zP3tZ<$F/7$$!3T&ppp4K'=yF2$ 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\\KGT)F/Fcis7$$\"3eB@rB,)3U)F/Fcis7$F`_mFcis7$$\"31wvvN_j%[)F/Fcis7$Fe _mFcis7$$\"3wsssP#Qjb)F/Fcis7$$\"3cXXXbPLs&)F/Fcis7$$\"3)>==V^J.e)F/Fc is7$$\"3Q===t#H$)e)F/Fcis7$$\"3oNOh_\"GBf)F/Fcis7$$\"3yaa/KqK'f)F/Fcis 7$$\"3)QFx9\"fK+')F/Fgeq7$$\"3>\"444zCVg)F/Fgeq7$$\"3#ojjj#eJO')F/Fgeq 7$Fj_mFgeq7$$\"33777sLXG()F/Fgeq7$F_`mFgeq7$$\"3g>@'*HFK#z)F/Fgeq7$$\" 3E****\\xb/'z)F/Fgeq7$$\"3#*yy.D%o(*z)F/Fgeq7$$\"3fedds7\\.))F/F]`o7$$ \"38;:lnp$4\"))F/F]`o7$$\"3ntssiEQ=))F/F]`o7$$\"3v)yyG0uK$))F/F]`o7$$ \"3%QIIIWl\"[))F/F]`o7$$\"3+MLLB#[z())F/F]`o7$Fd`mF]`o7$$\"3FPOOwu`P*) F/F]`o7$$\"3Q544\\RMn*)F/F]`o7$$\"3#paaa=ZA)*)F/F]`o7$$\"3[$===U]r**)F /F]`o7$$\"3h#444Bw3+*F/$\"3'y************f\"F/7$$\"3v,++S?g/!*F/Fa_t7$ $\"3*3\"44\\yK3!*F/Fa_t7$$\"3.?==eO07!*F/Fa_t7$$\"3IQOOw_]>!*F/Fa_t7$F i`mFa_t7$$\"3)faaaKex3*F/Fa_t7$F^amFa_t7$$\"3!**344)=nk\"*F/Fa_t7$$\"3 UWXX0Sy!=*F/Fa_t7$$\"3orssn+%))=*F/Fa_t7$$\"3%*)*****Hh*o>*F/Fa_t7$$\" 3Zjj8hT#4?*F/Fb[r7$$\"3@EFF#>_\\?*F/Fb[r7$$\"3&*)34MA!)*3#*F/Fb[r7$$\" 3Z`aaa#3I@*F/Fb[r7$$\"3_ijj.DBX#*F/Fb[r7$FcamFb[r7$$\"3o`aa9(\\lL*F/Fb [r7$FhamFb[r7$$\"3U$R*oZ%)G*R*F/Fb[r7$$\"3.^^,>U$HS*F/$\"3cr********** **zF27$$\"3k34M!**zlS*F/Fjbt7$$\"3CmmmhdA5%*F/Fjbt7$$\"3Y\"==VIRR%*F/Fjbt7$$\"3addd<](RX*F/Fjbt 7$$\"3S===)=TJ[*F/Fjbt7$F]bmFjbt7$$\"3+8:::b[V&*F/Fjbt7$$\"3I]^^rOmu&* F/Fjbt7$$\"3ifgg5#eCe*F/Fjbt7$$\"3%*opp\\FD!f*F/Fjbt7$$\"3gBCC>+:%f*F/ Fjbt7$$\"3Eyyy)GZ!)f*F/Fjbt7$$\"3%HLL$eX%>g*F/Fe_r7$$\"3g(yyy#=%eg*F/F e_r7$$\"3D11114V@'*F/Fe_r7$FbbmFe_r7$$\"3Otss_0C*p*F/Fe_r7$FgbmFe_r7$$ \"3'******\\yAax*F/Fe_r7$$\"3%)zyy[WQ*y*F/Fe_r7$$\"3Y[[tk[(Gz*F/Fe_r7$ $\"3))==o!Gljz*F/Fe_r7$$\"3I*yGmpb)*z*F/Fe_r7$$\"3sfdd7hM.)*F/F+7$$\"3 x)ppW%pK5)*F/F+7$$\"3$yjjjx2t\")*F/F+7$$\"3/%RRR5J_%)*F/F+7$F\\cmF+7$$ \"3\"pddd@xl$**F/F+F`cm-%&COLORG6&Ffcm$F[gn!\"\"F*$\"\"*Feht-%+AXESLAB ELSG6$Q\"x6\"Q!F\\it-%%VIEWG6$;F(Facm%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 261 4 "Note" }{TEXT -1 15 " : The function " }{XPPEDIT 18 0 "s(x)" "6#-%\"sG6#%\"xG" }{TEXT -1 53 " could be used to approximate a definite integral of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 84 ", as a sum of areas of rectan gles, in the manner suggested by the following picture." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 321 "f := x -> piecewise(x<0,4-x^2,4-2*x):\nf_ := x -> f(x-4*floor((x+2)/4)):\np1 := plot(f_(x),x=-2..10,color=red):\nn := 20:\nd := 4/n:\nboxes := [se q([[-2+(j-1)*d,0],[-2+j*d,0],[-2+j*d,f_(-2+j*d)],\n [-2+(j-1)*d ,f_(-2+j*d)]],j=0..3*n)]:\np2 := plots[polygonplot](boxes,color=COLOR( RGB,0.65,0.6,1)):\nplots[display]([p1,p2]);" }}{PARA 13 "" 1 "" {GLPLOT2D 646 251 251 {PLOTDATA 2 "6&-%'CURVESG6$7ar7$$!\"#\"\"!$F*F*7 $$!3&*****\\P&3Y$>!#<$\"3G96Km!)*Gd#!#=7$$!3!******\\2<#p=F/$\"3[[WGl_ Fg]F27$$!3%)****\\7c#Q!=F/$\"3g-+*ofJ@Y(F27$$!3,+++]TVQt#F/7$$!37******\\lfs**F2$\"39\"G40 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++++)*Fj`mF+7$FcgnFdfm7$F]gnFdfm7&FbgnFi_mFi_mFbgn-%&COLORG6&F_`m$\"#l F)$Fe`nF_cmF]hm-%+AXESLABELSG6$Q\"x6\"Q!Fbhn-%%VIEWG6$;F(Fj_m%(DEFAULT G" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" " Curve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 "The maximum difference between " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6# %\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "s(x)" "6#-%\"sG6#%\"xG" } {TEXT -1 19 " over the interval " }{XPPEDIT 18 0 "-2<=x" "6#1,$\"\"#! \"\"%\"xG" }{XPPEDIT 18 0 "``<2" "6#2%!G\"\"#" }{TEXT -1 14 " occurs w here " }{XPPEDIT 18 0 "x = -9/5;" "6#/%\"xG,$*&\"\"*\"\"\"\"\"&!\"\"F* " }{XPPEDIT 18 0 "``=-1.8" "6#/%!G,$-%&FloatG6$\"#=!\"\"F*" }{TEXT -1 18 ", and is equal to " }{XPPEDIT 18 0 "f(-9/5) = 19/25;" "6#/-%\"fG6# ,$*&\"\"*\"\"\"\"\"&!\"\"F,*&\"#>F*\"#DF," }{XPPEDIT 18 0 "``=0" "6#/% !G\"\"!" }{TEXT -1 5 ".76. 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H(***\\(Q0u[Fhu7$$\"3=++]Z\"Rdy&F/$\"3I1,++0<_GFhu7$$\"3S*\\Pf&Qz!z&F/ $\"3$=E+D\")G7%=Fhu7$$\"3]**\\Pk&[ez&F/$\"37(R-+DrGI)Fhv7$$\"3i*\\7GF. 4!eF/$\"3c2+vVX$>)RF27$$\"3t***\\7)z&f!eF/$\"3M0++v.%3)QF27$$\"3'***\\ 7)Rng\"eF/$\"3!4++v._'yOF27$$\"3=+++:o/$F27$$\"3 !3](=i'o(eeF/$\"3$Q)*\\ivEY#GF27$$\"37++D6EjpeF/$\"3P(****\\xZtg#F27$$ \"3Y*\\7.c'\\!)eF/$\"3#4,]Pzo+R#F27$$\"3m**\\P40O\"*eF/$\"3o1+]7)*ys@F 27$$\"3))*\\P%eWA-fF/$\"3[-+DJ3^b>F27$$\"34++]2%)38fF/$\"3D)*****\\=BQ 4/(2-sqrt(4-epsilon))" "6#2*&\"\"%\"\"\",&\"\"#F&-%%sqrtG6#,&F%F &%(epsilonG!\"\"F.F.%\"nG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "ee := 19/25;\n4/(2 -sqrt(4-ee));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#eeG#\"#>\"#D" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"#?" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 3 "If " }{XPPEDIT 18 0 "n>=20" "6#1\"#?% \"nG" }{TEXT -1 7 ", then " }{XPPEDIT 18 0 "abs(s(x)-f(x)) <= 0;" "6#1 -%$absG6#,&-%\"sG6#%\"xG\"\"\"-%\"fG6#F+!\"\"\"\"!" }{TEXT -1 5 ".76. \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "For a nother example, if " }{XPPEDIT 18 0 "16*`.`*10^6 <= n;" "6#1*(\"#;\"\" \"%\".GF&\"#5\"\"'%\"nG" }{TEXT -1 7 ", then " }{XPPEDIT 18 0 "abs(s(x )-f(x)) <= 10^(-6);" "6#1-%$absG6#,&-%\"sG6#%\"xG\"\"\"-%\"fG6#F+!\"\" )\"#5,$\"\"'F0" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "ee := 10^(-6);\nevalf(4/(2-s qrt(4-ee)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#eeG#\"\"\"\"(+++\" " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+++++;!\"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "s[n,j] = p hi(2*j*L/n-L);" "6#/&%\"sG6$%\"nG%\"jG-%$phiG6#,&**\"\"#\"\"\"F(F/%\"L GF/F'!\"\"F/F0F1" }{TEXT -1 11 ", for each " }{TEXT 289 1 "j" }{TEXT -1 10 ", so that " }{XPPEDIT 18 0 "s[n](x)" "6#-&%\"sG6#%\"nG6#%\"xG" }{TEXT -1 23 " is a sum of \"pulses\": " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "s[n](x) = Sum(s[n,j]*phi[[2*(j-1)*L/n-L, 2*j*L/ n-L]],j = 1 .. n);" "6#/-&%\"sG6#%\"nG6#%\"xG-%$SumG6$*&&F&6$F(%\"jG\" \"\"&%$phiG6#7$,&**\"\"#F2,&F1F2F2!\"\"F2%\"LGF2F(F;F2Finfinity" " 6#f*6#%\"kG7\"6$%)operatorG%&arrowG6\"%)infinityGF*F*F*" }{TEXT -1 4 " or " }{XPPEDIT 18 0 "k -> -infinity" "6#f*6#%\"kG7\"6$%)operatorG%&ar rowG6\",$%)infinityG!\"\"F*F*F*" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 63 "Thus the Riemann - Lebesgue lemma holds for stepped funct ions. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "Now, for a general continuous function " }{XPPEDIT 18 0 "phi(x)" "6#- %$phiG6#%\"xG" }{TEXT -1 59 ", suppose that we are given a (small) pos itive real number " }{XPPEDIT 18 0 "epsilon" "6#%(epsilonG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 7 "Choose " }{XPPEDIT 18 0 "N[1]" "6# &%\"NG6#\"\"\"" }{TEXT -1 75 " to be a sufficiently large positive int eger so that the stepped functions " }{XPPEDIT 18 0 "s[n](x)" "6#-&%\" sG6#%\"nG6#%\"xG" }{TEXT -1 11 " satisfies " }{XPPEDIT 18 0 "abs(s[n]( x)-phi(x)) < epsilon/2;" "6#2-%$absG6#,&-&%\"sG6#%\"nG6#%\"xG\"\"\"-%$ phiG6#F.!\"\"*&%(epsilonGF/\"\"#F3" }{TEXT -1 7 " when " }{XPPEDIT 18 0 "N[1] <= n;" "6#1&%\"NG6#\"\"\"%\"nG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 31 "If the Fourier coefficients of " }{XPPEDIT 18 0 " s[n](x)" "6#-&%\"sG6#%\"nG6#%\"xG" }{TEXT -1 5 " are " }{XPPEDIT 18 0 "d[k]" "6#&%\"dG6#%\"kG" }{TEXT -1 21 ", we can also choose " } {XPPEDIT 18 0 "N[2]" "6#&%\"NG6#\"\"#" }{TEXT -1 9 " so that " } {XPPEDIT 18 0 "abs(d[k]) 0" "6#f*6#-%$absG6#&%\"cG6#%\"k G7\"6$%)operatorG%&arrowG6\"\"\"!F0F0F0" }{TEXT -1 4 " as " }{XPPEDIT 18 0 "k -> infinity" "6#f*6#%\"kG7\"6$%)operatorG%&arrowG6\"%)infinity GF*F*F*" }{TEXT -1 4 " or " }{XPPEDIT 18 0 "k -> -infinity" "6#f*6#%\" kG7\"6$%)operatorG%&arrowG6\",$%)infinityG!\"\"F*F*F*" }{TEXT -1 1 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 16 "code for \+ picture" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 29 "code for narrow pulse picture" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 326 "fr \+ := x -> piecewise(x<4/3 and x>= 1,1,0):\nf := x -> fr(x-4*floor((x+2)/ 4)):\np1 := plot(f(x),x=-2..10,color=COLOR(RGB,.4,0,.9),xtickmarks=0, \nytickmarks=[1=`1`]):\nt1 := plots[textplot]([[.9,-.07,`a`],[1.5,-.07 ,`b`],[4.7,-.07,`a+2L`],\n[5.7,-.07,`b+2L`],[8.7,-.07,`a+4L`],[9.7,-.0 7,`b+4L`]],color=black):\nplots[display]([p1,t1]);" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "4 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }