{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 12 102 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 18 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 " " 259 266 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 259 267 "" 0 1 0 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 0 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 270 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 271 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 272 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 275 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 276 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 277 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 278 "" 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 "" 260 288 "" 1 24 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 289 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 128 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 "Time s" 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 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 13 "Distributions" }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nanaimo, B.C., Canada" }}{PARA 0 "" 0 "" {TEXT -1 19 "Version: 27.3.2007" }}{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 14 "Te st functions" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 2 "A " }{TEXT 261 13 "test function" }{TEXT -1 1 " " }{XPPEDIT 18 0 "xi(x);" "6#-%#xiG6# %\"xG" }{TEXT -1 155 " is a function which has derivatives of all orde rs, that is, an infinitely differentiable function, and which vanishes outside some finite closed interval " }{XPPEDIT 18 0 "[-a,a]" "6#7$,$ %\"aG!\"\"F%" }{TEXT -1 11 ", that is, " }{XPPEDIT 18 0 "xi(x) = 0;" " 6#/-%#xiG6#%\"xG\"\"!" }{TEXT -1 6 " when " }{XPPEDIT 18 0 "a < abs(x) ;" "6#2%\"aG-%$absG6#%\"xG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "Note that any test function " } {XPPEDIT 18 0 "xi(x);" "6#-%#xiG6#%\"xG" }{TEXT -1 18 " is automatical ly " }{TEXT 261 7 "bounded" }{TEXT -1 34 ", that is, there is a real n umber " }{TEXT 268 1 "B" }{TEXT -1 11 " such that " }{XPPEDIT 18 0 "ab s(xi(x)) < B;" "6#2-%$absG6#-%#xiG6#%\"xG%\"BG" }{TEXT -1 9 " for all \+ " }{TEXT 269 1 "x" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 8 "Examples" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 33 "An example of a test function is " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "phi[a](x) = PIECEWISE ([exp(-a^2/(a^2-x^2)), abs(x) < a],[0, a <= abs(x)]);" "6#/-&%$phiG6#% \"aG6#%\"xG-%*PIECEWISEG6$7$-%$expG6#,$*&F(\"\"#,&*$F(F4\"\"\"*$F*F4! \"\"F9F92-%$absG6#F*F(7$\"\"!1F(-F<6#F*" }{TEXT -1 2 " " }}{PARA 0 " " 0 "" {TEXT -1 20 "where a is positive." }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "phi[a](x)" "6#-&%$phiG6#%\"aG6#%\"xG" }{TEXT -1 68 " is infinite ly differentiable, for example, its first derivative is " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "phi[a]*`'`(x) = PIECEWISE([``(- 2*a^2*x/((a^2-x^2)^2))*exp(-a^2/(a^2-x^2)), abs(x) < a],[0, a <= abs(x )]);" "6#/*&&%$phiG6#%\"aG\"\"\"-%\"'G6#%\"xGF)-%*PIECEWISEG6$7$*&-%!G 6#,$**\"\"#F)*$F(F8F)F-F)*$,&*$F(F8F)*$F-F8!\"\"F8F>F>F)-%$expG6#,$*&F (F8,&*$F(F8F)*$F-F8F>F>F>F)2-%$absG6#F-F(7$\"\"!1F(-FI6#F-" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "F or any a, " }{XPPEDIT 18 0 "phi[a](0)=1/exp(1)" "6#/-&%$phiG6#%\"aG6# \"\"!*&\"\"\"F,-%$expG6#F,!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 13 "The function " }{XPPEDIT 18 0 "psi[a](x)=exp(1)*phi[a](x) " "6#/-&%$psiG6#%\"aG6#%\"xG*&-%$expG6#\"\"\"F/-&%$phiG6#F(6#F*F/" } {TEXT -1 59 " is also a test function, with the additional proprty tha t " }{XPPEDIT 18 0 "psi[a](0)=1" "6#/-&%$psiG6#%\"aG6#\"\"!\"\"\"" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "The following graph shows " }{XPPEDIT 18 0 "psi[2](x);" " 6#-&%$psiG6#\"\"#6#%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "psi[2]*` '`(x);" "6#*&&%$psiG6#\"\"#\"\"\"-%\"'G6#%\"xGF(" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 206 "psi := x -> piecewise(abs(x)<2,exp(-x^2/(4-x^2))):\n'psi(x)'=psi( x);\npsi1 := x -> piecewise(abs(x)<2,-8*x/(4-x^2)^2*exp(-x^2/(4-x^2))) :\n'psi1(x)'=psi1(x);\nplot([psi(x),psi1(x)],x=-2.5..2.5,color=[red,bl ue]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$psiG6#%\"xG-%*PIECEWISEG6 $7$-%$expG6#,$*&F'\"\"#,&\"\"%\"\"\"*$)F'F1F4!\"\"F7F72-%$absGF&F17$\" \"!%*otherwiseG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%psi1G6#%\"xG-%* PIECEWISEG6$7$,$**\"\")\"\"\"F'F/,&\"\"%F/*$)F'\"\"#F/!\"\"!\"#-%$expG 6#,$*&F'F4F0F5F5F/F52-%$absGF&F47$\"\"!%*otherwiseG" }}{PARA 13 "" 1 " " {GLPLOT2D 481 328 328 {PLOTDATA 2 "6&-%'CURVESG6$7eo7$$!3++++++++D!# <$\"\"!F,7$$!3smm;HU,\"R#F*F+7$$!3=L$3FH'='H#F*F+7$$!3gmmTgBa*=#F*F+7$ $!3wmm\"H_\">#3#F*F+7$$!3ML$3_!4Nv>F*$\"376qR'Gk63&!#N7$$!3))**\\iSM#e #>F*$\"3ed_*pCGI%H!#B7$$!3km;/wfHw=F*$\"3'odq]!***)yk!#@7$$!3ILL$3FvM' =F*$\"3kpP&G.!)HQ\"!#?7$$!3?+]ilXl]=F*$\"3#)o6x;]L\"f#FP7$$!3&om;/'Q$y $=F*$\"3eGppUs\"fR%FP7$$!3_L$3_:8]#=F*$\"3eF.,VFc+pFP7$$!3%om\"zW7$$!3;+]PM.ttP:F_p7$$!3%)**\\(oWB>c\"F*$\"3W9@]9n6%4#F_p7$$!3]mTNrNa2:F* $\"3-:F%Q:nFo#F_p7$$!3;LL$epjJX\"F*$\"3!\\ake*4QoKF_p7$$!3u**\\(=(eE09 F*$\"3w()**o[vfrPF_p7$$!3amm\"z/otN\"F*$\"3cAfj**oGdUF_p7$$!3@L$3FWYMI \"F*$\"3CIKN'>***yZF_p7$$!3))****\\P[_\\7F*$\"3'R=n9SI\"F*$\"3TIFImGEOdF_p7$$!3#*****\\7)Q79\"F*$\"3;N>0moWqhF_p7$$ !3'*****\\i^)o.\"F*$\"3p.-#*><1CpF_p7$$!3vlmT50A@%*F_p$\"3VIpXgQz=vF_p 7$$!3OKLLeaR%H)F_p$\"3X0m6LpNC\")F_p7$$!3kJLLLo#)RtF_p$\"3;*)*eBYE'e&) F_p7$$!3f***\\PfO%HiF_p$\"3Y]yivVK\")*)F_p7$$!3/MLL$3`lC&F_p$\"3kD_sLG j(G*F_p7$$!3q'**\\P4u\"oTF_p$\"3/ype]14c&*F_p7$$!3*z**\\7G-89$F_p$\"3+ VEfPxB](*F_p7$$!3%)GL$3Fp)p?F_p$\"3w%)o>r`J#*)*F_p7$$!3YKL3-$ff3\"F_p$ \"39zVs0QZq**F_p7$$!3)Grmm\"z%zY#FP$\"2)362tZ)*****F*7$$\"3!fL$e*)>px5 F_p$\"3Seqg0B#4(**F_p7$$\"3w++v$f4t.#F_p$\"3wpvs;Sp&*)*F_p7$$\"3OPL$e* GstIF_p$\"3ADQ!=[%*4w*F_p7$$\"3Y+++]#RW9%F_p$\"3]i'z`lS7c*F_p7$$\"3Y,+ ]7j#>>&F_p$\"3$*>SaD7*GI*F_p7$$\"3t-+D1RU0iF_p$\"3g)=1FyV&*)*)F_p7$$\" 3+++](=S2L(F_p$\"3?ln`I^Vi&)F_p7$$\"3:jmm;p)=M)F_p$\"3cVZ5Q&G45)F_p7$$ \"3O-++v=]@%*F_p$\"3lnX')f#H'=vF_p7$$\"35L$e*[$z*R5F*$\"3[!eaI^UK!pF_p 7$$\"3e++]iC$p9\"F*$\"3R*\\*44L?EhF_p7$$\"3_Le*[t\\s>\"F*$\"342!Q8*Qu? 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Hence " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG6#%\"xG" }{TEXT -1 25 " is also a test function." }}{PARA 0 "" 0 "" {TEXT -1 22 "If we take a value of " }{TEXT 271 1 "a" }{TEXT -1 28 " with large magnitud e, then " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG6#%\"xG" }{TEXT -1 36 " will \+ have values close to those of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG " }{TEXT -1 6 " when " }{TEXT 272 1 "x" }{TEXT -1 11 " is near 0." }} {PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 142 "phi := x -> piecewise(abs(x)<30,exp(-x^2/(900-x^2))):\n'phi'(x) =phi(x);\ng := x -> phi(x)*cos(x):\n'g(x)'='phi(x)'*cos(x);\nplot(g(x) ,x=-30..30);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$phiG6#%\"xG-%*PI ECEWISEG6$7$-%$expG6#,$*&F'\"\"#,&\"$+*\"\"\"*$)F'F1F4!\"\"F7F72-%$abs GF&\"#I7$\"\"!%*otherwiseG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6 #%\"xG*&-%$phiGF&\"\"\"-%$cosGF&F+" }}{PARA 13 "" 1 "" {GLPLOT2D 550 196 196 {PLOTDATA 2 "6%-%'CURVESG6$7iz7$$!#I\"\"!$F*F*7$$!3!******\\2< #pG!#;$!3,!y=B\"pu4?!#A7$$!38+]78.K7GF/$!3Ei?H%3%*y,(!#@7$$!3+++D^NUbF F/$!3%3$olU.%oT$!#?7$$!3***\\i:PIMs#F/$!3Ua#\\boG/n%F>7$$!3'***\\(=>P9 p#F/$!3!fd=)QLI$Q$F>7$$!39]7.-1WvEF/$!36,Z.4qzV5F>7$$!3'**\\(=7SWfEF/$ \"3)[](oq3+vFF>7$$!3x\\PMAuWVEF/$\"3?EX)pE&4\\#)F>7$$!3%*****\\K3XFEF/ $\"3U.C(41$GX:!#>7$$!3w******H./jDF/$\"3=s4/Vx\"4*eFgn7$$!3%*****\\F)H ')\\#F/$\"30?;&z:,w-\"!#=7$$!3C]P%[t.E[#F/$\"3q=RmmB`&3\"Fbo7$$!3=+v=U wdmCF/$\"35.;$G^0%36Fbo7$$!37]7`\\:b]CF/$\"3%*=u)G%z0!4\"Fbo7$$!31+](o XDXV#F/$\"34HpNp\")eD5Fbo7$$!3&**\\i:FtCS#F/$\"31EzK3nxouFgn7$$!3%)*** \\i3@/P#F/$\"3[-ln%=D4p#Fgn7$$!3')***\\(G\"))4J#F/$!3*y\")[(H[+:5Fbo7$ $!3!****\\7#F/$!3g]t\"Gv!3#=$Fbo7$$!37+++S4ju@F/$!3.;&HO4Dh?$Fbo7$ $!3/++v$4Y#f@F/$!3'ep*3LQwXJFbo7$$!3%*****\\Z7'Q9#F/$!3O'e*oM\\f(*HFbo 7$$!3%)***\\7Sw%G@F/$!3+Xy\\7QogFFbo7$$!3%)***\\7Mam4#F/$!33@&=2_W#**> Fbo7$$!3#)***\\7GK[1#F/$!3w`*yFQPr=*Fgn7$$!3#)***\\7A5I.#F/$\"3%e(fnWL 7bQFgn7$$!3\")***\\7;)=,?F/$\"3OZ%R4*yq\"y\"Fbo7$$!3')*****\\:o%p>F/$ \"3vX-RYF/$\"37;'Gc$>?AUFbo7$$!3&*****\\U\" Gg!>F/$\"3Iq8Zu4]p\\Fbo7$$!3+++DO\"3V(=F/$\"3F$H(*eh`HnS/r4%Fbo7$$!3I+DJg VUwFFbo7$$!3:+++NkzVjo\"F/$!3)Hc8*QapVDFbo7$$!3++](=N!ed;F/$! 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>$RFbo7$$\"3q+++b*=jP\"F/$\"3I+(3xQO$)z#Fbo7$$\"3g](=n899R\"F/$\"3Y.9+ y:f\"o\"Fbo7$$\"3^+vV=$4lS\"F/$\"3V>C?ks1LaFgn7$$\"3U]i:+Xg@9F/$!3ULd& yvU&)*eFgn7$$\"3L+](=o*pO9F/$!37I-y*4J[\"F/$!3#Q( *\\le%GoXFbo7$$\"3%****\\(3/3(\\\"F/$!3A,%[\"*)zA9`Fbo7$$\"3.++]P!Q'G: F/$!35]?]5PWDkFbo7$$\"35++Dmc>g:F/$!3%zeS$>$yM'oFbo7$$\"3=+++&H`dFbo7$$\"35+D1uU;a;F/$! 3yNtCGciUVFbo7$$\"3#***\\PCw,&o\"F/$!3*>7be6E7i#Fbo7$$\"3$)\\7`*HW/q\" F/$!3/^f8(>&Q(o\"Fbo7$$\"3u*\\(ou4(er\"F/$!3ck;A]^)\\O(Fgn7$$\"3m\\P%) \\wHJ7\"Fbo7$$\" 3q*\\PM#\\-zF/$\"3u]rNA.e_\\Fbo7$$\"39+]P\\`9Q>F/$\"3S+bvqv E5UFbo7$$\"3=+vokYDp>F/$\"3QX[8jG1?JFbo7$$\"3?+++!)RO+?F/$\"3'ol!**=Xt <=Fbo7$$\"3D+]7[x6uAu>Fbo7$$\"3T++]_!>w7#F/$!3)*fPyr>*[u# Fbo7$$\"3S]7`>URV@F/$!3]+]%fQ2<*HFbo7$$\"3S+Dc'Qp\"f@F/$!39'>zSrW_9$Fb o7$$\"3S]Pf`X%\\<#F/$!3o'Gl&3@Y1KFbo7$$\"3Q+]i?(>2>#F/$!3KYi)G]X!zJFbo 7$$\"3O+voa+FAAF/$!3ML'e)oDD&)GFbo7$$\"3O++v)Q?QD#F/$!3g9kv!p1vL#Fbo7$ $\"3]+]P\\L!=J#F/$!3)o%\\mF_/a**Fgn7$$\"3k+++5jypBF/$\"3)ysQYW'Rqv5Fbo7$$\"3`++]Ujp-DF/$\"3Pb%>M\"Q:35Fb o7$$\"35++D,X8iDF/$\"3#H5FPJ#fhfFgn7$$\"3l******fEd@EF/$\"3KxV/cv\"H&= Fgn7$$\"3$*\\PfGgTPEF/$\"3/,'[Wh\"=w5Fgn7$$\"3')*\\(=(RfKl#F/$\"3?-Ilk ]b)o%F>7$$\"3!)\\7ylF5pEF/$\"3SEwJX5([!GF87$$\"33+]PMh%\\o#F/$!3)H382E *f(f#F>7$$\"3H+DcrGj;FF/$!3#*[bVx$\\+p%F>7$$\"3]++v3'>$[FF/$!3Qt)>k'[= )z$F>7$$\"3S+++5h(*3GF/$!3&=&>b*>W]-)F87$$\"3K++D6EjpGF/$!3Bq(=/LXS$>F 27$$\"#IF*F+-%'COLOURG6&%$RGBG$\"#5!\"\"F+F+-%+AXESLABELSG6$Q\"x6\"Q!F b[p-%%VIEWG6$;F(Fejo%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "The sum " }{XPPEDIT 18 0 "xi+eta ;" "6#,&%#xiG\"\"\"%$etaGF%" }{TEXT -1 23 " of two test functions " } {XPPEDIT 18 0 "xi;" "6#%#xiG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "eta; " "6#%$etaG" }{TEXT -1 24 " defined \"pointwise\" by " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``(xi+eta)(x) = xi(x)+eta(x);" "6# /--%!G6#,&%#xiG\"\"\"%$etaGF*6#%\"xG,&-F)6#F-F*-F+6#F-F*" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 41 "is also a test function, and the pr oduct " }{XPPEDIT 18 0 "r*xi;" "6#*&%\"rG\"\"\"%#xiGF%" }{TEXT -1 8 ", where " }{TEXT 270 1 "r" }{TEXT -1 34 " is a real number, and defined by " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(r*phi)(x) = \+ r*``(phi(x));" "6#/-*&%\"rG\"\"\"%$phiGF'6#%\"xG*&F&F'-%!G6#-F(6#F*F' " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 19 "is a test function." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 "The set of all test functions T has the structure of a " }{TEXT 261 12 "vecto r space" }{TEXT -1 85 " with the addition of functions (vectors) and s calar multiplication defined as above." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 16 "Weak convergence" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 50 "A sequence of infin itely differentiable functions " }{XPPEDIT 18 0 "g[n](x);" "6#-&%\"gG6 #%\"nG6#%\"xG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "n = 0, 1, 2, ` . . . ` " "6&/%\"nG\"\"!\"\"\"\"\"#%(~.~.~.~G" }{TEXT -1 17 ", is said to be \+ " }{TEXT 261 17 "weakly convergent" }{TEXT -1 26 " if for any test fun ction " }{XPPEDIT 18 0 "xi(x);" "6#-%#xiG6#%\"xG" }{TEXT -1 30 ", the \+ sequence of real numbers" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Int(g[n](x)*xi(x),x = -infinity .. infinity);" "6#-%$In tG6$*&-&%\"gG6#%\"nG6#%\"xG\"\"\"-%#xiG6#F-F./F-;,$%)infinityG!\"\"F5 " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 10 "converges." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 "For example, fo r any the sequence of functions " }{XPPEDIT 18 0 "g[n](x) = n/(Pi*(1+n ^2*x^2));" "6#/-&%\"gG6#%\"nG6#%\"xG*&F(\"\"\"*&%#PiGF,,&F,F,*&F(\"\"# F*F1F,F,!\"\"" }{TEXT -1 51 ", converges weakly, because, for any test function " }{XPPEDIT 18 0 "xi" "6#%#xiG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "Int(g[n](x)*xi(x),x = -infinity .. infinity);" "6#-%$IntG6$*&-&% \"gG6#%\"nG6#%\"xG\"\"\"-%#xiG6#F-F./F-;,$%)infinityG!\"\"F5" }{TEXT -1 17 " converges to to " }{XPPEDIT 18 0 "xi(0);" "6#-%#xiG6#\"\"!" } {TEXT -1 49 ", by the sampling property of the delta function " } {XPPEDIT 18 0 "delta(x)" "6#-%&deltaG6#%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "Note that weak \+ convergence does " }{TEXT 260 3 "not" }{TEXT -1 55 " imply pointwise c onvergence, as can be seen by taking " }{XPPEDIT 18 0 "x = 0" "6#/%\"x G\"\"!" }{TEXT -1 30 " in the sequence of functions " }{XPPEDIT 18 0 " g[n](x) = n/(Pi*(1+n^2*x^2));" "6#/-&%\"gG6#%\"nG6#%\"xG*&F(\"\"\"*&%# PiGF,,&F,F,*&F(\"\"#F*F1F,F,!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 13 "In this case " }{XPPEDIT 18 0 "Limit(g[n](0),n = infinity ) = Limit(n/Pi,n = infinity);" "6#/-%&LimitG6$-&%\"gG6#%\"nG6#\"\"!/F+ %)infinityG-F%6$*&F+\"\"\"%#PiG!\"\"/F+F/" }{TEXT -1 3 " = " } {XPPEDIT 18 0 "infinity" "6#%)infinityG" }{TEXT -1 1 "." }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{TEXT 264 26 "__________________________" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 65 "Suppose that the sequence of infinitely differentiable fu nctions " }{XPPEDIT 18 0 "g[n](x);" "6#-&%\"gG6#%\"nG6#%\"xG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "n = 0, 1, 2, ` . . . `" "6&/%\"nG\"\"!\"\"\" \"\"#%(~.~.~.~G" }{TEXT -1 37 ", converges to a continuous function " }{XPPEDIT 18 0 "g(x);" "6#-%\"gG6#%\"xG" }{TEXT -1 62 ", such that the convergence is uniform on any closed interval " }{XPPEDIT 18 0 "[-a,a ]" "6#7$,$%\"aG!\"\"F%" }{TEXT -1 33 ". Then the sequence of functions " }{XPPEDIT 18 0 "g[n](x);" "6#-&%\"gG6#%\"nG6#%\"xG" }{TEXT -1 22 " \+ converges weakly to " }{XPPEDIT 18 0 "Int(g(x),x = -infinity .. infin ity);" "6#-%$IntG6$-%\"gG6#%\"xG/F);,$%)infinityG!\"\"F-" }{TEXT -1 1 "." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 263 26 "________________ __________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "To see why this is true let " }{XPPEDIT 18 0 "xi( x);" "6#-%#xiG6#%\"xG" }{TEXT -1 30 " be a test function such that " } {XPPEDIT 18 0 "xi(x) = 0;" "6#/-%#xiG6#%\"xG\"\"!" }{TEXT -1 6 " when \+ " }{XPPEDIT 18 0 "abs(x)>=a" "6#1%\"aG-%$absG6#%\"xG" }{TEXT -1 16 ", \+ and such that " }{XPPEDIT 18 0 "abs(xi(x)) < B;" "6#2-%$absG6#-%#xiG6# %\"xG%\"BG" }{TEXT -1 6 " when " }{XPPEDIT 18 0 "abs(x)N" "6#2%\"NG%\"nG" }{TEXT -1 9 " and all " }{TEXT 275 1 "x" }{TEXT -1 6 " with " }{XPPEDIT 18 0 "abs(x) <= a;" "6#1-%$absG6#%\"xG%\"aG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 4 "Then" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "abs(Int(g[n](x)*xi(x),x = -infinity .. \+ infinity)-Int(g(x)*xi(x),x = -infinity .. infinity)) = abs(Int((g[n](x )-g(x))*xi(x),x = -infinity .. infinity));" "6#/-%$absG6#,&-%$IntG6$*& -&%\"gG6#%\"nG6#%\"xG\"\"\"-%#xiG6#F2F3/F2;,$%)infinityG!\"\"F:F3-F)6$ *&-F.6#F2F3-F56#F2F3/F2;,$F:F;F:F;-F%6#-F)6$*&,&-&F.6#F06#F2F3-F.6#F2F ;F3-F56#F2F3/F2;,$F:F;F:" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = abs(Int((g[n](x)-g(x))*xi(x),x = -a .. a ));" "6#/%!G-%$absG6#-%$IntG6$*&,&-&%\"gG6#%\"nG6#%\"xG\"\"\"-F/6#F3! \"\"F4-%#xiG6#F3F4/F3;,$%\"aGF7F>" }{TEXT -1 2 " " }}{PARA 256 "" 0 " " {TEXT -1 2 " " }{XPPEDIT 18 0 "`` <= Int(abs(g[n](x)-g(x))*abs(xi(x )),x = -a .. a);" "6#1%!G-%$IntG6$*&-%$absG6#,&-&%\"gG6#%\"nG6#%\"xG\" \"\"-F/6#F3!\"\"F4-F*6#-%#xiG6#F3F4/F3;,$%\"aGF7F@" }{TEXT -1 2 " " } }{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "`` <= Int(``(epsilo n/(2*a*B))*B,x = -a .. a);" "6#1%!G-%$IntG6$*&-F$6#*&%(epsilonG\"\"\"* (\"\"#F-%\"aGF-%\"BGF-!\"\"F-F1F-/%\"xG;,$F0F2F0" }{XPPEDIT 18 0 "`` = epsilon;" "6#/%!G%(epsilonG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Hence " }{XPPEDIT 18 0 "Int(g[n](x)*xi(x),x = -infinity .. infinity);" "6#-%$IntG6$*&-&%\"gG6#%\"nG6#%\"xG\"\"\"-%#xiG6#F-F./F-; ,$%)infinityG!\"\"F5" }{TEXT -1 14 " converges to " }{XPPEDIT 18 0 "In t(g(x)*xi(x),x = -infinity .. infinity);" "6#-%$IntG6$*&-%\"gG6#%\"xG \"\"\"-%#xiG6#F*F+/F*;,$%)infinityG!\"\"F2" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 30 "Construction of test function s" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 44 "Given an infinitely differentiable function " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 28 ", and positive real number s " }{TEXT 276 1 "a" }{TEXT -1 5 " and " }{TEXT 277 1 "b" }{TEXT -1 6 " with " }{XPPEDIT 18 0 "ab" "6#2%\"bG-%$absG6#%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 13 "" 1 "" {GLPLOT2D 455 227 227 {PLOTDATA 2 "6(-%'CURVESG6%7a]l7$$!#7\"\"!$\"1&y ;t\\I'QZ!#:7$$!1+](oUIn>\"!#9$\"1Qn9?S)Rs%F-7$$!1++v`3Y$>\"F1$\"15dB'> =fr%F-7$$!1+]i!G\">!>\"F1$\"1W@9PJ]9ZF-7$$!1++]2<#p=\"F1$\"1rt'Q@J'>ZF -7$$!1++DhDQ!=\"F1$\"1P[ZE%>#[ZF-7$$!1+++:M%Q<\"F1$\"1`$4I#3\\(z%F-7$$ !1++]A^wg6F1$\"1E\\W^I+J\\F-7$$!1+++IooZ6F1$\"1pF-7***e0&F-7$$!1+D\"Q: (*>9\"F1$\"1O9C^9M\"4&F-7$$!1+]ixuIO6F1$\"1SN4ySh4^F-7$$!1+vV,yhI6F1$ \"1HJ%>d^%3^F-7$$!1++DD\"G\\7\"F1$\"1%o>F4Yr3&F-7$$!1+](Gx[N6\"F1$\"12 ,([3y#*)\\F-7$$!1++]?%p@5\"F1$\"1kSm'['[S[F-7$$!1+]i[@P*3\"F1$\"1oiSI2 )Gm%F-7$$!1++vw[dw5F1$\"1ZU&)z&GK`%F-7$$!1]7yebPt5F1$\"1@Y&4RTR^%F-7$$ !1+D\"3Cw,2\"F1$\"1U#)=uJ!3]%F-7$$!1]P%G#p(p1\"F1$\"1;3Fy\")*R\\%F-7$$ !1+]([gxP1\"F1$\"1HPM!\\[N\\%F-7$$!1]i!pGy01\"F1$\"1=!y$pMJ*\\%F-7$$!1 +v$*o*yt0\"F1$\"1O\"\\ZL$*4^%F-7$$!1](o4lzT0\"F1$\"1Q&\\$=o8GXF-7$$!1+ ++L.)40\"F1$\"1@a%[o_,b%F-7$$!1++]K#)4Q5F1$\"1Ox&\\&H/sYF-7$$!1+++Kh@D 5F1$\"1+?Y!*R]+[F-7$$!1++v\"3v(=5F1$\"15-K_tSZ[F-7$$!1++]JSL75F1$\"1Cw '\\OOT([F-7$$!1+]P1N645F1$\"1P!e\"z)y&y[F-7$$!1++D\")H*e+\"F1$\"15T>$) )\\m([F-7$$!1+]7cCn-5F1$\"1gO4m`Bo[F-7$$!1******4$>X***F-$\"1Oyt.')Q`[ F-7$$!1++]F=5Q(*F-$\"1Ru.)3Q:c%F-7$$!1+++XVo\"[*F-$\"1c0yLu$)pUF-7$$!1 ,+Dmy'>X*F-$\"1P,N*p$)eD%F-7$$!1,+](Q^AU*F-$\"1oi*=0ptC%F-7$$!1,+v3\\` #R*F-$\"1fF^mPMWUF-7$$!1,++I%=GO*F-$\"1lz-HntYUF-7$$!1++]saQ.$*F-$\"15 !Q?-)*oE%F-7$$!1+++:D&RC*F-$\"1VH9'*3)\\I%F-7$$!1*******f'3D\"*F-$\"1 \"=\\clBfT%F-7$$!1,++&o?i+*F-$\"130#G%etKXF-7$$!1*******H\"oW*)F-$\"1$ ))Gdep*yXF-7$$!1*****\\\">9$)))F-$\"1`g%G6)y2YF-7$$!1+++IDg@))F-$\"1vE VRB?:YF-7$$!1+++XJ1g()F-$\"1.?(4/D#*f%F-7$$!1,++vV)pj)F-$\"1p2F%**p+]% F-7$$!1+++0c!R^)F-$\"1P+e'\\XkL%F-7$$!1*****\\O+FQc!*RF-7$$!1, ++0]o&>)F-$\"1xt/Y2OuRF-7$$!1-++XH'Q;)F-$\"1RO'yq%QkRF-7$$!1+++&)3/K\" )F-$\"1e,`_QpgRF-7$$!1*****\\#)=-5)F-$\"1`L%H!o=jRF-7$$!1*****\\w'Ro!) 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%+AXESLABELSG6$Q\"x6\"%!G-%%VIEWG6$;F(F[gp%(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" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 23 "Explanation and example" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG6#%\"xG" }{TEXT -1 25 " \+ can be constructed from " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" } {TEXT -1 58 " by making use of the infinitely differentiable functions " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "h[n](x)=PIECEWIS E([0 , x<=-1/n],[(1+tanh(n*x/(1-n^2*x^2)))/2 , -1/n=1/n])" "6#/-&%\"hG6#%\"nG6#%\"xG-%*PIECEWISEG6%7$\"\"!1F*,$*&\"\"\" F3F(!\"\"F47$*&,&F3F3-%%tanhG6#*(F(F3F*F3,&F3F3*&F(\"\"#F*F>F4F4F3F3F> F432,$*&F3F3F(F4F4F*2F**&F3F3F(F47$F31*&F3F3F(F4F*" }{TEXT -1 2 ". " } }{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 154 "h := (n,x) ->\n piecewise(x<-1/n,0,abs(x)<1/n,(1+tanh(n*x/(1-n^ 2*x^2)))/2,1):\n'h(n,x)'=h(n,x);\nplot([h(1,x),h(2,x),h(3,x)],x=-3..3, color=[red,blue,green]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"hG6$% \"nG%\"xG-%*PIECEWISEG6%7$\"\"!2F(,$*&\"\"\"F1F'!\"\"F27$,&#F1\"\"#F1- %%tanhG6#*&*&F'F1F(F1F1,&F1F1*&)F'F6F1)F(F6F1F2F2F52-%$absG6#F(F07$F1% *otherwiseG" }}{PARA 13 "" 1 "" {GLPLOT2D 382 190 190 {PLOTDATA 2 "6'- %'CURVESG6$7]o7$$!\"$\"\"!F*7$$!1+++vq@pG!#:F*7$$!1++D^NUbFF.F*7$$!1++ ]K3XFEF.F*7$$!1++]F)H')\\#F.F*7$$!1++D'3@/P#F.F*7$$!1++Dr^b^AF.F*7$$!1 ++D,kZG@F.F*7$$!1++Dh\")=,?F.F*7$$!1++DO\"3V(=F.F*7$$!1+++NkzViUC\"F. 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Let " }{XPPEDIT 18 0 "d = (a+b)/2" "6#/%\"dG*&,&%\"aG\"\"\"%\"bGF(F(\"\"#!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 7 "Define " }{XPPEDIT 18 0 "g(x)" "6# -%\"gG6#%\"xG" }{TEXT -1 3 " by" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "g(x) = [h[n](x)(x+d)-h[n](x-d)]*f(x);" "6#/-%\"gG6#%\"x G*&7#,&--&%\"hG6#%\"nG6#F'6#,&F'\"\"\"%\"dGF4F4-&F.6#F06#,&F'F4F5!\"\" F;F4-%\"fG6#F'F4" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 16 "For example let " }{XPPEDIT 18 0 "f(x)=ln (1+x^2)+cos(5*x)/4" "6#/-%\"fG6#%\"xG,&-%#lnG6#,&\"\"\"F-*$F'\"\"#F-F- *&-%$cosG6#*&\"\"&F-F'F-F-\"\"%!\"\"F-" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a = 3*Pi" "6#/%\"aG*&\"\"$\"\"\"%#PiGF'" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "b=10" "6#/%\"bG\"#5" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 191 "c := evalf( (10-3*Pi)/2);\nn := ceil(1/c);\nd := evalf((10+3*Pi)/2);\nf := x ->ln( 1+x^2)+cos(5*x)/4;\ng := x -> (h(n,x+d)-h(n,x-d))*f(x);\nplot(g(x),x=- 11..11,thickness=1,color=COLOR(RGB,.4,0,.9));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"cG$\"*>5h(G!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%\"nG\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"dG$\"+\")*)Q7(*!\"* " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&a rrowGF(,&-%#lnG6#,&\"\"\"F1*$)9$\"\"#F1F1F1-%$cosG6#,$F4\"\"&#F1\"\"%F (F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%)operato rG%&arrowGF(*&,&-%\"hG6$%\"nG,&9$\"\"\"%\"dGF4F4-F/6$F1,&F3F4F5!\"\"F9 F4-%\"fG6#F3F4F(F(F(" }}{PARA 13 "" 1 "" {GLPLOT2D 401 161 161 {PLOTDATA 2 "6'-%'CURVESG6#7c[l7$$!#6\"\"!F*7$$!1LL$3EY?0\"!#9F*7$$!1+ D\"[6%=J5F.F*7$$!1n;zo>K55F.F*7$$!1Dc^_lX/5F.F*7$$!1MeRi8\"f)**!#:F*7$ $!1Ec,\"G%ec**F;F*7$$!1ds#**F;$\"1!>+]fsxm'!#=7$$!14_D=,$z*)*F;$ \"1mCc^bk!p\"!#;7$$!1,](o..'o)*F;$\"1F\\K8*z\\S&FJ7$$!1#z%\\bfFR)*F;$ \"1NL?.X4]&*FJ7$$!1%e9T()[*4)*F;$\"1i#*edASG8F;7$$!1wVt#z@1y*F;$\"1UV3 8Y\"*[;F;7$$!1mTN6ZH^(*F;$\"18jHF;7$$!1fR(*Hw'>s*F;$\"1$HiW#R.#=# F;7$$!1^Pf[0k#p*F;$\"1Kz[,%*HECF;7$$!1TN@nMJj'*F;$\"1yb+*R\"4wEF;7$$!1 LL$eQ')Rj*F;$\"1T:_6O[YHF;7$$!1K3_0\\Y/'*F;$\"1Y-(\\w(f`KF;7$$!1K$3_UV \\d*F;$\"1Uyypn\\,OF;7$$!1Ke*[%>UX&*F;$\"1,o=\"H*>)F;$ \"1lP4Rn)e(RF;7$$!1M$3-k&3r\")F;$\"1XhOWr4mRF;7$$!1,+v)3!)G9)F;$\"12Y! 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HJwmZ$F;7$$\"1mmm'3LCh*F;$\"1TxE3WimJF;7$$\"1v=Ui#z9k*F;$\"11\"p?F=Y(G F;7$$\"1%3x\"Qa_q'*F;$\"1vyaaML8EF;7$$\"1#HKRhr&*p*F;$\"1>joBF;7$$ \"1*\\(o*y<'G(*F;$\"1%)4gzc&e7#F;7$$\"11FWlRmd(*F;$\"1w:QN`?q=F;7$$\"1 :z>T,r'y*F;$\"1u$Q%['zie\"F;7$$\"1CJ&pJcd\")*F;$\"1dQZtG#)e7F;7$$\"1K$ 3F\\-[%)*F;$\"1NA\"y[**))y)FJ7$$\"1TNYo'[Q()*F;$\"1nBfh&3mm%FJ7$$\"1]( =U%[*G!**F;$\"1I+@@2))G7FJ7$$\"1aj4KzT<**F;$\"1#f*)eY+t<$Fjem7$$\"1fR( *>5%>$**F;$\"1#yUNF(\\BAFD7$$\"1j:&y5kk%**F;$\"1)e&z()esG7!#@7$$\"1n\" Hd>()4'**F;F*7$$\"1eRsaz!>+\"F.F*7$$\"1+]()*=_5F.F*7$$\"#6F*F*-%*THICKNESSG6#\"\"\"-%&COLORG6&%$RGB G$\"\"%!\"\"F*$\"\"*F[]p-%+AXESLABELSG6$Q\"x6\"%!G-%%VIEWG6$;F(F_\\p%( DEFAULTG" 1 2 0 1 10 1 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Cur ve 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 30 "The concept of a distribution " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 68 "A weakly convergent sequence of infinitely differentiable functions " }{XPPEDIT 18 0 "g[n](x);" "6#-&%\"gG6#%\"nG6#%\"xG" } {TEXT -1 2 ", " }{XPPEDIT 18 0 "n = 0, 1, 2, ` . . . `" "6&/%\"nG\"\"! \"\"\"\"\"#%(~.~.~.~G" }{TEXT -1 37 ", gives rise to a linear mapping, or " }{TEXT 261 10 "functional" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "mu" "6#%#muG" }{TEXT -1 71 " from the space of test functions T to the set of real numbers given by" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "mu(xi) = Limit(Int(g[n](x)*xi(x),x = -infinity .. infin ity),n = infinity);" "6#/-%#muG6#%#xiG-%&LimitG6$-%$IntG6$*&-&%\"gG6#% \"nG6#%\"xG\"\"\"-F'6#F5F6/F5;,$%)infinityG!\"\"F " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 29 "Operations with distributions" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 56 "Addition of distributions and multiplication by a scala r" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT 266 25 "Addition of distributions" }{TEXT -1 2 " " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "If " }{XPPEDIT 18 0 "g[n](x);" "6#-&%\"gG6#%\"nG6#%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "h[n](x);" "6#-&%\"hG6#%\"nG6#%\"xG" }{TEXT -1 107 " are two weak ly convergent sequences of infinitely differentiable functions, which \+ represent distributions " }{XPPEDIT 18 0 "G(x)" "6#-%\"GG6#%\"xG" } {TEXT -1 5 " and " }{XPPEDIT 18 0 "H(x)" "6#-%\"HG6#%\"xG" }{TEXT -1 43 " respectively, then, for any test function " }{XPPEDIT 18 0 "xi(x) ;" "6#-%#xiG6#%\"xG" }{TEXT -1 3 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Limit(Int((g[n](x)+h[n](x))*phi(x),x = -infinity . . infinity),n = infinity) = Limit(Int(g[n](x)*xi(x),x = -infinity .. i nfinity),n = infinity)+Limit(Int(h[n](x)*xi(x),x = -infinity .. infini ty),n = infinity);" "6#/-%&LimitG6$-%$IntG6$*&,&-&%\"gG6#%\"nG6#%\"xG \"\"\"-&%\"hG6#F06#F2F3F3-%$phiG6#F2F3/F2;,$%)infinityG!\"\"F?/F0F?,&- F%6$-F(6$*&-&F.6#F06#F2F3-%#xiG6#F2F3/F2;,$F?F@F?/F0F?F3-F%6$-F(6$*&-& F66#F06#F2F3-FM6#F2F3/F2;,$F?F@F?/F0F?F3" }{TEXT -1 2 " " }}{PARA 0 " " 0 "" {TEXT -1 8 "that is," }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Int((G(x)+H(x))*phi(x),x = -infinity .. infinity) = Int (G(x)*xi(x),x = -infinity .. infinity)+Int(H(x)*xi(x),x = -infinity .. infinity);" "6#/-%$IntG6$*&,&-%\"GG6#%\"xG\"\"\"-%\"HG6#F,F-F--%$phiG 6#F,F-/F,;,$%)infinityG!\"\"F7,&-F%6$*&-F*6#F,F--%#xiG6#F,F-/F,;,$F7F8 F7F--F%6$*&-F/6#F,F--F@6#F,F-/F,;,$F7F8F7F-" }{TEXT -1 3 ". " }} {PARA 0 "" 0 "" {TEXT -1 2 "or" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "(mu[G]+mu[F])(phi) = mu[G](phi)+mu[H](phi);" "6#/-,&&%# muG6#%\"GG\"\"\"&F'6#%\"FGF*6#%$phiG,&-&F'6#F)6#F/F*-&F'6#%\"HG6#F/F* " }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 267 44 "Multiplication of a distribution by a scalar" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "If " }{XPPEDIT 18 0 "g[n](x);" "6#-&%\"gG6#%\"nG6#%\"xG" }{TEXT -1 2 ", \+ " }{XPPEDIT 18 0 "n = 0, 1, 2, ` . . . `" "6&/%\"nG\"\"!\"\"\"\"\"#%(~ .~.~.~G" }{TEXT -1 106 ", is a weakly convergent sequence of infinitel y differentiable functions, which represents a distribution " } {XPPEDIT 18 0 "G(x)" "6#-%\"GG6#%\"xG" }{TEXT -1 6 ", and " }{TEXT 279 1 "r" }{TEXT -1 47 " is a real number, then, for any test function " }{XPPEDIT 18 0 "xi(x);" "6#-%#xiG6#%\"xG" }{TEXT -1 3 ", " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Limit(Int(r*g[n](x)*x i(x),x = -infinity .. infinity),n = infinity) = r*Limit(Int(g[n](x)*xi (x),x = -infinity .. infinity),n = infinity);" "6#/-%&LimitG6$-%$IntG6 $*(%\"rG\"\"\"-&%\"gG6#%\"nG6#%\"xGF,-%#xiG6#F3F,/F3;,$%)infinityG!\" \"F:/F1F:*&F+F,-F%6$-F(6$*&-&F/6#F16#F3F,-F56#F3F,/F3;,$F:F;F:/F1F:F, " }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 8 "that is," }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(r*G(x)*xi(x),x = -infin ity .. infinity) = r*Int(G(x)*xi(x),x = -infinity .. infinity);" "6#/- %$IntG6$*(%\"rG\"\"\"-%\"GG6#%\"xGF)-%#xiG6#F-F)/F-;,$%)infinityG!\"\" F4*&F(F)-F%6$*&-F+6#F-F)-F/6#F-F)/F-;,$F4F5F4F)" }{TEXT -1 2 ", " }} {PARA 0 "" 0 "" {TEXT -1 2 "or" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "r*mu[G](xi) = r*``(mu[G](xi));" "6#/*&%\"rG\"\"\"-&%#mu G6#%\"GG6#%#xiGF&*&F%F&-%!G6#-&F)6#F+6#F-F&" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 88 "This means that distributions form a vector space with ordinary functions as a subspa ce." }}{PARA 0 "" 0 "" {TEXT -1 210 "( A subspace of a vector space is a subset which is itself a vector space with the addition and scalar \+ multiplication inherited from the containing space. For example, in th e space of 3-dimensional real vectors" }{XPPEDIT 18 0 " ``(x,y,z)" "6# -%!G6%%\"xG%\"yG%\"zG" }{TEXT -1 62 " any plane through the origin for ms a 2-dimensional subspace.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 15 "Differentiation" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "g[n](x); " "6#-&%\"gG6#%\"nG6#%\"xG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "n = 0, 1, 2, ` . . . `" "6&/%\"nG\"\"!\"\"\"\"\"#%(~.~.~.~G" }{TEXT -1 106 ", b e a weakly convergent sequence of infinitely differentiable functions, which represents a distribution " }{XPPEDIT 18 0 "G(x)" "6#-%\"GG6#% \"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 28 "The sequence of d erivatives " }{XPPEDIT 18 0 "g*`'`(x)" "6#*&%\"gG\"\"\"-%\"'G6#%\"xGF% " }{TEXT -1 71 " is also weakly convergent. This follows by using inte gration by parts:" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Int(g[n]*`'`(x)*xi(x),x = -infinity .. infinity) = Limit(``,R = infin ity);" "6#/-%$IntG6$*(&%\"gG6#%\"nG\"\"\"-%\"'G6#%\"xGF,-%#xiG6#F0F,/F 0;,$%)infinityG!\"\"F7-%&LimitG6$%!G/%\"RGF7" }{XPPEDIT 18 0 "Int(g[n] *`'`(x)*xi(x),x = -R .. R);" "6#-%$IntG6$*(&%\"gG6#%\"nG\"\"\"-%\"'G6# %\"xGF+-%#xiG6#F/F+/F/;,$%\"RG!\"\"F6" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Limit(``,R = infinity);" "6#/ %!G-%&LimitG6$F$/%\"RG%)infinityG" }{XPPEDIT 18 0 "g[n](x)*xi(x);" "6# *&-&%\"gG6#%\"nG6#%\"xG\"\"\"-%#xiG6#F*F+" }{XPPEDIT 18 0 "PIECEWISE([ R, ``],[-R, ``]);" "6#-%*PIECEWISEG6$7$%\"RG%!G7$,$F'!\"\"F(" } {XPPEDIT 18 0 "``-``" "6#,&%!G\"\"\"F$!\"\"" }{XPPEDIT 18 0 "Limit(``, R = infinity);" "6#-%&LimitG6$%!G/%\"RG%)infinityG" }{XPPEDIT 18 0 "In t(g[n](x)*xi*`'`(x),x = -R .. R);" "6#-%$IntG6$*(-&%\"gG6#%\"nG6#%\"xG \"\"\"%#xiGF.-%\"'G6#F-F./F-;,$%\"RG!\"\"F6" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 49 "The first term is zero because the test functio n " }{XPPEDIT 18 0 "xi(x);" "6#-%#xiG6#%\"xG" }{TEXT -1 32 " vanishes \+ outside some interval " }{XPPEDIT 18 0 "[-a,a]" "6#7$,$%\"aG!\"\"F%" } {TEXT -1 4 ", so" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "I nt(g*`'`(x)*xi(x),x = -infinity .. infinity) = Int(g[n](x)*xi*`'`(x),x = -infinity .. infinity);" "6#/-%$IntG6$*(%\"gG\"\"\"-%\"'G6#%\"xGF)- %#xiG6#F-F)/F-;,$%)infinityG!\"\"F4-F%6$*(-&F(6#%\"nG6#F-F)F/F)-F+6#F- F)/F-;,$F4F5F4" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 9 "for each " }{TEXT 280 1 "n" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 4 "Then " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Limit(Int(g[n]*`' `(x)*xi(x),x = -infinity .. infinity),n = infinity) = -``;" "6#/-%&Lim itG6$-%$IntG6$*(&%\"gG6#%\"nG\"\"\"-%\"'G6#%\"xGF/-%#xiG6#F3F//F3;,$%) infinityG!\"\"F:/F.F:,$%!GF;" }{XPPEDIT 18 0 "Limit(Int(g[n](x)*xi*`'` (x),x = -infinity .. infinity),n = infinity)" "6#-%&LimitG6$-%$IntG6$* (-&%\"gG6#%\"nG6#%\"xG\"\"\"%#xiGF1-%\"'G6#F0F1/F0;,$%)infinityG!\"\"F 9/F.F9" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 57 "The limit in t he second term exists because the sequence " }{XPPEDIT 18 0 "g[n](x); " "6#-&%\"gG6#%\"nG6#%\"xG" }{TEXT -1 41 " is weakly convergent and th e derivative " }{XPPEDIT 18 0 "xi*`'`(x)" "6#*&%#xiG\"\"\"-%\"'G6#%\"x GF%" }{TEXT -1 22 " of the test function " }{XPPEDIT 18 0 "xi(x);" "6# -%#xiG6#%\"xG" }{TEXT -1 25 " is also a test function." }}{PARA 0 "" 0 "" {TEXT -1 19 "Hence the sequence " }{XPPEDIT 18 0 "g[n]*`'`(x)" "6 #*&&%\"gG6#%\"nG\"\"\"-%\"'G6#%\"xGF(" }{TEXT -1 27 " represents a dis tribution " }{XPPEDIT 18 0 "G*`'`(x)" "6#*&%\"GG\"\"\"-%\"'G6#%\"xGF% " }{TEXT -1 54 " which we can call the derivative of the distribution \+ " }{XPPEDIT 18 0 "G(x)" "6#-%\"GG6#%\"xG" }{TEXT -1 6 ", and " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(G*`'`(x)*xi(x),x \+ = -infinity .. infinity) = -Int(G(x)*xi*`'`(x),x = -infinity .. infini ty);" "6#/-%$IntG6$*(%\"GG\"\"\"-%\"'G6#%\"xGF)-%#xiG6#F-F)/F-;,$%)inf inityG!\"\"F4,$-F%6$*(-F(6#F-F)F/F)-F+6#F-F)/F-;,$F4F5F4F5" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 57 "In terms of the corresponding li near functionals, we have" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "mu[G*`' `](xi) = -mu[G](xi);" "6#/-&%#muG6#*&%\"GG\"\" \"%#'~GF*6#%#xiG,$-&F&6#F)6#F-!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 " " {TEXT -1 36 "For example, the delta distribution " }{XPPEDIT 18 0 "d elta(x)" "6#-%&deltaG6#%\"xG" }{TEXT -1 45 " is represented by the seq uence of functions " }{XPPEDIT 18 0 "g[n](x) = n/(Pi*(1+n^2*x^2));" "6 #/-&%\"gG6#%\"nG6#%\"xG*&F(\"\"\"*&%#PiGF,,&F,F,*&F(\"\"#F*F1F,F,!\"\" " }{TEXT -1 64 ", so its derivative is represented by the sequence of \+ functions " }{XPPEDIT 18 0 "g[n]*`'`(x) = -2*n^3*x/(Pi*(1+n^2*x^2)^2); " "6#/*&&%\"gG6#%\"nG\"\"\"-%\"'G6#%\"xGF),$**\"\"#F)*$F(\"\"$F)F-F)*& %#PiGF)*$,&F)F)*&F(F0F-F0F)F0F)!\"\"F8" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 8 "We have " }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "Int(delt a*`'`(x)*phi(x),x = -infinity .. infinity) = -Int(delta(x)*xi*`'`(x),x = -infinity .. infinity);" "6#/-%$IntG6$*(%&deltaG\"\"\"-%\"'G6#%\"xG F)-%$phiG6#F-F)/F-;,$%)infinityG!\"\"F4,$-F%6$*(-F(6#F-F)%#xiGF)-F+6#F -F)/F-;,$F4F5F4F5" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=-xi*`'`(0)" "6#/%!G,$*&%#xiG\"\"\"-%\"'G6#\"\"!F(! \"\"" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 25 "by the sampling \+ property." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 46 "Multiplica tion of a distribution by a function" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 60 "The product of two \+ distributions is not defined in general. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 7 "Example" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 26 "The sequen ce of functions " }{XPPEDIT 18 0 "g[n](x) = n/(Pi*(1+n^2*x^2));" "6#/- &%\"gG6#%\"nG6#%\"xG*&F(\"\"\"*&%#PiGF,,&F,F,*&F(\"\"#F*F1F,F,!\"\"" } {TEXT -1 2 ", " }{XPPEDIT 18 0 "n = 0, 1, 2, ` . . . `" "6&/%\"nG\"\"! \"\"\"\"\"#%(~.~.~.~G" }{TEXT -1 36 ", represents the delta distributi on " }{XPPEDIT 18 0 "delta(x)" "6#-%&deltaG6#%\"xG" }{TEXT -1 19 ", bu t the sequence " }{XPPEDIT 18 0 "h[n](x) = g[n](x)^2;" "6#/-&%\"hG6#% \"nG6#%\"xG*$-&%\"gG6#F(6#F*\"\"#" }{XPPEDIT 18 0 "`` = n^2/(Pi^2*(1+n ^2*x^2)^2);" "6#/%!G*&%\"nG\"\"#*&%#PiGF',&\"\"\"F+*&F&F'%\"xGF'F+F'! \"\"" }{TEXT -1 26 " fails to converge weakly." }}{PARA 0 "" 0 "" {TEXT -1 22 "For any test function " }{XPPEDIT 18 0 "xi(x);" "6#-%#xiG 6#%\"xG" }{TEXT -1 8 " we have" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "B*a bs(Int(n^2/(Pi^2*(1+n^2*x^2)^2),x = -infinity .. infinity))*`` <= ``*a bs(Int(n^2*xi(x)/(Pi^2*(1+n^2*x^2)^2),x = -infinity .. infinity));" "6 #1*(%\"BG\"\"\"-%$absG6#-%$IntG6$*&%\"nG\"\"#*&%#PiGF/,&F&F&*&F.F/%\"x GF/F&F/!\"\"/F4;,$%)infinityGF5F9F&%!GF&*&F:F&-F(6#-F+6$*(F.F/-%#xiG6# F4F&*&F1F/,&F&F&*&F.F/F4F/F&F/F5/F4;,$F9F5F9F&" }{TEXT -1 3 " , " }} {PARA 0 "" 0 "" {TEXT -1 7 "where " }{XPPEDIT 18 0 "abs(xi(x)) < B;" "6#2-%$absG6#-%#xiG6#%\"xG%\"BG" }{TEXT -1 9 " for all " }{TEXT 281 1 "x" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 18 "Then substituting \+ " }{XPPEDIT 18 0 "n*x = tan*u,n*``(dx/du) = sec^2*u;" "6$/*&%\"nG\"\" \"%\"xGF&*&%$tanGF&%\"uGF&/*&F%F&-%!G6#*&%#dxGF&%#duG!\"\"F&*&%$secG\" \"#F*F&" }{TEXT -1 6 " gives" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Int(n^2/(Pi^2*(1+n^2*x^2)^2),x) = Int(n*sec^2*u/(Pi^2*s ec^4*u),u);" "6#/-%$IntG6$*&%\"nG\"\"#*&%#PiGF),&\"\"\"F-*&F(F)%\"xGF) F-F)!\"\"F/-F%6$**F(F-*$%$secGF)F-%\"uGF-*(F+F)F5\"\"%F6F-F0F6" } {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = n/(Pi^2);" "6#/%!G*&%\"nG\"\"\"*$%#PiG\"\"#!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "Int(cos^2*u,u);" "6#-%$IntG6$*&%$cosG\"\"#%\"uG\"\"\"F) " }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "` ` = n/(Pi^2);" "6#/%!G*&%\"nG\"\"\"*$%#PiG\"\"#!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(1/2+cos*2*u/2,u);" "6#-%$IntG6$,&*&\"\"\"F(\"\"#! \"\"F(**%$cosGF(F)F(%\"uGF(F)F*F(F-" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = n/(Pi^2);" "6#/%!G*&%\"nG\"\"\" *$%#PiG\"\"#!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``(u/2+sin*2*u/4)+c ;" "6#,&-%!G6#,&*&%\"uG\"\"\"\"\"#!\"\"F***%$sinGF*F+F*F)F*\"\"%F,F*F* %\"cGF*" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = n/(2*Pi^2);" "6#/%!G*&%\"nG\"\"\"*&\"\"#F'*$%#PiGF)F'!\"\" " }{TEXT -1 1 " " }{XPPEDIT 18 0 "``(u+sin*u*cos*u)+c;" "6#,&-%!G6#,&% \"uG\"\"\"**%$sinGF)F(F)%$cosGF)F(F)F)F)%\"cGF)" }{TEXT -1 1 " " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = n/(2*Pi^2);" "6# /%!G*&%\"nG\"\"\"*&\"\"#F'*$%#PiGF)F'!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``(arctan*n*x+n*x/(1+n^2*x^2))+c;" "6#,&-%!G6#,&*(%'arctanG\"\" \"%\"nGF*%\"xGF*F**(F+F*F,F*,&F*F**&F+\"\"#F,F0F*!\"\"F*F*%\"cGF*" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Hence" }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "Int(n^2/(Pi^2*(1+n^2*x^2)^2),x = -infinity .. infinity) = n/(2*Pi^2 );" "6#/-%$IntG6$*&%\"nG\"\"#*&%#PiGF),&\"\"\"F-*&F(F)%\"xGF)F-F)!\"\" /F/;,$%)infinityGF0F4*&F(F-*&F)F-*$F+F)F-F0" }{TEXT -1 1 " " } {XPPEDIT 18 0 "Limit(``(arctan(n*x)+n*x/(1+n^2*x^2)),R = infinity);" " 6#-%&LimitG6$-%!G6#,&-%'arctanG6#*&%\"nG\"\"\"%\"xGF/F/*(F.F/F0F/,&F/F /*&F.\"\"#F0F4F/!\"\"F//%\"RG%)infinityG" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([R, ``],[``, ``],[-R, ``]);" "6#-%*PIECEWISEG6%7$%\"RG %!G7$F(F(7$,$F'!\"\"F(" }{TEXT -1 2 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = n/(2*Pi^2);" "6#/%!G*&%\"nG\"\"\"*&\"\"#F'* $%#PiGF)F'!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Limit(``(arctan*n*R-a rctan(-n*R)),R = infinity);" "6#-%&LimitG6$-%!G6#,&*(%'arctanG\"\"\"% \"nGF,%\"RGF,F,-F+6#,$*&F-F,F.F,!\"\"F3/F.%)infinityG" }{TEXT -1 1 " \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = n/(2*Pi^2); " "6#/%!G*&%\"nG\"\"\"*&\"\"#F'*$%#PiGF)F'!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "Limit(``(2*arctan*n*R),R = infinity);" "6#-%&LimitG6$-% !G6#**\"\"#\"\"\"%'arctanGF+%\"nGF+%\"RGF+/F.%)infinityG" }{TEXT -1 2 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = n/(2*Pi^2 )*Pi;" "6#/%!G*(%\"nG\"\"\"*&\"\"#F'*$%#PiGF)F'!\"\"F+F'" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = n/(2*Pi); " "6#/%!G*&%\"nG\"\"\"*&\"\"#F'%#PiGF'!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "It follows that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "B*n/(2*Pi) <= ab s(Int(``(n^2/(Pi^2*(1+n^2*x^2)^2))*xi(x),x = -infinity .. infinity)); " "6#1*(%\"BG\"\"\"%\"nGF&*&\"\"#F&%#PiGF&!\"\"-%$absG6#-%$IntG6$*&-%! G6#*&F'F)*&F*F),&F&F&*&F'F)%\"xGF)F&F)F+F&-%#xiG6#F:F&/F:;,$%)infinity GF+FA" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 4 "for " }{XPPEDIT 18 0 "n=0,1,2,` . . . `" "6&/%\"nG\"\"!\"\"\"\"\"#%(~.~.~.~G" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 6 "Hence " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Limit (Int(``(n^2/(Pi^2*(1+n^2*x^2)^2))*xi(x),x = -infinity .. infinity),n = infinity) = infinity;" "6#/-%&LimitG6$-%$IntG6$*&-%!G6#*&%\"nG\"\"#*& %#PiGF0,&\"\"\"F4*&F/F0%\"xGF0F4F0!\"\"F4-%#xiG6#F6F4/F6;,$%)infinityG F7F>/F/F>F>" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "assume(n_,posint);\nInt(n^2 /(Pi*(1+n^2*x^2))^2,x=-infinity..infinity);\nsubs(n_=n,value(subs(n=n_ ,%)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*(%\"nG\"\"#%#PiG! \"#,&\"\"\"F,*&)F'F(F,)%\"xGF(F,F,F*/F0;,$%)infinityG!\"\"F4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\"\"#!\"\"%\"nG\"\"\"%#PiGF&F(" }}} {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 53 "The product of an infinitely differentiable function \+ " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG6#%\"xG" }{TEXT -1 20 " and a distrib ution " }{XPPEDIT 18 0 "H(x)" "6#-%\"HG6#%\"xG" }{TEXT -1 17 " is well -defined." }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "h[n](x) ;" "6#-&%\"hG6#%\"nG6#%\"xG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "n = 0, 1 , 2, ` . . . `" "6&/%\"nG\"\"!\"\"\"\"\"#%(~.~.~.~G" }{TEXT -1 88 ", b e a sequence of infinitely differentiable functions which represent th e distribution " }{XPPEDIT 18 0 "H(x);" "6#-%\"HG6#%\"xG" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 27 "Then for any test function " } {XPPEDIT 18 0 "xi(x);" "6#-%#xiG6#%\"xG" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Limit(Int(``(g(x)*h[n](x))*xi( x),x = -infinity .. infinity),n = infinity);" "6#-%&LimitG6$-%$IntG6$* &-%!G6#*&-%\"gG6#%\"xG\"\"\"-&%\"hG6#%\"nG6#F1F2F2-%#xiG6#F1F2/F1;,$%) infinityG!\"\"F?/F7F?" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Limit(Int(h[n](x)*[g(x)*xi(x)],x = -infinity \+ .. infinity),n = infinity);" "6#/%!G-%&LimitG6$-%$IntG6$*&-&%\"hG6#%\" nG6#%\"xG\"\"\"7#*&-%\"gG6#F2F3-%#xiG6#F2F3F3/F2;,$%)infinityG!\"\"F?/ F0F?" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 26 "This limit exist s because " }{XPPEDIT 18 0 "g(x)*xi(x);" "6#*&-%\"gG6#%\"xG\"\"\"-%#xi G6#F'F(" }{TEXT -1 21 " is a test function. " }}{PARA 0 "" 0 "" {TEXT -1 19 "Hence the sequence " }{XPPEDIT 18 0 "g(x)*h[n](x);" "6#*&-%\"gG 6#%\"xG\"\"\"-&%\"hG6#%\"nG6#F'F(" }{TEXT -1 76 " converges weakly, an d so represents a distribution, which we can denote by " }{XPPEDIT 18 0 "g(x)*H(x);" "6#*&-%\"gG6#%\"xG\"\"\"-%\"HG6#F'F(" }{TEXT -1 1 "." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 56 "For exam ple, for any infinitely differentiable function " }{XPPEDIT 18 0 "g(x) " "6#-%\"gG6#%\"xG" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "g(x)*delta(x) = g(0)*delta(x)" "6#/*&-%\"gG6#%\"xG\" \"\"-%&deltaG6#F(F)*&-F&6#\"\"!F)-F+6#F(F)" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 262 12 "____________" }{TEXT -1 1 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "This \+ follows, because, for any test function " }{XPPEDIT 18 0 "xi(x);" "6#- %#xiG6#%\"xG" }{TEXT -1 1 "," }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Int(g(x)*delta(x)*xi(x),x = -infinity .. infinity) = In t(delta(x)*[g(x)*xi(x)],x = -infinity .. infinity);" "6#/-%$IntG6$*(-% \"gG6#%\"xG\"\"\"-%&deltaG6#F+F,-%#xiG6#F+F,/F+;,$%)infinityG!\"\"F6-F %6$*&-F.6#F+F,7#*&-F)6#F+F,-F16#F+F,F,/F+;,$F6F7F6" }{TEXT -1 1 " " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = g(0)*xi(0);" "6# /%!G*&-%\"gG6#\"\"!\"\"\"-%#xiG6#F)F*" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = g(0)*Int(delta(x)*xi(x),x = - infinity .. infinity);" "6#/%!G*&-%\"gG6#\"\"!\"\"\"-%$IntG6$*&-%&delt aG6#%\"xGF*-%#xiG6#F2F*/F2;,$%)infinityG!\"\"F9F*" }{TEXT -1 2 ", " }} {PARA 0 "" 0 "" {TEXT -1 48 "by two applications of the sampling prope rty of " }{XPPEDIT 18 0 "delta(x)" "6#-%&deltaG6#%\"xG" }{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 26 "Functions as distributions" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 38 "A n infinitely differentiable function " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG 6#%\"xG" }{TEXT -1 76 " can be considered as the distribution represen ted by the constant sequence " }{XPPEDIT 18 0 "g(x), g(x), g(x),` . . \+ . `" "6&-%\"gG6#%\"xG-F$6#F&-F$6#F&%(~.~.~.~G" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 80 "However, more general functions such as \+ continuous functions are distributions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 39 "A continuous function as a d istribution" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "g(x);" "6#-%\"gG6#%\"xG" } {TEXT -1 34 " be a bounded continuous function." }}{PARA 0 "" 0 "" {TEXT -1 33 "Then we can construct a sequence " }{XPPEDIT 18 0 "g[n](x );" "6#-&%\"gG6#%\"nG6#%\"xG" }{TEXT -1 66 " of infinitely differentia ble functions which converges weakly to " }{XPPEDIT 18 0 "g(x)" "6#-% \"gG6#%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 91 "The constr uction of such a sequence is based on the fact that, by the sampling p roperty of " }{XPPEDIT 18 0 "delta(x" "6#-%&deltaG6#%\"xG" }{TEXT -1 10 ", we have " }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "I nt(g(x-y)*delta(y),y = -infinity .. infinity) = eval(g(x-y),y = 0);" " 6#/-%$IntG6$*&-%\"gG6#,&%\"xG\"\"\"%\"yG!\"\"F--%&deltaG6#F.F-/F.;,$%) infinityGF/F6-%%evalG6$-F)6#,&F,F-F.F//F.\"\"!" }{TEXT -1 13 " ------ - (i)" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = g(x);" "6#/%!G-%\"gG6#%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "The integral in (i) defines the " } {TEXT 261 11 "convolution" }{TEXT -1 5 " g * " }{XPPEDIT 18 0 "delta" "6#%&deltaG" }{TEXT -1 10 " of g and " }{XPPEDIT 18 0 "delta;" "6#%&de ltaG" }{TEXT -1 14 ", where ( f * " }{XPPEDIT 18 0 "delta" "6#%&deltaG " }{TEXT -1 2 " )" }{XPPEDIT 18 0 "``(x) = Int(f(x-y)*delta(y),y = -in finity .. infinity);" "6#/-%!G6#%\"xG-%$IntG6$*&-%\"fG6#,&F'\"\"\"%\"y G!\"\"F0-%&deltaG6#F1F0/F1;,$%)infinityGF2F9" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } {XPPEDIT 18 0 "delta(x)" "6#-%&deltaG6#%\"xG" }{TEXT -1 75 " can be re presented by the sequence of infinitely differentiable functions " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "d[n](x) = PIECEWISE([ ``(n/A)*exp(1/(n^2*x^2-1)), abs(x) < 1/n],[0, 1/n <= abs(x)]);" "6#/-& %\"dG6#%\"nG6#%\"xG-%*PIECEWISEG6$7$*&-%!G6#*&F(\"\"\"%\"AG!\"\"F4-%$e xpG6#*&F4F4,&*&F(\"\"#F*F=F4F4F6F6F42-%$absG6#F**&F4F4F(F67$\"\"!1*&F4 F4F(F6-F@6#F*" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 7 " where " }{XPPEDIT 18 0 "A =`` " "6#/%\"AG %!G" }{XPPEDIT 18 0 "Int(exp(1/(x^2-1)),x = -1 .. 1) = Limit(``,epsilo n = 0);" "6#/-%$IntG6$-%$expG6#*&\"\"\"F+,&*$%\"xG\"\"#F+F+!\"\"F0/F.; ,$F+F0F+-%&LimitG6$%!G/%(epsilonG\"\"!" }{XPPEDIT 18 0 "Int( exp(1/(x^ 2-1)),x=-1+epsilon..1-epsilon)" "6#-%$IntG6$-%$expG6#*&\"\"\"F*,&*$%\" xG\"\"#F*F*!\"\"F//F-;,&F*F/%(epsilonGF*,&F*F*F3F/" }{TEXT -1 2 ". " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "g[n](x);" "6#-&%\"gG 6#%\"nG6#%\"xG" }{TEXT -1 15 " be defined by " }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "g[n](x);" "6#-&%\"gG6#%\"nG6#%\"xG" }{TEXT -1 3 " = " } {XPPEDIT 18 0 "d[n](x)*`*`*g(x) = Int(d[n](x-y)*g(y),y = -infinity .. \+ infinity);" "6#/*(-&%\"dG6#%\"nG6#%\"xG\"\"\"%\"*GF,-%\"gG6#F+F,-%$Int G6$*&-&F'6#F)6#,&F+F,%\"yG!\"\"F,-F/6#F:F,/F:;,$%)infinityGF;FA" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 5 "Then " }{XPPEDIT 18 0 "g[ n](x);" "6#-&%\"gG6#%\"nG6#%\"xG" }{TEXT -1 38 " is infinitely differe ntiable because " }{XPPEDIT 18 0 "d[n](x);" "6#-&%\"dG6#%\"nG6#%\"xG" }{TEXT -1 80 " is infinitely differentiable, and the sequence of these functions converges to " }{XPPEDIT 18 0 "g(x);" "6#-%\"gG6#%\"xG" } {TEXT -1 27 " uniformly on any interval " }{XPPEDIT 18 0 "[-a,a]" "6#7 $,$%\"aG!\"\"F%" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 27 "To se e this first note that" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "d[n](x)*`*`*g(x) = Int(d[n](x-y) *g(y),y = -infinity .. infinity);" "6#/*(-&%\"dG6#%\"nG6#%\"xG\"\"\"% \"*GF,-%\"gG6#F+F,-%$IntG6$*&-&F'6#F)6#,&F+F,%\"yG!\"\"F,-F/6#F:F,/F:; ,$%)infinityGF;FA" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=-Int(d[n](u)*g(x-u),u = infinity .. -infinity)" "6# /%!G,$-%$IntG6$*&-&%\"dG6#%\"nG6#%\"uG\"\"\"-%\"gG6#,&%\"xGF1F0!\"\"F1 /F0;%)infinityG,$F:F7F7" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "u = x \+ -y" "6#/%\"uG,&%\"xG\"\"\"%\"yG!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "du/dy = -1" "6#/*&%#duG\"\"\"%#dyG!\"\",$F&F(" }{TEXT -1 2 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Int(g(x-u)*d[n ](u),u = -infinity .. infinity);" "6#/%!G-%$IntG6$*&-%\"gG6#,&%\"xG\" \"\"%\"uG!\"\"F.-&%\"dG6#%\"nG6#F/F./F/;,$%)infinityGF0F:" }{TEXT -1 2 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = g(x)*`* `*d[n](x);" "6#/%!G*(-%\"gG6#%\"xG\"\"\"%\"*GF*-&%\"dG6#%\"nG6#F)F*" } {TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 42 "which is the commutativ ity of convolution." }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "Int(d[n](y),u = -infinity .. infinity) = 1;" "6#/-%$IntG6$-&%\"d G6#%\"nG6#%\"yG/%\"uG;,$%)infinityG!\"\"F2\"\"\"" }{TEXT -1 10 ", we h ave " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "abs(g[n](x)-g (x)) = abs(Int(g(x-y)*d[n](y),u = -infinity .. infinity)-Int(g(x)*d[n] (y),u = -infinity .. infinity));" "6#/-%$absG6#,&-&%\"gG6#%\"nG6#%\"xG \"\"\"-F*6#F.!\"\"-F%6#,&-%$IntG6$*&-F*6#,&F.F/%\"yGF2F/-&%\"dG6#F,6#F =F//%\"uG;,$%)infinityGF2FGF/-F76$*&-F*6#F.F/-&F@6#F,6#F=F//FD;,$FGF2F GF2" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 3 " = " }{XPPEDIT 18 0 "abs(Int((g(x-y)-g(x))*d[n](y),u = -infinity .. infinity));" "6#- %$absG6#-%$IntG6$*&,&-%\"gG6#,&%\"xG\"\"\"%\"yG!\"\"F0-F,6#F/F2F0-&%\" dG6#%\"nG6#F1F0/%\"uG;,$%)infinityGF2F?" }{TEXT -1 1 " " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` <= Int(abs(g(x-y)-g(x))*d[n]( y),u = -infinity .. infinity);" "6#1%!G-%$IntG6$*&-%$absG6#,&-%\"gG6#, &%\"xG\"\"\"%\"yG!\"\"F2-F.6#F1F4F2-&%\"dG6#%\"nG6#F3F2/%\"uG;,$%)infi nityGF4FA" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`` = Int(abs(g(x-y)-g(x))*d[n](y),u = -1/n .. 1/n);" "6 #/%!G-%$IntG6$*&-%$absG6#,&-%\"gG6#,&%\"xG\"\"\"%\"yG!\"\"F2-F.6#F1F4F 2-&%\"dG6#%\"nG6#F3F2/%\"uG;,$*&F2F2F;F4F4*&F2F2F;F4" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 65 "It i s a result from basic analysis that, on any closed interval " } {XPPEDIT 18 0 " [-a,a" "6#7$,$%\"aG!\"\"F%" }{TEXT -1 24 ", a continuo us function " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG6#%\"xG" }{TEXT -1 13 " i s actually " }{TEXT 261 20 "uniformly continuous" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 46 "This means that we can find a positive nu mber " }{XPPEDIT 18 0 "delta" "6#%&deltaG" }{TEXT -1 24 " so that for \+ any number " }{TEXT 282 1 "x" }{TEXT -1 17 " in the interval " } {XPPEDIT 18 0 "[-a, a];" "6#7$,$%\"aG!\"\"F%" }{TEXT -1 12 " and numbe r " }{TEXT 283 1 "y" }{TEXT -1 11 " such that " }{XPPEDIT 18 0 "abs(y) =N" "6#1%\" NG%\"nG" }{TEXT -1 24 " and x in the interval " }{XPPEDIT 18 0 "[-a,a ]" "6#7$,$%\"aG!\"\"F%" }{TEXT -1 1 "," }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "abs(g[n](x)-g(x)) <= Int(abs(g(x-y)-g(x))*d[n]( y),u = -1/n .. 1/n);" "6#1-%$absG6#,&-&%\"gG6#%\"nG6#%\"xG\"\"\"-F*6#F .!\"\"-%$IntG6$*&-F%6#,&-F*6#,&F.F/%\"yGF2F/-F*6#F.F2F/-&%\"dG6#F,6#F= F//%\"uG;,$*&F/F/F,F2F2*&F/F/F,F2" }{TEXT -1 1 " " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "`` <= Int(epsilon*d[n](y),u = -1/n .. 1/n);" "6#1%!G-%$IntG6$*&%(epsilonG\"\"\"-&%\"dG6#%\"nG6#%\"yGF*/%\"u G;,$*&F*F*F/!\"\"F7*&F*F*F/F7" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` <= epsilon*Int(d[n](y),u = -infinity .. infinity);" "6#1%!G*&%(epsilonG\"\"\"-%$IntG6$-&%\"dG6#%\"nG6#%\"y G/%\"uG;,$%)infinityG!\"\"F6F'" }{XPPEDIT 18 0 "`` = epsilon;" "6#/%!G %(epsilonG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 32 "Since the sequence of functions " }{XPPEDIT 18 0 "g[n](x);" "6#-&%\"gG6#%\"nG6#%\"xG" }{TEXT -1 24 " converges unifor mly to " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG6#%\"xG" }{TEXT -1 17 " on any interval " }{XPPEDIT 18 0 "[-a,a]" "6#7$,$%\"aG!\"\"F%" }{TEXT -1 15 ", the sequence " }{XPPEDIT 18 0 "g[n](x);" "6#-&%\"gG6#%\"nG6#%\"xG" }{TEXT -1 16 " also converges " }{TEXT 261 6 "weakly" }{TEXT -1 31 ", \+ that is, for a test function " }{XPPEDIT 18 0 "xi(x);" "6#-%#xiG6#%\"x G" }{TEXT -1 40 " which vanishes outside of the interval " }{XPPEDIT 18 0 "[-a, a]" "6#7$,$%\"aG!\"\"F%" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Limit(Int(g[n](x)*xi(x),y = -infinit y .. infinity),n = infinity) = Int(g(x)*xi(x),x = -infinity .. infinit y);" "6#/-%&LimitG6$-%$IntG6$*&-&%\"gG6#%\"nG6#%\"xG\"\"\"-%#xiG6#F1F2 /%\"yG;,$%)infinityG!\"\"F:/F/F:-F(6$*&-F-6#F1F2-F46#F1F2/F1;,$F:F;F: " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "Thus the continuous function " }{XPPEDIT 18 0 "g(x)" "6#- %\"gG6#%\"xG" }{TEXT -1 4 " is " }{TEXT 261 29 "represented as a distr ibution" }{TEXT -1 56 " by the sequence of infinitely differentiable f unctions " }{XPPEDIT 18 0 "g[n](x);" "6#-&%\"gG6#%\"nG6#%\"xG" }{TEXT -1 1 "." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 265 20 "___________ _________" }{TEXT 288 1 " " }}{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 4 "" 0 "" {TEXT -1 9 "Example 1" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 5 "Let " }{XPPEDIT 18 0 "g(x) = PIECEWISE([1-abs(x), abs(x) < 1],[ 0, 1 <= abs(x)]);" "6#/-%\"gG6#%\"xG-%*PIECEWISEG6$7$,&\"\"\"F--%$absG 6#F'!\"\"2-F/6#F'F-7$\"\"!1F--F/6#F'" }{TEXT -1 1 "." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "g := x -> \+ piecewise(abs(x)<1,1-abs(x),0):\n'g(n,x)'=g(n,x);\nplot(g(x),x=-2..2,t hickness=2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6$%\"nG%\"xG-%* PIECEWISEG6$7$,&\"\"\"F.-%$absG6#F'!\"\"2F/F.7$\"\"!%*otherwiseG" }} {PARA 13 "" 1 "" {GLPLOT2D 471 201 201 {PLOTDATA 2 "6&-%'CURVESG6$7Y7$ $!\"#\"\"!$F*F*7$$!3MLLL$Q6G\">!#'***!#=$\"3HI#* ******H,Q!#@7$$!3_++++Y0j&*FT$\"3%[*******RXpV!#>7$$!3E++++0\"*H\"*FT$ \"3Q(*******\\*3q)Fgn7$$!35++++83&H)FT$\"3!********p=\\q\"FT7$$!3\\LLL 3k(p`(FT$\"3_mmm\"fBIY#FT7$$!3Anmmmj^NmFT$\"3yKLLLO[kLFT7$$!3)zmmmYh=( eFT$\"3.KLLL&Q\"GTFT7$$!3+,++v#\\N)\\FT$\"3+*****\\s]k,&FT7$$!3commmCC (>%FT$\"3WJLLLvv-eFT7$$!39*****\\FRXL$FT$\"3'3++]sgam'FT7$$!3t*****\\# =/8DFT$\"3G+++v\"ep[(FT7$$!3=mmm;a*el\"FT$\"3#QLLLe/TM)FT7$$!3komm;Wn( o)Fgn$\"39LLLeDBJ\"*FT7$$!3$G++]7bDW%Fgn$\"3s****\\([Wdb*FT7$$!3IqLLL$ eV(>!#?$\"3Immm;kD!)**FT7$$\"3V[mmT+07UFgn$\"39NL$e*\\zy&*FT7$$\"3)Qjm m\"f`@')Fgn$\"3hOLL3k%y8*FT7$$\"3%z****\\nZ)H;FT$\"31-++DB:q$)FT7$$\"3 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#-%$IntG6$-%$expG6#*&\"\"\"F*,&*$)%\"xG\"\"#F*F*F*!\"\"F0/F.;F0F*" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG$\"+i\"Q*RW!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 126 "d := (n, x) -> n/A*exp(1/(n^2*x^2-1));\ng := x -> piecewise(abs(x)<1,1-abs(x),0 ):\ngn := (n,x) -> Int(d(n,y)*g(x-y),y=-1/n..1/n);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%\"dGf*6$%\"nG%\"xG6\"6$%)operatorG%&arrowGF)*(9$\" \"\"%\"AG!\"\"-%$expG6#*&F/F/,&*&)F.\"\"#F/)9%F9F/F/F/F1F1F/F)F)F)" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#gnGf*6$%\"nG%\"xG6\"6$%)operatorG%& arrowGF)-%$IntG6$*&-%\"dG6$9$%\"yG\"\"\"-%\"gG6#,&9%F6F5!\"\"F6/F5;,$* &F6F6F4F " 0 "" {MPLTEXT 1 0 40 "xx := 0:\nevalf(g(xx));\nevalf(gn(32,xx));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"\"\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+DJ[&*)*!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "xx := 0.5:\nevalf(g(xx));\nevalf(gn(32,xx ));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"\"&!\"\"" }}{PARA 11 "" 1 " " 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color=[red,COLOR(RGB,.4,0,.9)],thickness=[1,2]):\nt1 := pl ots[textplot]([12,4.5,`f(x)`],color=red):\nt2 := plots[textplot]([11,2 ,`g(x)`],color=COLOR(RGB,.4,0,.9)):\nplots[display]([p1,t1,t2]);" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{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 0 "" }}}{MARK "4 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }