{VERSION 3 0 "IBM INTEL LINUX" "3.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Plot" 0 13 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "Daubechies Polynomial, Sca ling Function and Wavelet" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 27 "First define the polynomial" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 75 "p0:=(1+sqrt(3))/4; p1:=(3+sqrt(3))/4; p2:=(3-s qrt(3))/4; p3:=(1-sqrt(3))/4;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p0 G,&#\"\"\"\"\"%F'*$-%%sqrtG6#\"\"$\"\"\"F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p1G,&#\"\"$\"\"%\"\"\"*$-%%sqrtG6#F'\"\"\"#F)F(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p2G,&#\"\"$\"\"%\"\"\"*$-%%sqrtG6#F '\"\"\"#!\"\"F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p3G,&#\"\"\"\"\" %F'*$-%%sqrtG6#\"\"$\"\"\"#!\"\"F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "P:=z->(p0+p1*z+p2*z^2+p3*z^3)/2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"PGR6#%\"zG6\"6$%)operatorG%&arrowGF(,*%#p0G#\"\"\" \"\"#*&%#p1GF/9$F/F.*&%#p2GF/)F3F0\"\"\"F.*&%#p3GF/)F3\"\"$F7F.F(F(F( " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "Now show that P(z) satisfies \+ the key conditions" }}{PARA 0 "" 0 "" {TEXT -1 43 "needed so that the \+ scaling function and its" }}{PARA 0 "" 0 "" {TEXT -1 27 "translates ar e orthonormal." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "z:=exp(I* t); zc:=exp(-I*t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"zG-%$expG6#* &%\"IG\"\"\"%\"tGF*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#zcG-%$expG6# ,$*&%\"IG\"\"\"%\"tGF+!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "expr:=P(z)*P(zc) +P(-z)*P(-zc):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\" \"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "f:=(Re(P(z)))^2+(Im(P (z)))^2;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"fG,&*$),(#\"\"\"\"\")F **$-%%sqrtG6#\"\"$\"\"\"F)-%#ReG6#,(*&,&#F0\"\"%F*F,#F*F9F*-%$expG6#*& %\"IGF*%\"tGF*F*F**&,&F8F*F,#!\"\"F9F*)F;\"\"#F1F**&,&F:F*F,FCF*)F;F0F 1F*#F*FFFFF1F**$)-%#ImGF4FFF1F:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "plot(f,t=-Pi/2..Pi/2,y=0..5);" }}{PARA 13 "" 1 "" {INLPLOT "6( -%'CURVESG6$7W7$$!+Cjzq:!\"*$\"+D+++]!#57$$!+NubO:F*$\"+L7kc_F-7$$!+X& =B]\"F*$\"+%>\"Q7bF-7$$!+qu_s9F*$\"+L)pMt&F-7$$!+&ROFW\"F*$\"+r\">E&fF -7$$!+O+tv8F*$\"+83`NkF-7$$!+;&z#38F*$\"+Ymk-pF-7$$!+w&\\6C\"F*$\"+PY, VtF-7$$!+H:\"*y6F*$\"+C1?DxF-7$$!+OwY96F*$\"+jE+\"4)F-7$$!+p'>y/\"F*$ \"+!znWV)F-7$$!+NW&Q\")*F-$\"+#fM'R()F-7$$!+nn\\I\"*F-$\"+%)p#Q,*F-7$$ !+g\\fG&)F-$\"+H.1A#*F-7$$!+'3'*4&yF-$\"+&p3-U*F-7$$!+E\\hqrF-$\"+_e]# e*F-7$$!+2;%\\^'F-$\"+3K%oq*F-7$$!+>u_>fF-$\"+/')=&z*F-7$$!*OA:@&F*$\" +#)>nt)*F-7$$!*?\\F*$\"+>7>(***F-7$$!* AP0I\"F*$\"+7mY****F-7$$!)NGBoF*$\"+D%f*****F-7$$!(d1b\"F*$\"+,+++5F*7 $$\")&Q8x'F*$\"+Y1'*****F-7$$\"*Pz+G\"F*$\"++$*\\****F-7$$\"*0x7$>F*$ \"+4QU(***F-7$$\"*(z-/EF*$\"+A7d\"***F-7$$\"*^$=iKF*$\"+(>1&z**F-7$$\" *#G)*)*QF*$\"+n0\")e**F-7$$\"**)Rgg%F*$\"+.FP@**F-7$$\"*7i8C&F*$\"+_$) )3()*F-7$$\"*?/(>fF*$\"+mb;&z*F-7$$\"*&HQMlF*$\"+&zzNq*F-7$$\"*=*Q1sF* $\"++R%[d*F-7$$\"*n%pQyF*$\"+k;YB%*F-7$$\"*5O'*\\)F*$\"+(Q/8B*F-7$$\"* ;Fe9*F*$\"+;%=\"3!*F-7$$\"*JHB#)*F*$\"+E/)ft)F-7$$\"+H\")QZ5F*$\"+q9dO %)F-7$$\"+H(=S6\"F*$\"+(HPM4)F-7$$\"+%f(4!=\"F*$\"+eN<=xF-7$$\"+3s\"3C \"F*$\"+rc7XtF-7$$\"+Q(3/J\"F*$\"+\"*3D))oF-7$$\"+KAls8F*$\"+ndIdkF-7$ $\"+zn,R9F*$\"+Yo\")zfF-7$$\"+\\kxq9F*$\"+g*4ku&F-7$$\"+>h`-:F*$\"+(zg 2^&F-7$$\"+AimO:F*$\"+gn#eD&F-7$$\"+Cjzq:F*F+-%'COLOURG6&%$RGBG$\"#5! \"\"\"\"!Fa\\l-%*THICKNESSG6#\"\"$-%&TITLEG6$%!G-%%FONTG6%%&TIMESG%'IT ALICG\"#=-%+AXESLABELSG6%Q\"t6\"Q\"yFd]l-F[]l6%F]]l%&ROMANGF_]l-%*AXES TICKSG6%%(DEFAULTGF\\^lFf]l-%%VIEWG6$;$!+Fjzq:F*$\"+Fjzq:F*;Fa\\l$\"\" &Fa\\l" 2 333 333 333 2 0 1 3 2 9 0 4 2 1.000000 45.000000 45.000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "Here is the iterative process for constru cting" }}{PARA 0 "" 0 "" {TEXT -1 62 "the scaling function. 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