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{SECT 0 {PARA 3 "" 0 "" {TEXT -1 52 "Fejer's theorem on the convergenc
e of Fourier series" }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Na
naimo, B.C., Canada" }}{PARA 0 "" 0 "" {TEXT -1 19 "Version:  26.3.200
7" }}{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 17 "The Fejer kernel " }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT 
-1 4 "Let " }{XPPEDIT 18 0 "phi(x)" "6#-%$phiG6#%\"xG" }{TEXT -1 51 " \+
be a complex valued periodic function with period " }{XPPEDIT 18 0 "2*
L" "6#*&\"\"#\"\"\"%\"LGF%" }{TEXT -1 27 " which has a Fourier series
" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(c[k]*exp(k*Pi
*i*x/L),k = -infinity .. infinity);" "6#-%$SumG6$*&&%\"cG6#%\"kG\"\"\"
-%$expG6#*,F*F+%#PiGF+%\"iGF+%\"xGF+%\"LG!\"\"F+/F*;,$%)infinityGF4F8
" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }}{PARA 256 "
" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[k]=1/(2*L)" "6#/&%\"cG6#%\"kG*
&\"\"\"F)*&\"\"#F)%\"LGF)!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(f(
x)*exp(k*Pi*i*x/L),x = -L .. L);" "6#-%$IntG6$*&-%\"fG6#%\"xG\"\"\"-%$
expG6#*,%\"kGF+%#PiGF+%\"iGF+F*F+%\"LG!\"\"F+/F*;,$F3F4F3" }{TEXT -1 
1 "." }}{PARA 0 "" 0 "" {TEXT -1 3 "Let" }}{PARA 256 "" 0 "" {TEXT -1 
3 "   " }{XPPEDIT 18 0 "s[m](x) = Sum(c[k]*exp(k*Pi*i*x/L),k = -m .. m
);" "6#/-&%\"sG6#%\"mG6#%\"xG-%$SumG6$*&&%\"cG6#%\"kG\"\"\"-%$expG6#*,
F2F3%#PiGF3%\"iGF3F*F3%\"LG!\"\"F3/F2;,$F(F;F(" }{TEXT -1 1 "," }}
{PARA 257 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "m=0,1,2,` . . . \+
`" "6&/%\"mG\"\"!\"\"\"\"\"#%(~.~.~.~G" }{TEXT -1 1 "." }}{PARA 0 "" 
0 "" {TEXT -1 39 "Consider the corresponding sequence of " }{TEXT 261 
12 "Cesaro means" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 2 "  " }
{XPPEDIT 18 0 "sigma[0](x) = s[0](x);" "6#/-&%&sigmaG6#\"\"!6#%\"xG-&%
\"sG6#F(6#F*" }{XPPEDIT 18 0 "``=c[0]" "6#/%!G&%\"cG6#\"\"!" }{TEXT 
-1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "sigma[1](x)
=(s[-1](x)+s[0](x)+s[1](x))/2" "6#/-&%&sigmaG6#\"\"\"6#%\"xG*&,(-&%\"s
G6#,$F(!\"\"6#F*F(-&F/6#\"\"!6#F*F(-&F/6#F(6#F*F(F(\"\"#F2" }{XPPEDIT 
18 0 "``=(c[-1]*exp(-Pi*i*x/L)+c[0]+c[1]*exp(Pi*i*x/L))/2" "6#/%!G*&,(
*&&%\"cG6#,$\"\"\"!\"\"F,-%$expG6#,$**%#PiGF,%\"iGF,%\"xGF,%\"LGF-F-F,
F,&F)6#\"\"!F,*&&F)6#F,F,-F/6#**F3F,F4F,F5F,F6F-F,F,F,\"\"#F-" }{TEXT 
-1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "sigma[2](x)
 = (s[-2](x)+s[-1](x)+s[0](x)+s[1](x)+s[2](x))/3;" "6#/-&%&sigmaG6#\"
\"#6#%\"xG*&,,-&%\"sG6#,$F(!\"\"6#F*\"\"\"-&F/6#,$F4F26#F*F4-&F/6#\"\"
!6#F*F4-&F/6#F46#F*F4-&F/6#F(6#F*F4F4\"\"$F2" }{XPPEDIT 18 0 "``=(c[-2
]*exp(-2*Pi*i*x/L)+c[-1]*exp(-Pi*i*x/L)+c[0]+c[1]*exp(Pi*i*x/L)+c[2]*e
xp(2*Pi*i*x/L))/3" "6#/%!G*&,,*&&%\"cG6#,$\"\"#!\"\"\"\"\"-%$expG6#,$*
,F,F.%#PiGF.%\"iGF.%\"xGF.%\"LGF-F-F.F.*&&F)6#,$F.F-F.-F06#,$**F4F.F5F
.F6F.F7F-F-F.F.&F)6#\"\"!F.*&&F)6#F.F.-F06#**F4F.F5F.F6F.F7F-F.F.*&&F)
6#F,F.-F06#*,F,F.F4F.F5F.F6F.F7F-F.F.F.\"\"$F-" }{TEXT -1 1 " " }}
{PARA 0 "" 0 "" {TEXT -1 14 "           :  " }}{PARA 0 "" 0 "" {TEXT 
-1 15 "           :   " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 
0 "sigma[n](x) = 1/(n+1)" "6#/-&%&sigmaG6#%\"nG6#%\"xG*&\"\"\"F,,&F(F,
F,F,!\"\"" }{XPPEDIT 18 0 "Sum(s[m](x),m = 0 .. n) = Sum((n-abs(k)+1)*
c[k]*exp(k*Pi*i*x/L)/(n+1),k = -n .. n);" "6#/-%$SumG6$-&%\"sG6#%\"mG6
#%\"xG/F+;\"\"!%\"nG-F%6$**,(F1\"\"\"-%$absG6#%\"kG!\"\"F6F6F6&%\"cG6#
F:F6-%$expG6#*,F:F6%#PiGF6%\"iGF6F-F6%\"LGF;F6,&F1F6F6F6F;/F:;,$F1F;F1
" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 17 "At a fixed value " }{XPPEDIT 18 0 "x=x[0]" "6#/%\"xG&F$6#
\"\"!" }{TEXT -1 10 " we have: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "sigma[n](x[0]) = Sum((n-abs(k)+1)*c[k]*exp(k*Pi*i*x[0]/
L)/(n+1),k = -n .. n);" "6#/-&%&sigmaG6#%\"nG6#&%\"xG6#\"\"!-%$SumG6$*
*,(F(\"\"\"-%$absG6#%\"kG!\"\"F3F3F3&%\"cG6#F7F3-%$expG6#*,F7F3%#PiGF3
%\"iGF3&F+6#F-F3%\"LGF8F3,&F(F3F3F3F8/F7;,$F(F8F(" }{TEXT -1 1 " " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "`` = Sum(``((n-abs(k)+1)/(n+1)),k = -n .. n);" "6#/%!G-
%$SumG6$-F$6#*&,(%\"nG\"\"\"-%$absG6#%\"kG!\"\"F-F-F-,&F,F-F-F-F2/F1;,
$F,F2F," }{TEXT -1 1 " " }{XPPEDIT 18 0 "1/(2*L)" "6#*&\"\"\"F$*&\"\"#
F$%\"LGF$!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``(Int(phi(x)*exp(-k*P
i*i*x/L),x = -L .. L))*exp(k*Pi*i*x[0]/L);" "6#*&-%!G6#-%$IntG6$*&-%$p
hiG6#%\"xG\"\"\"-%$expG6#,$*,%\"kGF/%#PiGF/%\"iGF/F.F/%\"LG!\"\"F9F//F
.;,$F8F9F8F/-F16#*,F5F/F6F/F7F/&F.6#\"\"!F/F8F9F/" }{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)*Sum(``((n-abs(k)+1)/(n+1))*exp(k*Pi*i*(x[0]-x)/L),k = -n .. n),
x = -L .. L);" "6#-%$IntG6$*&-%$phiG6#%\"xG\"\"\"-%$SumG6$*&-%!G6#*&,(
%\"nGF+-%$absG6#%\"kG!\"\"F+F+F+,&F5F+F+F+F:F+-%$expG6#*,F9F+%#PiGF+%
\"iGF+,&&F*6#\"\"!F+F*F:F+%\"LGF:F+/F9;,$F5F:F5F+/F*;,$FFF:FF" }{TEXT 
-1 2 "  " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/(2
*L);" "6#/%!G*&\"\"\"F&*&\"\"#F&%\"LGF&!\"\"" }{TEXT -1 1 " " }
{XPPEDIT 18 0 "Int(phi(x)*Phi[n](Pi*(x[0]-x)/L),x = -L .. L);" "6#-%$I
ntG6$*&-%$phiG6#%\"xG\"\"\"-&%$PhiG6#%\"nG6#*(%#PiGF+,&&F*6#\"\"!F+F*!
\"\"F+%\"LGF8F+/F*;,$F9F8F9" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" 
{TEXT -1 1 " " }{TEXT 262 19 "___________________" }{TEXT -1 1 " " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }
{XPPEDIT 18 0 "Phi[n](x) = Sum(``((n-abs(k)+1)/(n+1))*exp(k*i*x),k = -
n .. n);" "6#/-&%$PhiG6#%\"nG6#%\"xG-%$SumG6$*&-%!G6#*&,(F(\"\"\"-%$ab
sG6#%\"kG!\"\"F4F4F4,&F(F4F4F4F9F4-%$expG6#*(F8F4%\"iGF4F*F4F4/F8;,$F(
F9F(" }{TEXT -1 8 " is the " }{TEXT 261 12 "Fejer kernel" }{TEXT -1 
10 " of order " }{TEXT 266 1 "n" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "Now " }}{PARA 256 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(r^k,k = 0 .. n)^2=(1+r+r^2+` . . . \+
`+r^n)*(1+r+r^2+` . . . `+r^n)" "6#/*$-%$SumG6$)%\"rG%\"kG/F*;\"\"!%\"
nG\"\"#*&,,\"\"\"F2F)F2*$F)F/F2%(~.~.~.~GF2)F)F.F2F2,,F2F2F)F2*$F)F/F2
F4F2)F)F.F2F2" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "``=(1+r+r^2+` . . . `+r^n)+r*(1+r+r^2+` . . . `+r^n)+r^
2*(1+r+r^2+` . . . `+r^n)+` . . . `+r^n*(1+r+r^2+` . . . `+r^n)" "6#/%
!G,4\"\"\"F&%\"rGF&*$F'\"\"#F&%(~.~.~.~GF&)F'%\"nGF&*&F'F&,,F&F&F'F&*$
F'F)F&F*F&)F'F,F&F&F&*&F'F),,F&F&F'F&*$F'F)F&F*F&)F'F,F&F&F&F*F&*&)F'F
,F&,,F&F&F'F&*$F'F)F&F*F&)F'F,F&F&F&" }{TEXT -1 1 " " }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 " ``=1+2*r+3*r^2+` .
 . . `+(n+1)*r^n+` . . . `+3*r^(2*n-2)+2*r^(2*n-1)+r^(2*n)" "6#/%!G,4
\"\"\"F&*&\"\"#F&%\"rGF&F&*&\"\"$F&*$F)F(F&F&%(~.~.~.~GF&*&,&%\"nGF&F&
F&F&)F)F0F&F&F-F&*&F+F&)F),&*&F(F&F0F&F&F(!\"\"F&F&*&F(F&)F),&*&F(F&F0
F&F&F&F6F&F&)F)*&F(F&F0F&F&" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=r^n*Sum(
[n-abs(k)+1]*r^k,k = -n .. n)" "6#/%!G*&)%\"rG%\"nG\"\"\"-%$SumG6$*&7#
,(F(F)-%$absG6#%\"kG!\"\"F)F)F))F'F3F)/F3;,$F(F4F(F)" }{TEXT -1 2 ". \+
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "Hence
, if " }{XPPEDIT 18 0 "r<>1" "6#0%\"rG\"\"\"" }{TEXT -1 2 ", " }}
{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum([n-abs(k)+1]*r^k,
k = -n .. n) = r^(-n)*Sum(r^k,k = 0 .. n)^2;" "6#/-%$SumG6$*&7#,(%\"nG
\"\"\"-%$absG6#%\"kG!\"\"F+F+F+)%\"rGF/F+/F/;,$F*F0F**&)F2,$F*F0F+*$-F
%6$)F2F//F/;\"\"!F*\"\"#F+" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT 
-1 1 " " }{XPPEDIT 18 0 "`` = r^(-n)*((1-r^(n+1))/(1-r))^2;" "6#/%!G*&
)%\"rG,$%\"nG!\"\"\"\"\"*$*&,&F+F+)F',&F)F+F+F+F*F+,&F+F+F'F*F*\"\"#F+
" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 10 "Hence, if " }
{XPPEDIT 18 0 "x <>0" "6#0%\"xG\"\"!" }{TEXT -1 3 ",  " }}{PARA 256 "
" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum([n-abs(k)+1]*exp(k*i*x),k = \+
-n .. n) = exp(-n*i*x)*((1-exp((n+1)*i*x))/(1-exp(i*x)))^2;" "6#/-%$Su
mG6$*&7#,(%\"nG\"\"\"-%$absG6#%\"kG!\"\"F+F+F+-%$expG6#*(F/F+%\"iGF+%
\"xGF+F+/F/;,$F*F0F**&-F26#,$*(F*F+F5F+F6F+F0F+*$*&,&F+F+-F26#*(,&F*F+
F+F+F+F5F+F6F+F0F+,&F+F+-F26#*&F5F+F6F+F0F0\"\"#F+" }{TEXT -1 1 " " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "``=(exp(-(n+1)*i*x/2)*(1-exp((n+1)*i*x))/(exp(-i*x/2)*(
1-exp(i*x))))^2" "6#/%!G*$*(-%$expG6#,$**,&%\"nG\"\"\"F.F.F.%\"iGF.%\"
xGF.\"\"#!\"\"F2F.,&F.F.-F(6#*(,&F-F.F.F.F.F/F.F0F.F2F.*&-F(6#,$*(F/F.
F0F.F1F2F2F.,&F.F.-F(6#*&F/F.F0F.F2F.F2F1" }{TEXT -1 1 " " }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 
0 "`` = ((exp(-(n+1)*i*x/2)-exp((n+1)*i*x/2))/(exp(-i*x/2)-exp(i*x/2))
)^2;" "6#/%!G*$*&,&-%$expG6#,$**,&%\"nG\"\"\"F/F/F/%\"iGF/%\"xGF/\"\"#
!\"\"F3F/-F)6#**,&F.F/F/F/F/F0F/F1F/F2F3F3F/,&-F)6#,$*(F0F/F1F/F2F3F3F
/-F)6#*(F0F/F1F/F2F3F3F3F2" }{TEXT -1 2 "  " }}{PARA 256 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "`` = (sin((n+1)*x/2)/sin(x/2))^2;" "6#/
%!G*$*&-%$sinG6#*(,&%\"nG\"\"\"F-F-F-%\"xGF-\"\"#!\"\"F--F(6#*&F.F-F/F
0F0F/" }{TEXT -1 3 " , " }}{PARA 0 "" 0 "" {TEXT -1 3 "so " }}{PARA 
256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Phi[n](x) = 1/(n+1);" "6#/-
&%$PhiG6#%\"nG6#%\"xG*&\"\"\"F,,&F(F,F,F,!\"\"" }{TEXT -1 1 " " }
{XPPEDIT 18 0 "(sin((n+1)*x/2)/sin(x/2))^2" "6#*$*&-%$sinG6#*(,&%\"nG
\"\"\"F+F+F+%\"xGF+\"\"#!\"\"F+-F&6#*&F,F+F-F.F.F-" }{TEXT -1 6 " , if
 " }{XPPEDIT 18 0 "x<>0" "6#0%\"xG\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Also  " }{XPPEDIT 
18 0 "Phi[n](0) = 1/(n+1);" "6#/-&%$PhiG6#%\"nG6#\"\"!*&\"\"\"F,,&F(F,
F,F,!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(n-abs(k)+1,k = -n .. n)
 = 1/(n+1);" "6#/-%$SumG6$,(%\"nG\"\"\"-%$absG6#%\"kG!\"\"F)F)/F-;,$F(
F.F(*&F)F),&F(F)F)F)F." }{TEXT -1 1 " " }{XPPEDIT 18 0 "[ 1 +2 + ` . .
 . ` + (n+1) + ` . . . ` + 2 + 1] = n+1" "6#/7#,0\"\"\"F&\"\"#F&%(~.~.
~.~GF&,&%\"nGF&F&F&F&F(F&F'F&F&F&,&F*F&F&F&" }{TEXT -1 2 ". " }}{PARA 
0 "" 0 "" {TEXT -1 12 "Note that   " }{XPPEDIT 18 0 "Limit(sin((2*n+1)
*x/2)^2/((n+1)*sin(x/2)^2),x = 0) = n+1;" "6#/-%&LimitG6$*&-%$sinG6#*(
,&*&\"\"#\"\"\"%\"nGF/F/F/F/F/%\"xGF/F.!\"\"F.*&,&F0F/F/F/F/*$-F)6#*&F
1F/F.F2F.F/F2/F1\"\"!,&F0F/F/F/" }{TEXT -1 6 ",  so " }{XPPEDIT 18 0 "
Phi[n](x);" "6#-&%$PhiG6#%\"nG6#%\"xG" }{TEXT -1 20 " is continuous at
 0." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "Th
e Fejer kernel of order " }{TEXT 267 1 "n" }{TEXT -1 30 " can be defin
ed by the formula" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "
Phi[n](x) = PIECEWISE([sin((n+1)*x/2)^2/((n+1)*sin(x/2)^2), x <> 0],[n
+1, x = 0]);" "6#/-&%$PhiG6#%\"nG6#%\"xG-%*PIECEWISEG6$7$*&-%$sinG6#*(
,&F(\"\"\"F5F5F5F*F5\"\"#!\"\"F6*&,&F(F5F5F5F5*$-F16#*&F*F5F6F7F6F5F70
F*\"\"!7$,&F(F5F5F5/F*F?" }{TEXT -1 3 "  ." }}{PARA 256 "" 0 "" {TEXT 
-1 1 " " }{TEXT 265 21 "_____________________" }{TEXT -1 1 " " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "We have" }
}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Phi[n](x) = Sum(``((
n-abs(k)+1)/(n+1))*exp(k*i*x),k = -n .. n);" "6#/-&%$PhiG6#%\"nG6#%\"x
G-%$SumG6$*&-%!G6#*&,(F(\"\"\"-%$absG6#%\"kG!\"\"F4F4F4,&F(F4F4F4F9F4-
%$expG6#*(F8F4%\"iGF4F*F4F4/F8;,$F(F9F(" }{TEXT -1 2 "  " }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 
0 "`` = Sum(``((n-k+1)/(n+1))*(cos*k*x-i*sin*k*x),k = 1 .. n)+1+Sum(``
((n-k+1)/(n+1))*(cos*k*x+i*sin*k*x),k = 1 .. n);" "6#/%!G,(-%$SumG6$*&
-F$6#*&,(%\"nG\"\"\"%\"kG!\"\"F/F/F/,&F.F/F/F/F1F/,&*(%$cosGF/F0F/%\"x
GF/F/**%\"iGF/%$sinGF/F0F/F6F/F1F//F0;F/F.F/F/F/-F'6$*&-F$6#*&,(F.F/F0
F1F/F/F/,&F.F/F/F/F1F/,&*(F5F/F0F/F6F/F/**F8F/F9F/F0F/F6F/F/F//F0;F/F.
F/" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "
" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1+2*Sum(``((n-k+1)/(n+1))*cos*k*
x,k = 1 .. n);" "6#/%!G,&\"\"\"F&*&\"\"#F&-%$SumG6$**-F$6#*&,(%\"nGF&%
\"kG!\"\"F&F&F&,&F1F&F&F&F3F&%$cosGF&F2F&%\"xGF&/F2;F&F1F&F&" }{TEXT 
-1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 2 "  " }}{PARA 0 "" 0 "" {TEXT 
-1 39 "From this we see that the Fejer kernel " }{XPPEDIT 18 0 "Phi[n]
(x);" "6#-&%$PhiG6#%\"nG6#%\"xG" }{TEXT -1 7 " is an " }{TEXT 261 13 "
even periodic" }{TEXT -1 22 " function with period " }{XPPEDIT 18 0 "2
*Pi" "6#*&\"\"#\"\"\"%#PiGF%" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 56 "The following picture sho
ws the graphs of Fejer kernels " }{XPPEDIT 18 0 "Phi[n](x);" "6#-&%$Ph
iG6#%\"nG6#%\"xG" }{TEXT -1 19 " of various orders " }{TEXT 268 1 "n" 
}{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 164 "Phi := (n,x) -> sin((n+1)*x/2)^2/((n+1)*sin(x/2
)^2);\nplot([Phi(1,x),Phi(2,x),Phi(4,x),Phi(8,x)],x=-Pi..3*Pi,\n      \+
                 color=[red,blue,green,magenta]);" }}{PARA 11 "" 1 "" 
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ugtn\"FY7$Fb[r$\"3*H?^=)**)=#=FY7$Fg[r$\"3-uHfY=O**=FY7$Fddl$\"3-gxqMh
#=!>FY7$$\"3$[fP!zem0YF*$\"3s=4uE5.F=FY7$F_\\r$\"3'e&QgXJ#*y;FY7$$\"37
A%43\"eMpYF*$\"3]!\\ZfV-xY\"FY7$Fidl$\"3st0#\\\\/%47FY7$$\"3'4!oz$[\"*
=x%F*$\"3y\\o'\\h]_y&FI7$F^el$\"3`TJVG\\i-5FI7$$\"3g(**[W/]z([F*$\"3#f
Z6L'3lMVF37$$\"3TI(**z*GI8\\F*$\"3MT#)pYJG%*QF97$$\"3Aj/b^dl[\\F*$\"3=
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rrigWFY7$$\"3swvi*fa?F&F*$\"3_-8AR$QNh%FY7$Ff^o$\"3/HMbV^/fXFY7$$\"3R%
4Gz!))))R`F*$\"3G!e:&)\\`*)G%FY7$Fbfl$\"3jb)y<\")*z9QFY7$F^_o$\"391<wu
*R$4CFY7$Fgfl$\"3A32Q)o*o'o)FI7$$\"3f!R7hE3-a&F*$\"3of;kX;h(Q$FI7$Fc_r
$\"3s`O7oA7\"o$F97$$\"3%4*=Es\"4je&F*$\"32;$Gl:eE/$F\\`l7$$\"31\"Ry49w
;g&F*$\"3c?%pH*z%fb&F97$$\"3=\"*[p4J/<cF*$\"3j.]G%e\\\"R@FI7$F\\gl$\"3
[#p95wiL'[FI7$F[`r$\"3h#*Qf)>e[\"HFY7$Fagl$\"3;Ej-REE_yFY7$$\"3eLVlRh%
*)y&F*$\"3Grlq#4r4<\"F*7$Fc`r$\"3A3='zNO#Q;F*7$$\"3E;inin9ceF*$\"3kM6r
(e!=#=#F*7$Ffgl$\"37D5HC-5%z#F*7$$\"3$*)4)p&QZL#fF*$\"3'e`#**4ndhMF*7$
F[ar$\"3Y=@f]U#)oTF*7$$\"3g\")*>(3![0*fF*$\"3aU\"fW7Fs*[F*7$F[hl$\"3-&
[4TAWgi&F*7$$\"3Y6#H[fjd0'F*$\"3C\"zHuTsCH'F*7$F`hl$\"39.WIs56@pF*7$$
\"3M*yDS9%**=hF*$\"3-0;jd#[P\\(F*7$Fehl$\"31vnJn1M$*zF*7$F_il$\"3eBZ4;
a7:()F*7$Fiil$\"3e;^v#ycx**)F*7$Fcjl$\"3[)H],0'>'y)F*7$F][m$\"33yQuB5h
&3)F*7$$\"32)zHW(eIUkF*$\"3]k)>jP9#zvF*7$Fb[m$\"3Qs#\\4-2-*pF*7$$\"3@C
[R<+S3lF*$\"3?#GiLX)zPjF*7$Fg[m$\"3l-eEX+pUcF*7$$\"3)Q@1<kcPd'F*$\"3C9
21#*yCU\\F*7$F_br$\"3]1z6&[p4C%F*7$$\"3>mROZdPQmF*$\"3'*e*pfsCub$F*7$F
\\\\m$\"3qquA$*>V3HF*7$$\"3Q><-`[*Hq'F*$\"3Jh.rOPY3BF*7$Fgbr$\"3MDBqH8
=p<F*7$$\"3or%z'eRhnnF*$\"3q:VjRu6*H\"F*7$Fa\\m$\"37X#)z=4;N!*FY7$F_cr
$\"3\\KE2y%\\B@$FY7$Ff\\m$\"3iC=1Hdz$f%FI7$$\"3#=zS`\\O@&pF*$\"3u/DF&[
dww\"FI7$$\"3g_**4\\!\\!ppF*$\"3))e93<O+#*HF97$$\"3Q8\"fGghf)pF*$\"319
$)Q$**yi4%F37$Fgcr$\"3t3#>l(3?1%)F97$$\"3'[fOTE*pOqF*$\"3KVL??\"yQ*\\F
I7$F[]m$\"3a?O;8x'H:\"FY7$Fbco$\"3A#p3]#\\:jEFY7$F`]m$\"3yiHY)Q3y%RFY7
$$\"3f!e-aO7MB(F*$\"3%)[!)fsrThVFY7$Fjco$\"3SY*Hk@yGe%FY7$$\"3aX;!Hcn&
)H(F*$\"3#>_B/.v`g%FY7$Fe]m$\"3!R*\\?^/`QWFY7$Fj]m$\"3u9i^8AJICFY7$F_^
m$\"3Wk;B(GQI&QFI7$$\"3ga2A%*pqIwF*$\"3#zXblcb7N\"FI7$$\"3)fRCnUYPm(F*
$\"3'=_\"))fR(fO\"F97$$\"3OP!G#fey'p(F*$\"3%3AF_#QR+;F97$Fd^m$\"33ermB
uC\"G\"FI7$$\"3_h*Qn:/fz(F*$\"388Zg*\\4r!fFI7$Fi^m$\"3yj1()*\\,;=\"FY7
$Fgfr$\"3XI(y:w\\Dk\"FY7$F^_m$\"3!)y0xa!R#))=FY7$F_gr$\"3$[>P$4tu3>FY7
$Fdgr$\"3sBCYK8tf=FY7$Figr$\"3[Pd<yJSY<FY7$Fc_m$\"3_-(R%eyEy:FY7$F^fo$
\"3/K&=HL'**f5FY7$Fh_m$\"3#H`'ye)fn9&FI7$Fffo$\"3>@(4$[&*f2HFI7$F[go$
\"3G6SVTEwR7FI7$F`go$\"3cw1MO)y!yDF97$F]`m$\"3')G=K\"4KT`*F\\`l7$Fb`m$
\"3@)Rg0\")o7A%FI7$Fg`m$\"3#*pzq`@n$3\"FY7$$\"33Y%G&\\QQl')F*$\"3)ooGu
V=E>\"FY7$F]jr$\"3]yg[u8K^7FY7$$\"363@d'R[<t)F*$\"31$\\^<c<`D\"FY7$F\\
am$\"3m+SaOJ#\\?\"FY7$Fjjr$\"3#e@P-Y\"))['*FI7$Faam$\"3/p0z>j\")[hFI7$
$\"3_Zui.\"z6'*)F*$\"3co#f%G\"R'\\GFI7$Ffam$\"3'e#pGHev*)fF97$$\"3IAnz
7\"ek0*F*$\"3/i#*)e)Q&*3')F37$$\"3]9)>Dy<#)3*F*$\"3-p&[%RWb;OF37$$\"3o
1HC_u(*>\"*F*$\"3!eHq4*yK`WF97$F[bm$\"33*G\"pTX5u7FI7$Febm$\"3]e.#=E**
RX(FI7$Ficm$\"39.6666666FY-F_dm6&FadmFbdmFedmFbdm-%+AXESLABELSG6$Q\"x6
\"Q!Fefv-%%VIEWG6$;$!+aEfTJ!\"*$\"+izxC%*F]gv%(DEFAULTG" 1 2 0 1 10 0 
2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve \+
3" "Curve 4" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 37 "Some \+
properties of the Fejer kernel  " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 26 "The Fejer kernel of order " 
}{TEXT 269 1 "n" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "Phi[n](x) = PIECEWISE([sin((n+1)*x/2)^2/((n+1)*sin(x/2)
^2), x <> 0],[n+1, x = 0]);" "6#/-&%$PhiG6#%\"nG6#%\"xG-%*PIECEWISEG6$
7$*&-%$sinG6#*(,&F(\"\"\"F5F5F5F*F5\"\"#!\"\"F6*&,&F(F5F5F5F5*$-F16#*&
F*F5F6F7F6F5F70F*\"\"!7$,&F(F5F5F5/F*F?" }{TEXT -1 2 "  " }}{PARA 0 "
" 0 "" {TEXT -1 30 "has the following properties. " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "0 <= \+
Phi[n](x);" "6#1\"\"!-&%$PhiG6#%\"nG6#%\"xG" }{TEXT -1 9 " for all " }
{TEXT 270 1 "x" }}{PARA 15 "" 0 "" {TEXT -1 3 "As " }{XPPEDIT 18 0 "n-
> infinity" "6#f*6#%\"nG7\"6$%)operatorG%&arrowG6\"%)infinityGF*F*F*" 
}{TEXT -1 2 ", " }{XPPEDIT 18 0 "Phi[n](x);" "6#-&%$PhiG6#%\"nG6#%\"xG
" }{TEXT -1 42 " converges to 0 uniformly in the interval " }{XPPEDIT 
18 0 "[-Pi,Pi]" "6#7$,$%#PiG!\"\"F%" }{TEXT -1 25 " outside of any int
erval " }{XPPEDIT 18 0 "[-delta,delta]" "6#7$,$%&deltaG!\"\"F%" }
{TEXT -1 31 ", for any positive real number " }{XPPEDIT 18 0 "delta" "
6#%&deltaG" }{TEXT -1 22 ", no matter how small." }}{PARA 15 "" 0 "" 
{TEXT -1 0 "" }{XPPEDIT 18 0 "1/(2*Pi)" "6#*&\"\"\"F$*&\"\"#F$%#PiGF$!
\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(Phi[n](x),x = -Pi .. Pi) = 1
;" "6#/-%$IntG6$-&%$PhiG6#%\"nG6#%\"xG/F-;,$%#PiG!\"\"F1\"\"\"" }
{TEXT -1 2 ". " }}{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 ";" }}}{SECT 1 {PARA 4 "" 0 "
" {TEXT -1 11 "Explanation" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 
";" }}}{PARA 0 "" 0 "" {TEXT -1 68 "The first property follows immedia
tely from the formula given above." }}{PARA 0 "" 0 "" {TEXT -1 49 "To \+
see why the 2nd property holds, note that for " }{XPPEDIT 18 0 "x<>0" 
"6#0%\"xG\"\"!" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "Phi[n](x) <= 1/((n+1)*sin(x/2)^2);" "6#1-&%$PhiG6#%\"nG
6#%\"xG*&\"\"\"F,*&,&F(F,F,F,F,*$-%$sinG6#*&F*F,\"\"#!\"\"F4F,F5" }
{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 29 "For any positive real nu
mber " }{XPPEDIT 18 0 "delta" "6#%&deltaG" }{TEXT -1 23 ", no matter h
ow small, " }}{PARA 0 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "delta<=a
bs(x)" "6#1%&deltaG-%$absG6#%\"xG" }{XPPEDIT 18 0 "``<=Pi" "6#1%!G%#Pi
G" }{TEXT -1 16 "  implies that  " }{XPPEDIT 18 0 "sin(delta/2)^2<=sin
(x/2)^2" "6#1*$-%$sinG6#*&%&deltaG\"\"\"\"\"#!\"\"F+*$-F&6#*&%\"xGF*F+
F,F+" }{XPPEDIT 18 0 "``<=1" "6#1%!G\"\"\"" }{TEXT -1 6 " and, " }
{XPPEDIT 18 0 "1 <= 1/(sin(x/2)^2);" "6#1\"\"\"*&F$F$*$-%$sinG6#*&%\"x
GF$\"\"#!\"\"F,F-" }{XPPEDIT 18 0 "``<=1/sin(delta/2)^2" "6#1%!G*&\"\"
\"F&*$-%$sinG6#*&%&deltaGF&\"\"#!\"\"F-F." }{TEXT -1 1 "." }}{PARA 0 "
" 0 "" {TEXT -1 12 "Hence, when " }{XPPEDIT 18 0 "delta<=abs(x)" "6#1%
&deltaG-%$absG6#%\"xG" }{XPPEDIT 18 0 "``<=Pi" "6#1%!G%#PiG" }{TEXT 
-1 9 ", we have" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Ph
i[n](x) <= 1/((n+1)*sin(delta/2)^2);" "6#1-&%$PhiG6#%\"nG6#%\"xG*&\"\"
\"F,*&,&F(F,F,F,F,*$-%$sinG6#*&%&deltaGF,\"\"#!\"\"F5F,F6" }{TEXT -1 
3 ",  " }}{PARA 0 "" 0 "" {TEXT -1 20 "which tends to 0 as " }
{XPPEDIT 18 0 "n -> infinity" "6#f*6#%\"nG7\"6$%)operatorG%&arrowG6\"%
)infinityGF*F*F*" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 53 "The 3rd property can be deduced by using \+
the formula " }{XPPEDIT 18 0 "Phi[n](x) = 1+2;" "6#/-&%$PhiG6#%\"nG6#%
\"xG,&\"\"\"F,\"\"#F," }{TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(``((n-k+1)/
(n+1))*cos*k*x,k = 1 .. n);" "6#-%$SumG6$**-%!G6#*&,(%\"nG\"\"\"%\"kG!
\"\"F-F-F-,&F,F-F-F-F/F-%$cosGF-F.F-%\"xGF-/F.;F-F," }{TEXT -1 2 ". " 
}}{PARA 0 "" 0 "" {TEXT -1 8 "We have " }}{PARA 256 "" 0 "" {XPPEDIT 
18 0 "1/(2*Pi);" "6#*&\"\"\"F$*&\"\"#F$%#PiGF$!\"\"" }{TEXT -1 1 " " }
{XPPEDIT 18 0 "Int(Phi[n](x),x = -Pi .. Pi) = 1/(2*Pi);" "6#/-%$IntG6$
-&%$PhiG6#%\"nG6#%\"xG/F-;,$%#PiG!\"\"F1*&\"\"\"F4*&\"\"#F4F1F4F2" }
{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(1,x = -pi .. Pi)+1/Pi;" "6#,&-%$Int
G6$\"\"\"/%\"xG;,$%#piG!\"\"%#PiGF'*&F'F'F.F-F'" }{XPPEDIT 18 0 "Sum(I
nt(``((n-k+1)/(n+1))*cos*k*x,x = -Pi .. Pi),k = 1 .. n);" "6#-%$SumG6$
-%$IntG6$**-%!G6#*&,(%\"nG\"\"\"%\"kG!\"\"F0F0F0,&F/F0F0F0F2F0%$cosGF0
F1F0%\"xGF0/F5;,$%#PiGF2F9/F1;F0F/" }{TEXT -1 1 " " }}{PARA 256 "" 0 "
" {TEXT -1 3 " = " }{XPPEDIT 18 0 "1+0 = 1;" "6#/,&\"\"\"F%\"\"!F%F%" 
}{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT 261 4 "Note" }{TEXT -1 97 ": Maple can check the 3rd property of
 the Fejer kernel directly by integration in specific cases." }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "Ph
i := (x,n) -> sin((n+1)*x/2)^2/((n+1)*sin(x/2)^2);\n1/(2*Pi)*Int(Phi(x
,10),x=-Pi..Pi);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$PhiG
f*6$%\"xG%\"nG6\"6$%)operatorG%&arrowGF)*(-%$sinG6#,$*&#\"\"\"\"\"#F4*
&9$F4,&F4F49%F4F4F4F4F5F8!\"\"-F/6#,$*&F3F4F7F4F4!\"#F)F)F)" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#,$*&#\"\"\"\"\"#F&,$-%$IntG6$,$*&#F&\"#6F&*&
-%$sinG6#,$*(F/F&F'!\"\"%\"xGF&F&F'-F26#,$*&F'F6F7F&F&!\"#F&F&/F7;,$%#
PiGF6F@*$F@F6F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" 
}}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "
" 0 "" {TEXT -1 16 "Fejer's theorem " }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 
0 "phi(x)" "6#-%$phiG6#%\"xG" }{TEXT -1 51 " be a complex valued perio
dic function with period " }{XPPEDIT 18 0 "2*L" "6#*&\"\"#\"\"\"%\"LGF
%" }{TEXT -1 35 " which is absolutely integrable on " }{XPPEDIT 18 0 "
[-L,L]" "6#7$,$%\"LG!\"\"F%" }{TEXT -1 29 ", and so has a Fourier seri
es" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(c[k]*exp(k*
Pi*i*x/L),k = -infinity .. infinity);" "6#-%$SumG6$*&&%\"cG6#%\"kG\"\"
\"-%$expG6#*,F*F+%#PiGF+%\"iGF+%\"xGF+%\"LG!\"\"F+/F*;,$%)infinityGF4F
8" }{TEXT -1 1 "," }}{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(ph
i(x)*exp(k*Pi*i*x/L),x = -L .. L);" "6#-%$IntG6$*&-%$phiG6#%\"xG\"\"\"
-%$expG6#*,%\"kGF+%#PiGF+%\"iGF+F*F+%\"LG!\"\"F+/F*;,$F3F4F3" }{TEXT 
-1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "sigma[n](x)
 = Sum((n-abs(k)+1)*c[k]*exp(k*Pi*i*x/L)/(n+1),k = -n .. n);" "6#/-&%&
sigmaG6#%\"nG6#%\"xG-%$SumG6$**,(F(\"\"\"-%$absG6#%\"kG!\"\"F0F0F0&%\"
cG6#F4F0-%$expG6#*,F4F0%#PiGF0%\"iGF0F*F0%\"LGF5F0,&F(F0F0F0F5/F4;,$F(
F5F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 7 "be the " }{TEXT 
271 1 "n" }{TEXT -1 4 " th " }{TEXT 261 10 "Cesaro sum" }{TEXT -1 1 ".
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "If " }
{XPPEDIT 18 0 "phi(x)" "6#-%$phiG6#%\"xG" }{TEXT -1 18 " is continuous
 at " }{XPPEDIT 18 0 "x=x[0]" "6#/%\"xG&F$6#\"\"!" }{TEXT -1 7 ", then
 " }{XPPEDIT 18 0 "sigma[n](x[0]);" "6#-&%&sigmaG6#%\"nG6#&%\"xG6#\"\"
!" }{TEXT -1 14 " converges to " }{XPPEDIT 18 0 "phi(x[0]);" "6#-%$phi
G6#&%\"xG6#\"\"!" }{TEXT -1 18 ". \nFurthemore, if " }{XPPEDIT 18 0 "p
hi(x)" "6#-%$phiG6#%\"xG" }{TEXT -1 33 " is continuous (everywhere) th
en " }{XPPEDIT 18 0 "sigma[n](x)" "6#-&%&sigmaG6#%\"nG6#%\"xG" }{TEXT 
-1 24 " converges uniformly to " }{XPPEDIT 18 0 "phi(x)" "6#-%$phiG6#%
\"xG" }{TEXT -1 48 " on any interval with width equal to the period." 
}}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 263 17 "_________________" }
{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 12 "Explan
ation " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 
"" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "phi(x)" "6#-%$phiG6#%\"xG" }
{TEXT -1 29 " is absolutely integrable on " }{XPPEDIT 18 0 "[-L,L]" "6
#7$,$%\"LG!\"\"F%" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "phi(x)" "6#-%$phiG
6#%\"xG" }{TEXT -1 57 " is bounded, that is, we can find a positive re
al number " }{TEXT 272 1 "M" }{TEXT -1 11 " such that " }{XPPEDIT 18 
0 "abs(phi(x)) <= M;" "6#1-%$absG6#-%$phiG6#%\"xG%\"MG" }{TEXT -1 6 " \+
when " }{XPPEDIT 18 0 "-L<=x" "6#1,$%\"LG!\"\"%\"xG" }{XPPEDIT 18 0 "`
`<=L" "6#1%!G%\"LG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 57 "Suppose that we are given a (small) posit
ive real number " }{XPPEDIT 18 0 "epsilon" "6#%(epsilonG" }{TEXT -1 1 
"." }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "phi(x)" "6#-
%$phiG6#%\"xG" }{TEXT -1 18 " is continuous at " }{TEXT 273 1 "x" }
{TEXT -1 37 ", we can find a positive real number " }{XPPEDIT 18 0 "de
lta" "6#%&deltaG" }{TEXT -1 9 " so that:" }}{PARA 256 "" 0 "" 
{XPPEDIT 18 0 "abs(phi(x)-phi(y)) < epsilon/2;" "6#2-%$absG6#,&-%$phiG
6#%\"xG\"\"\"-F)6#%\"yG!\"\"*&%(epsilonGF,\"\"#F0" }{TEXT -1 6 " when \+
" }{XPPEDIT 18 0 "abs(x-y)<delta" "6#2-%$absG6#,&%\"xG\"\"\"%\"yG!\"\"
%&deltaG" }{TEXT -1 14 "  ------- (i)." }}{PARA 0 "" 0 "" {TEXT -1 
115 "Then, from the 2nd property of the Fejer kernel considered in the
 previous section, we can find a positive integer " }{TEXT 274 1 "N" }
{TEXT -1 9 " so that:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 
18 0 "abs(Phi[n](Pi*x/L)) <= epsilon/(4*M);" "6#1-%$absG6#-&%$PhiG6#%
\"nG6#*(%#PiG\"\"\"%\"xGF/%\"LG!\"\"*&%(epsilonGF/*&\"\"%F/%\"MGF/F2" 
}{TEXT -1 8 "  when  " }{XPPEDIT 18 0 "delta <= abs(x);" "6#1%&deltaG-
%$absG6#%\"xG" }{XPPEDIT 18 0 "`` <= L;" "6#1%!G%\"LG" }{TEXT -1 7 "  \+
and  " }{XPPEDIT 18 0 "n>=N" "6#1%\"NG%\"nG" }{TEXT -1 15 "  ------- (
ii)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 " \+
From the 3rd property of the Fejer kernel " }}{PARA 256 "" 0 "" {TEXT 
-1 1 " " }{XPPEDIT 18 0 "1/(2*L)" "6#*&\"\"\"F$*&\"\"#F$%\"LGF$!\"\"" 
}{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(Phi[n](Pi*x/L),x = -L .. L) = 1/(2
*Pi);" "6#/-%$IntG6$-&%$PhiG6#%\"nG6#*(%#PiG\"\"\"%\"xGF/%\"LG!\"\"/F0
;,$F1F2F1*&F/F/*&\"\"#F/F.F/F2" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(Ph
i[n](u),u = -Pi .. Pi) = 1;" "6#/-%$IntG6$-&%$PhiG6#%\"nG6#%\"uG/F-;,$
%#PiG!\"\"F1\"\"\"" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 6 "wher
e " }{XPPEDIT 18 0 "u = Pi*x/L" "6#/%\"uG*(%#PiG\"\"\"%\"xGF'%\"LG!\"
\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 24 "Thus, for a fixed v
alue " }{XPPEDIT 18 0 "x=x[0]" "6#/%\"xG&F$6#\"\"!" }{TEXT -1 11 ", we
 have: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "phi(x[0]) \+
= 1/(2*L);" "6#/-%$phiG6#&%\"xG6#\"\"!*&\"\"\"F,*&\"\"#F,%\"LGF,!\"\"
" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(phi(x[0])*Phi[n](Pi*x/L),x = -L \+
.. L);" "6#-%$IntG6$*&-%$phiG6#&%\"xG6#\"\"!\"\"\"-&%$PhiG6#%\"nG6#*(%
#PiGF.F+F.%\"LG!\"\"F./F+;,$F7F8F7" }{TEXT -1 18 "  ------- (iii).  " 
}}{PARA 0 "" 0 "" {TEXT -1 24 "Making the substitution " }{XPPEDIT 18 
0 "t =  x[0] - x" "6#/%\"tG,&&%\"xG6#\"\"!\"\"\"F'!\"\"" }{TEXT -1 9 "
, we have" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "sigma[n]
(x[0]) = 1/(2*L);" "6#/-&%&sigmaG6#%\"nG6#&%\"xG6#\"\"!*&\"\"\"F/*&\"
\"#F/%\"LGF/!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(phi(x)*Phi[n](P
i*(x[0]-x)/L),x = -L .. L);" "6#-%$IntG6$*&-%$phiG6#%\"xG\"\"\"-&%$Phi
G6#%\"nG6#*(%#PiGF+,&&F*6#\"\"!F+F*!\"\"F+%\"LGF8F+/F*;,$F9F8F9" }
{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` =
 -1/(2*L);" "6#/%!G,$*&\"\"\"F'*&\"\"#F'%\"LGF'!\"\"F+" }{TEXT -1 1 " \+
" }{XPPEDIT 18 0 "Int(phi(x[0]-t)*Phi[n](Pi*t/L),t = x+L .. x[0]-L);" 
"6#-%$IntG6$*&-%$phiG6#,&&%\"xG6#\"\"!\"\"\"%\"tG!\"\"F/-&%$PhiG6#%\"n
G6#*(%#PiGF/F0F/%\"LGF1F//F0;,&F,F/F:F/,&&F,6#F.F/F:F1" }{TEXT -1 1 " \+
" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/(2*L);" "6
#/%!G*&\"\"\"F&*&\"\"#F&%\"LGF&!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "
Int(phi(x[0]-t)*Phi[n](Pi*t/L),t = x[0]-L .. x[0]+L);" "6#-%$IntG6$*&-
%$phiG6#,&&%\"xG6#\"\"!\"\"\"%\"tG!\"\"F/-&%$PhiG6#%\"nG6#*(%#PiGF/F0F
/%\"LGF1F//F0;,&&F,6#F.F/F:F1,&&F,6#F.F/F:F/" }{TEXT -1 1 " " }}{PARA 
256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/(2*L);" "6#/%!G*&\"
\"\"F&*&\"\"#F&%\"LGF&!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(phi(x
[0]-t)*Phi[n](Pi*t/L),t = -L .. L);" "6#-%$IntG6$*&-%$phiG6#,&&%\"xG6#
\"\"!\"\"\"%\"tG!\"\"F/-&%$PhiG6#%\"nG6#*(%#PiGF/F0F/%\"LGF1F//F0;,$F:
F1F:" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 8 "because " }
{XPPEDIT 18 0 "f(x[0]-t)*Phi[n](Pi*t/L);" "6#*&-%\"fG6#,&&%\"xG6#\"\"!
\"\"\"%\"tG!\"\"F,-&%$PhiG6#%\"nG6#*(%#PiGF,F-F,%\"LGF.F," }{TEXT -1 
27 " is a periodic function of " }{TEXT 275 1 "t" }{TEXT -1 13 " with \+
period " }{XPPEDIT 18 0 "2*L" "6#*&\"\"#\"\"\"%\"LGF%" }{TEXT -1 1 ".
" }}{PARA 0 "" 0 "" {TEXT -1 22 "Changing the variable " }{TEXT 276 1 
"t" }{TEXT -1 20 " in the integral to " }{TEXT 277 1 "x" }{TEXT -1 8 "
 we have" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "sigma[n](
x[0]) = 1/(2*L);" "6#/-&%&sigmaG6#%\"nG6#&%\"xG6#\"\"!*&\"\"\"F/*&\"\"
#F/%\"LGF/!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(phi(x[0]-x)*Phi[n
](Pi*x/L),x = -L .. L);" "6#-%$IntG6$*&-%$phiG6#,&&%\"xG6#\"\"!\"\"\"F
,!\"\"F/-&%$PhiG6#%\"nG6#*(%#PiGF/F,F/%\"LGF0F//F,;,$F9F0F9" }{TEXT 
-1 16 "  ------- (iv). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 7 " When  " }{XPPEDIT 18 0 "n>= N" "6#1%\"NG%\"nG" }
{TEXT -1 8 " we have" }}{PARA 256 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 
18 0 "abs(sigma[n](x[0])-phi(x[0])) = 1/(2*L);" "6#/-%$absG6#,&-&%&sig
maG6#%\"nG6#&%\"xG6#\"\"!\"\"\"-%$phiG6#&F/6#F1!\"\"*&F2F2*&\"\"#F2%\"
LGF2F8" }{TEXT -1 1 " " }{XPPEDIT 18 0 "abs(Int(phi(x[0]-x)*Phi[n](Pi*
x/L),x = -L .. L)-Int(phi(x[0])*Phi[n](Pi*x[0]/L),x = -L .. L));" "6#-
%$absG6#,&-%$IntG6$*&-%$phiG6#,&&%\"xG6#\"\"!\"\"\"F0!\"\"F3-&%$PhiG6#
%\"nG6#*(%#PiGF3F0F3%\"LGF4F3/F0;,$F=F4F=F3-F(6$*&-F,6#&F06#F2F3-&F76#
F96#*(F<F3&F06#F2F3F=F4F3/F0;,$F=F4F=F4" }{TEXT -1 22 "  using (iii) a
nd (iv)" }}{PARA 256 "" 0 "" {TEXT -1 3 " = " }{XPPEDIT 18 0 "1/(2*L);
" "6#*&\"\"\"F$*&\"\"#F$%\"LGF$!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "
abs(Int([phi(x[0]-x)-phi(x[0])]*Phi[n](Pi*x[0]/L),x = -L .. L));" "6#-
%$absG6#-%$IntG6$*&7#,&-%$phiG6#,&&%\"xG6#\"\"!\"\"\"F1!\"\"F4-F-6#&F1
6#F3F5F4-&%$PhiG6#%\"nG6#*(%#PiGF4&F16#F3F4%\"LGF5F4/F1;,$FDF5FD" }
{TEXT -1 2 "  " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" 
{TEXT -1 3 "   " }{XPPEDIT 18 0 "`` <= 1/(2*L);" "6#1%!G*&\"\"\"F&*&\"
\"#F&%\"LGF&!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "abs(Int([phi(x[0]-x
)-phi(x[0])]*Phi[n](Pi*x[0]/L),x = -L .. -delta))+abs(Int([phi(x[0]-x)
-phi(x[0])]*Phi[n](Pi*x[0]/L),x = -delta .. delta));" "6#,&-%$absG6#-%
$IntG6$*&7#,&-%$phiG6#,&&%\"xG6#\"\"!\"\"\"F2!\"\"F5-F.6#&F26#F4F6F5-&
%$PhiG6#%\"nG6#*(%#PiGF5&F26#F4F5%\"LGF6F5/F2;,$FEF6,$%&deltaGF6F5-F%6
#-F(6$*&7#,&-F.6#,&&F26#F4F5F2F6F5-F.6#&F26#F4F6F5-&F=6#F?6#*(FBF5&F26
#F4F5FEF6F5/F2;,$FJF6FJF5" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT 
-1 48 "                                              + " }{XPPEDIT 18 
0 "abs(Int([phi(x[0]-x)-phi(x[0])]*Phi[n](Pi*x[0]/L),x = delta .. L));
" "6#-%$absG6#-%$IntG6$*&7#,&-%$phiG6#,&&%\"xG6#\"\"!\"\"\"F1!\"\"F4-F
-6#&F16#F3F5F4-&%$PhiG6#%\"nG6#*(%#PiGF4&F16#F3F4%\"LGF5F4/F1;%&deltaG
FD" }{TEXT -1 2 "  " }}{PARA 256 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 
0 "``<= 1/(2*L)" "6#1%!G*&\"\"\"F&*&\"\"#F&%\"LGF&!\"\"" }{TEXT -1 1 "
 " }{XPPEDIT 18 0 "``(Int(2*M*Phi[n](Pi*x[0]/L),x = -L .. -delta)+Int(
``(epsilon/2)*Phi[n](Pi*x[0]/L),x = -delta .. delta)+Int(2*M*Phi[n](Pi
*x[0]/L),x = delta .. L));" "6#-%!G6#,(-%$IntG6$*(\"\"#\"\"\"%\"MGF,-&
%$PhiG6#%\"nG6#*(%#PiGF,&%\"xG6#\"\"!F,%\"LG!\"\"F,/F7;,$F:F;,$%&delta
GF;F,-F(6$*&-F$6#*&%(epsilonGF,F+F;F,-&F06#F26#*(F5F,&F76#F9F,F:F;F,/F
7;,$F@F;F@F,-F(6$*(F+F,F-F,-&F06#F26#*(F5F,&F76#F9F,F:F;F,/F7;F@F:F," 
}{TEXT -1 10 " using (i)" }}{PARA 256 "" 0 "" {TEXT -1 2 "  " }
{XPPEDIT 18 0 "``<= M/L" "6#1%!G*&%\"MG\"\"\"%\"LG!\"\"" }{TEXT -1 1 "
 " }{XPPEDIT 18 0 "Int(Phi[n](Pi*x[0]/L),x = -L .. -delta)+epsilon/2;
" "6#,&-%$IntG6$-&%$PhiG6#%\"nG6#*(%#PiG\"\"\"&%\"xG6#\"\"!F/%\"LG!\"
\"/F1;,$F4F5,$%&deltaGF5F/*&%(epsilonGF/\"\"#F5F/" }{TEXT -1 1 " " }
{XPPEDIT 18 0 "Int(Phi[n](Pi*x[0]/L),x = -delta .. delta)+M/L;" "6#,&-
%$IntG6$-&%$PhiG6#%\"nG6#*(%#PiG\"\"\"&%\"xG6#\"\"!F/%\"LG!\"\"/F1;,$%
&deltaGF5F9F/*&%\"MGF/F4F5F/" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(Phi[
n](Pi*x[0]/L),x = delta .. L);" "6#-%$IntG6$-&%$PhiG6#%\"nG6#*(%#PiG\"
\"\"&%\"xG6#\"\"!F.%\"LG!\"\"/F0;%&deltaGF3" }{TEXT -1 1 " " }}{PARA 
256 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "``<= M/L" "6#1%!G*&%\"MG\"
\"\"%\"LG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(epsilon/(4*M),x = \+
-L .. -delta)+``(epsilon/2);" "6#,&-%$IntG6$*&%(epsilonG\"\"\"*&\"\"%F
)%\"MGF)!\"\"/%\"xG;,$%\"LGF-,$%&deltaGF-F)-%!G6#*&F(F)\"\"#F-F)" }
{TEXT -1 1 " " }{XPPEDIT 18 0 "1+M/L;" "6#,&\"\"\"F$*&%\"MGF$%\"LG!\"
\"F$" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(epsilon/(4*M),x = delta .. L
);" "6#-%$IntG6$*&%(epsilonG\"\"\"*&\"\"%F(%\"MGF(!\"\"/%\"xG;%&deltaG
%\"LG" }{TEXT -1 13 ",  using (ii)" }}{PARA 256 "" 0 "" {TEXT -1 2 "  \+
" }{XPPEDIT 18 0 "``<= epsilon/4 + epsilon/2 + epsilon/4" "6#1%!G,(*&%
(epsilonG\"\"\"\"\"%!\"\"F(*&F'F(\"\"#F*F(*&F'F(F)F*F(" }{TEXT -1 3 " \+
= " }{XPPEDIT 18 0 "epsilon" "6#%(epsilonG" }{TEXT -1 2 ". " }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "If " }{XPPEDIT 
18 0 "phi(x)" "6#-%$phiG6#%\"xG" }{TEXT -1 31 " is continuous on the i
nterval " }{XPPEDIT 18 0 "[-L,L]" "6#7$,$%\"LG!\"\"F%" }{TEXT -1 2 ", \+
" }{XPPEDIT 18 0 "phi(x)" "6#-%$phiG6#%\"xG" }{TEXT -1 13 " is actuall
y " }{TEXT 261 20 "uniformly continuous" }{TEXT -1 29 ". In this case \+
we can choose " }{XPPEDIT 18 0 "delta" "6#%&deltaG" }{TEXT -1 46 " and
 N in the argument above independently of " }{XPPEDIT 18 0 "x[0]" "6#&
%\"xG6#\"\"!" }{TEXT -1 38 ", which shows that the convergence of " }
{XPPEDIT 18 0 "sigma[n](x);" "6#-&%&sigmaG6#%\"nG6#%\"xG" }{TEXT -1 4 
" to " }{XPPEDIT 18 0 "phi(x)" "6#-%$phiG6#%\"xG" }{TEXT -1 4 " is " }
{TEXT 261 7 "uniform" }{TEXT -1 17 " in the interval " }{XPPEDIT 18 0 
"[-L,L]" "6#7$,$%\"LG!\"\"F%" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{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 }