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$$\"3O+vV)e#3'[#F/$\"3#**>YW'Rqv5Fbo7$$\"3`++]Ujp-DF/$\"3Pb%>M\"Q:35Fb
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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 ""
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$Fbs$\"11zo^(\\lQ$F]q7$$!1)*\\PM,n)=#Ffn$\"1/+mo\"4G/*F]q7$Fg_l$\"1L5U
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*\\i:sw%)*F]q$\"1*y4X#R%eV$Ffn7$F_`l$\"1\"G7F31O(RFfn7$$!1***\\(=K**zM
F]q$\"1^\\&31+UZ%Ffn7$F\\t$\"17Qt*fub&\\Ffn7$$\"1&**\\ilg4,$F]q$\"1eJ*
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fn7$Fat$\"1KD/@*4q8(Ffn7$$\"1***\\(oc6\"e\"Ffn$\"1p/s_g!ys(Ffn7$F_al$
\"1!)[eT+Pp$)Ffn7$$\"1***\\7B')o:#Ffn$\"1!*o1f4#e-*Ffn7$Fft$\"1f$zr?H'
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QY**Ffn7$$\"1]7y+lULGFfn$\"19M(*QXAy**Ffn7$$\"1+vV)\\d6\"HFfn$\"1GMw*[
`O***Ffn7$$\"1]P4'\\)))))HFfn$\"1Vim#4Q*)***Ffn7$Fgal$\"1VvqDr$*****Ff
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}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "c = (b-a)/
2" "6#/%\"cG*&,&%\"bG\"\"\"%\"aG!\"\"F(\"\"#F*" }{TEXT -1 12 " and cho
ose " }{TEXT 289 1 "n" }{TEXT -1 9 " so that " }{XPPEDIT 18 0 "1/n< c
" "6#2*&\"\"\"F%%\"nG!\"\"%\"cG" }{TEXT -1 7 ". Let " }{XPPEDIT 18 0
"d = (a+b)/2" "6#/%\"dG*&,&%\"aG\"\"\"%\"bGF(F(\"\"#!\"\"" }{TEXT -1
1 "." }}{PARA 0 "" 0 "" {TEXT -1 7 "Define " }{XPPEDIT 18 0 "g(x)" "6#
-%\"gG6#%\"xG" }{TEXT -1 3 " by" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "g(x) = [h[n](x)(x+d)-h[n](x-d)]*f(x);" "6#/-%\"gG6#%\"x
G*&7#,&--&%\"hG6#%\"nG6#F'6#,&F'\"\"\"%\"dGF4F4-&F.6#F06#,&F'F4F5!\"\"
F;F4-%\"fG6#F'F4" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 16 "For example let " }{XPPEDIT 18 0 "f(x)=ln
(1+x^2)+cos(5*x)/4" "6#/-%\"fG6#%\"xG,&-%#lnG6#,&\"\"\"F-*$F'\"\"#F-F-
*&-%$cosG6#*&\"\"&F-F'F-F-\"\"%!\"\"F-" }{TEXT -1 2 ", " }{XPPEDIT 18
0 "a = 3*Pi" "6#/%\"aG*&\"\"$\"\"\"%#PiGF'" }{TEXT -1 5 " and " }
{XPPEDIT 18 0 "b=10" "6#/%\"bG\"#5" }{TEXT -1 1 "." }}{PARA 0 "" 0 ""
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 191 "c := evalf(
(10-3*Pi)/2);\nn := ceil(1/c);\nd := evalf((10+3*Pi)/2);\nf := x ->ln(
1+x^2)+cos(5*x)/4;\ng := x -> (h(n,x+d)-h(n,x-d))*f(x);\nplot(g(x),x=-
11..11,thickness=1,color=COLOR(RGB,.4,0,.9));" }}{PARA 11 "" 1 ""
{XPPMATH 20 "6#>%\"cG$\"*>5h(G!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>%\"nG\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"dG$\"+\")*)Q7(*!\"*
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&a
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$\"1JL3K[H*[&F;$\"1cHmAV8pKF;7$$\"1)\\P%eEA=bF;$\"1J:VCz`aKF;7$$\"1l;z
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XDuNF;7$$\"1Le*[+p_1'F;$\"1VVjP@pZPF;7$$\"1m;/czRyhF;$\"1\"zUrkTW)QF;7
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TmF;$\"1ir%)p9KaPF;7$$\"1(**\\(ozRyoF;$\"1pl=#e/6j$F;7$$\"1L$3x%H`1rF;
$\"15sh&[u8!QF;7$$\"1ommEzmMtF;$\"1.<4z:DLTF;7$$\"1o;z4%=8X(F;$\"1F8\"
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;$\"1ti6%p\\^'RF;7$$\"1voH%R-%>\")F;$\"1%Qiwia4'RF;7$$\"1oT&Q`B$[\")F;
$\"1n#)))>(*zhRF;7$$\"1](oH\"e;1#)F;$\"1OtH1q,zRF;7$$\"1LL3#43SE)F;$\"
1!)*\\S_Il,%F;7$$\"1o;/\"G7mZ)F;$\"1F-!*3UL\"G%F;7$$\"1-++qk@*o)F;$\"1
-Q$fNNAb%F;7$$\"1p\"H#oN8]()F;$\"1.4C8wV%f%F;7$$\"1N$ekm]5\"))F;$\"1#)
)=<&p99YF;7$$\"1=Hd:#4:%))F;$\"1SQBQ4J:YF;7$$\"1,vokx'>())F;$\"1dT())>
63h%F;7$$\"1&3-QJEC!*)F;$\"1*p?mD44g%F;7$$\"1om\"H'[)G$*)F;$\"1]eVf6*f
e%F;7$$\"1+]Pf!>Z0*F;$\"13+M*F;$\"1=[;]S8_UF;7$$\"1K!)H0qmUF;7$$\"1&e*F;$\"1]9!
HJwmZ$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$$\"3
ckmm;$y*eCFT$\"3XNLL$o@5a(FT7$$\"3f)******R^bJ$FT$\"3S,+++'[Wo'FT7$$\"
3'e*****\\5a`TFT$\"3;/++]*ek%eFT7$$\"3'o****\\7RV'\\FT$\"39.++v3mN]FT7
$$\"3Y'*****\\@fkeFT$\"3b.++]ySNTFT7$$\"3_ILLL&4Nn'FT$\"3[pmmm/\\ELFT7
$$\"3A*******\\,s`(FT$\"3y++++&)ziCFT7$$\"3%[mm;zM)>$)FT$\"3;NLL3_;!o
\"FT7$$\"3M*******pfa<*FT$\"3q1+++ISX#)Fgn7$$\"3Ckm;zy*zd*FT$\"3qdLL37
-?UFgn7$$\"39HLLeg`!)**FT$\"3)p3nm;%RY>Fgr7$$\"3Lmm;W/8S5F/F+7$$\"3w**
**\\#G2A3\"F/F+7$$\"3;LLL$)G[k6F/F+7$$\"3#)****\\7yh]7F/F+7$$\"3xmmm')
fdL8F/F+7$$\"3bmmm,FT=9F/F+7$$\"3FLL$e#pa-:F/F+7$$\"3!*******Rv&)z:F/F
+7$$\"3ILLLGUYo;F/F+7$$\"3_mmm1^rZF/F+7$$\"\"#F*F+-%'COLOURG6&%$RGBG$\"#5!\"\"F+F+-%*THICKNESSG6#Ffy
-%+AXESLABELSG6$Q\"x6\"Q!Fez-%%VIEWG6$;F(Fey%(DEFAULTG" 1 2 0 1 10 2
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 43 "Int(exp(1
/(x^2-1)),x=-1..1);\nA := evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#-%$IntG6$-%$expG6#*&\"\"\"F*,&*$)%\"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 "
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