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{SECT 0 {PARA 256 "" 0 "" {TEXT -1 29 "Module 8: General Convergence"
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 99 "In this m
odule, we discuss three types of convergence in C([0, 1]): normed, poi
ntwise, and uniform." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 53 " We suppose that we have a sequence of functions "
}{XPPEDIT 18 0 "f[1](x);" "6#-&%\"fG6#\"\"\"6#%\"xG" }{TEXT -1 3 " , \+
" }{XPPEDIT 18 0 "f[2](x);" "6#-&%\"fG6#\"\"#6#%\"xG" }{TEXT -1 3 " , \+
" }{XPPEDIT 18 0 "f[3](x);" "6#-&%\"fG6#\"\"$6#%\"xG" }{TEXT -1 53 ", \+
... converging to a function g(x). We say that the " }{XPPEDIT 18 0 "f
;" "6#%\"fG" }{TEXT -1 33 " 's converge to g in the sense of" }}{PARA
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "1. " }{TEXT 256
16 "norm convergence" }{TEXT -1 6 " if " }{XPPEDIT 18 0 "int(abs(f[n
](t)-g(t))^2,t = 0 .. 1);" "6#-%$intG6$*$-%$absG6#,&-&%\"fG6#%\"nG6#%
\"tG\"\"\"-%\"gG6#F1!\"\"\"\"#/F1;\"\"!F2" }{TEXT -1 17 " -> 0 as n
-> " }{XPPEDIT 18 0 "infinity;" "6#%)infinityG" }{TEXT -1 2 " ." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "2. " }
{TEXT 257 9 "pointwise" }{TEXT -1 22 " if, for each x, " }
{XPPEDIT 18 0 "f[n](x);" "6#-&%\"fG6#%\"nG6#%\"xG" }{TEXT -1 19 " -> g
(x) as n -> " }{XPPEDIT 18 0 "infinity;" "6#%)infinityG" }{TEXT -1
2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "3
. " }{TEXT 258 9 "uniformly" }{TEXT -1 59 " if the maximum for all x \+
in [0, 1] of the difference in " }{XPPEDIT 18 0 "f[n](x);" "6#-&%\"fG
6#%\"nG6#%\"xG" }{TEXT -1 34 " and g(x) goes to zero as n -> " }
{XPPEDIT 18 0 "infinity;" "6#%)infinityG" }{TEXT -1 2 " ." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "These methods of conver
gence can be contrasted. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 86 "1. Uniform Convergence implies pointwise convergen
ce. To see this, note only that if " }}{PARA 0 "" 0 "" {TEXT -1 2 " \+
" }}{PARA 0 "" 0 "" {TEXT -1 30 " " }
{XPPEDIT 18 0 "MAX[x]*abs(f[n](x)-g(x));" "6#*&&%$MAXG6#%\"xG\"\"\"-%$
absG6#,&-&%\"fG6#%\"nG6#F'F(-%\"gG6#F'!\"\"F(" }{TEXT -1 5 " -> 0" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "then for \+
each x, " }{XPPEDIT 18 0 "f[n](x)-g(x);" "6#,&-&%\"fG6#%\"nG6#%\"
xG\"\"\"-%\"gG6#F*!\"\"" }{TEXT -1 9 " -> 0." }}{PARA 0 "" 0 ""
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT
-1 74 "2. Uniform Convergence implies normed convergence. To see this,
note that " }}{PARA 0 "" 0 "" {TEXT -1 27 " \+
" }}{PARA 0 "" 0 "" {TEXT -1 26 " " }
{XPPEDIT 18 0 "abs(int((f[n](x)-g(x))^2,x = 0 .. 1)) <= MAX[x]*abs(f[n
](x)-g(x))^2;" "6#1-%$absG6#-%$intG6$*$,&-&%\"fG6#%\"nG6#%\"xG\"\"\"-%
\"gG6#F2!\"\"\"\"#/F2;\"\"!F3*&&%$MAXG6#F2F3*$-F%6#,&-&F.6#F06#F2F3-F5
6#F2F7F8F3" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA
0 "" 0 "" {TEXT -1 241 "3. Pointwise convergence does not imply unifor
m convergence. To see this, just note that the following sequence of f
unctions converges to zero pointwise, but the maximum of each function
differs from the zero function by 1/e = 0.367689... . " }}{PARA 0 ""
0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "plot([s
eq(n*x*exp(-n*x),n=1..10)],x=0..1);" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 ""
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 207 "4. Pointwise convergence
does not imply normed convergence. To see this, just note that the pr
evious sequence of functions converges to zero pointwise, but the inte
gral of the difference squared goes to 1/2." }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 46 "plot([seq(sqrt(n)*exp(-n*x),n=1..10)],x=0..1);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "int((sqrt(n)*exp(-n*x))^2,x=
0..1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "limit(%,n=infinit
y);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 ""
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 188 "5. Norm convergence does
not imply point wise convergence. The following is nine terms of an i
nfinite sequence. The sequence does not converge point wise. To see th
is, after executing the " }{TEXT 259 7 "display" }{TEXT -1 320 " comma
nd, touch the graph with the mouse, see a new tool bar above; the tool
bar looks like a CD player control. Push the ->| symbol with the mous
e. The graphs will progress through in order. You should see that you \+
do not have pointwise convergence. To see that the norm converges to z
ero, compute the norm in the next " }{TEXT 260 7 "do loop" }{TEXT -1
36 ". See how the continuation might go." }}{PARA 0 "" 0 "" {TEXT -1
0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 383 "f[1]:=x->(1+signum(1/2-x))/
2:\nf[2]:=x->(1+signum(x-1/2))/2:\nf[3]:=x->(1+signum(1/3-x))/2:\nf[4]
:=x->(1+signum(2/3-x))/2-(1+signum(1/3-x))/2:\nf[5]:=x->(1+signum(1-x)
)/2-(1+signum(2/3-x))/2:\nf[6]:=x->(1+signum(1/4-x))/2-(1+signum(0-x))
/2:\nf[7]:=x->(1+signum(1/2-x))/2-(1+signum(1/4-x))/2:\nf[8]:=x->(1+si
gnum(3/4-x))/2-(1+signum(1/2-x))/2:\nf[9]:=x->(1+signum(1-x))/2-(1+sig
num(3/4-x))/2:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "for n fro
m 1 to 9 do\np[n]:=plot([f[n](x)],x=0..1):\nod:" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 44 "display([seq(p[n],n=1..9)],insequence=true);"
}}{PARA 13 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT
1 0 47 "for n from 1 to 9 do\nint(f[n](x)^2,x=0..1);\nod;" }}}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 631 "These ideas are important for creating \+
solutions for more partial differential equations than just those in t
his course. What will often happen is that you will only be able to ap
proximate the solution. In this course, we will often find analytic ap
proximations. In other contexts you might find numerical approximation
s. The question will be: \"How good is your approximation?\" The respo
nse should be something such as this: \"Increasing the computation inc
reases computation costs, but the approximations will converge to the \+
real solution in the sense of ... .\" And, you will say, something suc
h as norm, pointwise, or uniformly. " }}{PARA 0 "" 0 "" {TEXT -1 0 ""
}}{PARA 0 "" 0 "" {TEXT -1 169 "Maybe you have already encountered the
question for how good approximations are. In any case it is good to d
istinguish these methods of convergence early in our studies." }}
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