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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "    \+
 An example of using Maple to compute " }{XPPEDIT 18 0 "T(f)" "-%\"TG6
#%\"fG" }{TEXT -1 30 " where the (bounded) operator " }{XPPEDIT 18 0 "
T:H->H" "C$%\"TG:6#%\"HG7\"6$%)operatorG%&arrowG6\"F&F+F+" }}{PARA 0 "
" 0 "" {TEXT -1 60 "     is defined using some complete set of basis f
unctions  " }{XPPEDIT 18 0 "phi[n](x)" "-&%$phiG6#%\"nG6#%\"xG" }
{TEXT -1 5 " for " }{XPPEDIT 18 0 "L[2](0,1)" "-&%\"LG6#\"\"#6$\"\"!\"
\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 76 "     The operator
 T defined below has a range which is not closed, hence the" }}{PARA 
0 "" 0 "" {TEXT -1 43 "      Fredholm alternative does not apply. " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 24 "phi:=(x,n)->sin(n*Pi*x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$p
hiG:6$%\"xG%\"nG6\"6$%)operatorG%&arrowGF)-%$sinG6#*(9%\"\"\"%#PiGF29$
F2F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "f:=x->x^2;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG:6#%\"xG6\"6$%)operatorG%&arrowG
F(*$9$\"\"#F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 80 "      The Four
ier coefficients of f are given by the function a[n] below so that" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "         \+
                              " }{XPPEDIT 18 0 "f = sum(a[n]*phi[n](x)
,n=1..infinity)" "/%\"fG-%$sumG6$*&&%\"aG6#%\"nG\"\"\"-&%$phiG6#F+6#%
\"xGF,/F+;\"\"\"%)infinityG" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "a
:=n->int(f(x)*phi(x,n),x=0..1)/int(phi(x,n)*phi(x,n),x=0..1):" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 83 "  \+
    Here's where T is defined: T(N) refers to an N-term approximation \+
of T(f)(x):" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 37 "                                     " }{XPPEDIT 18 0 "T[N](f)(
x) = sum(a[n]*phi[n](x)/n,n=1..N)" "/--&%\"TG6#%\"NG6#%\"fG6#%\"xG-%$s
umG6$*(&%\"aG6#%\"nG\"\"\"-&%$phiG6#F46#F,F5F4!\"\"/F4;\"\"\"F(" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 36 "T:=N->sum(1/n*a(n)*phi(x,n),n=1..N);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"TG:6#%\"NG6\"6$%)operatorG%&arrowGF(-%$sumG6$*(%\"n
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