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\begin{document}
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% !!! IMAGES START HERE !!!

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\stepcounter{section}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline117}%
$w(t)$%
\lthtmlindisplaymathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline119}%
$\tilde w (s)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay311}%
$\displaystyle \tilde w (s)$%
\lthtmlindisplaymathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay312}%
$\textstyle =$%
\lthtmlindisplaymathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay313}%
$\displaystyle \int e^{-i 2 \pi s t }  w(t) dt {\rm ~and}$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay314}%
$\displaystyle w(t)$%
\lthtmlindisplaymathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay316}%
$\displaystyle \int e^{i 2 \pi s t }  \tilde w (s) ds .$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay318}%
$\displaystyle C_w = \int { 1 \over |s| } \tilde w (s) ds .
$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline121}%
$\tilde w (s) / |s|^\epsilon $%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline123}%
$\rightarrow 0 $%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline125}%
$s \rightarrow 0$%
\lthtmlindisplaymathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline127}%
$\epsilon > 0 $%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline129}%
$w(t) = {\rm cos}(2\pi t) (e^{- \pi t^2 / c^2} )$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline131}%
$c$%
\lthtmlindisplaymathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay326}%
$\displaystyle
\int w(ut) du = C_w \cdot \delta (t) .
$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline133}%
$t=0$%
\lthtmlindisplaymathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay329}%
$\displaystyle {\bf Proof:~}  \int w(ut) du 
= \int \left \{ \int  {1 \over |u|} \tilde w ( { s \over u } ) 
e^{i 2 \pi s t } ds \right \} du \ldots
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline135}%
$(s,u)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline137}%
$(s'=s,u'=s/u)$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline139}%
$\partial (s,u) / \partial (s',u') = |s'/u'^2|$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay334}%
$\displaystyle \int w(ut) du$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay336}%
$\displaystyle \int \int  \left | {u' \over s'} \right | \tilde w ( u' ) e^{i 2 \pi s' t }
\left | { s' \over u'^2 } \right | ds' du'$%
\lthtmlindisplaymathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay338}%
$\displaystyle \int \int  {1 \over |u'|} \tilde w ( u' ) e^{i 2 \pi s' t } ds' du'$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay340}%
$\displaystyle \left \{ \int {1 \over |u'|} \tilde w ( u' ) du'  \right \}  \cdot
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay342}%
$\displaystyle C_w \cdot \delta (t) .$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay344}%
$\displaystyle w(t) = w(-t) .
$%
\lthtmlindisplaymathZ
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\stepcounter{section}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline141}%
$F(u,a)$%
\lthtmlindisplaymathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline143}%
$L^2$%
\lthtmlindisplaymathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline145}%
$f(t)$%
\lthtmlindisplaymathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay350}%
$\displaystyle
F(u,a) = \int w[u(a - t)] f(t) dt .
$%
\lthtmlindisplaymathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline147}%
$a$%
\lthtmlindisplaymathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline149}%
$u$%
\lthtmlindisplaymathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline155}%
$2^u$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline157}%
$u^{-1}$%
\lthtmlindisplaymathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay360}%
$\displaystyle
f(t) = C_w^{-1} \int F(u, a=t) du .
$%
\lthtmlindisplaymathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay362}%
$\displaystyle {\bf Proof:~}
C_w^{-1} \int F(u,a=t) du$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay364}%
$\displaystyle C_w^{-1} \int \left \{ \int w[u(t - t' )] f(t') dt' \right \} du$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay366}%
$\displaystyle C_w^{-1} \int \left \{ \int w[u(t - t' )] du \right \} f(t') dt'$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay368}%
$\displaystyle C_w^{-1} \int C_w \delta(t-t') f(t') dt'$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay370}%
$\displaystyle f(t).$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{section}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay373}%
$\displaystyle g(\tau) = f[t(\tau)].
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline163}%
$t(\tau)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline165}%
$t$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline167}%
$\tau$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay378}%
$\displaystyle t(\tau)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay379}%
$\textstyle \approx$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay380}%
$\displaystyle t(\tau_0 ) + (\tau - \tau_0 )
\cdot {dt \over d\tau} ( \tau_0 ) ~~~~{\rm and }$%
\lthtmlindisplaymathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay381}%
$\displaystyle \tau(t)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay383}%
$\displaystyle \tau_0 + [ t - t (\tau_0 ) ]
\cdot \left [ {dt \over d\tau} ( \tau_0 ) \right ]^{-1} .$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline169}%
$g(\tau)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay386}%
$\displaystyle G(v, b ) = \int w [ v ( b  - \tau ) ]  g (\tau) d\tau  ~~~~ {\rm and}  
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay388}%
$\displaystyle g(\tau) = C_w^{-1} \int G(v, b  = \tau ) dv.
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline171}%
$G(v, b)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay392}%
$\displaystyle
G(v,b) \approx 
\left | {dt \over d\tau} ( b ) \right |^{-1} 
 ~ F \left \{ u=v \cdot  
\left [ {dt \over d\tau} ( b ) \right ]^{-1} , ~ a = t(b) \right \} .
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay396}%
$\displaystyle {\bf Proof: ~}
G ( v, b)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay398}%
$\displaystyle \int w[v(b-\tau)] \cdot f[t(\tau)] d \tau$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay400}%
$\displaystyle \int w [ v(b-\tau)] \cdot f \left [ t(b) + (\tau-b)
\cdot {dt \over d\tau} ( b ) \right ]  d\tau \ldots$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay402}%
$\displaystyle t'$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay403}%
$\textstyle \equiv$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay404}%
$\displaystyle t(b) + (\tau-b) \cdot  {dt \over d\tau} ( b )
~~~~ {\rm and}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay405}%
$\displaystyle \tau$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay407}%
$\displaystyle b + [ t' - t(b) ]
\cdot \left [ {dt \over d\tau} ( b ) \right ]^{-1} .$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay409}%
$\displaystyle G ( v, b)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay411}%
$\displaystyle \int w \left \{ v \cdot [t(b) - t'] \cdot
\left [ {dt \over d\tau} ( b ) \right ]^{-1} \right \} \cdot f(t')
\cdot \left | {dt \over d\tau} ( b ) \right |^{-1} ~ dt'$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay413}%
$\displaystyle \left | {dt \over d\tau} ( b ) \right |^{-1}
\int w \left \{ v
\left [ {dt \over d\tau} ( b ) \right ]^{-1}
\cdot [t(b) - t']
\right \}  \cdot f(t') d t'$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay415}%
$\displaystyle \left | {dt \over d\tau} ( b ) \right |^{-1}
~ F \left \{ u=v \cdot
\left [ {dt \over d\tau} ( b ) \right ]^{-1} , ~ a = t(b) \right \} .$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}


\end{document}
