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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\centerline{\fbox{\,\textbf{TROISIEME PARTIE}$\ph{\big|\dot{l}}$\hs{-2mm}}}\vvbc
\centerline{\fbox{\:\Large{\textbf{\tsf{Int\e gration sur un rectangle compact  de\ha\ $\R^2\ph{\big|}$}}}\!}}\vs{12.5mm}

\initNum
\centerline{\textbf{CHAPITRE\, XV}}\vva
\centerline{\textbf{FONCTIONS ET PSEUDO\ha-MESURES}}\vvw
\centerline{\textbf{SUR UN RECTANGLE COMPACT}}\vvcd

\oui On g\e n\e ralise \ac\ grands traits pour deux dimensions les r\e sultats obtenus\ha\ sur\hb\ \ab\pt
Le\ha\ \emph{produit tensoriel}\ha\ de deux pseudo\ha-mesures unidimensionnelles constitue la principale
nouveaut\e\ de ce chapitre\vg qui nous conduit \ac\ l'apoth\e ose du \theo\ha\ de \mbox{Fubini\ha.}\vvc

\initnum
\quad\textbf{\Num\underline{Espaces fondamentaux}}\vvc

I\, et\, J\, sont des intervalles compacts de longueur non nulle de\, $\R$\pv on pose\hh K =\hb\ I\,$\times$\,J\vb

\Def{\num}Une fonction\, \underline{\e tag\e e}\, dans \ha K \ha est une combinaison lin\e aire de fonctions\vvw
de la forme\hh$\un\ha_{\I\ha'\tim\J\ha'}$\,,\: \ou\hh I\ha'\, et\: J\ha'\, sont des intervalles inclus
respectivement \ac\, I\, et\, J\vg\vvw (\ha\e ventuellement r\e duits \ac\ des singletons\ha)\vab

\underline{\emph{Notations}}\, :\vva
$\EE\PK$\ha\ =\: alg\ee bre de Riesz des fonctions \e tag\e es\hh:\: $\K\to\R$\vvw
$\CC\PK$\ha\ =\: alg\ee bre de Riesz des fonctions continues\hh:\: $\K\to\R$\vvw
$\FF\PK$\ha\ =\: alg\ee bre de Riesz des fonctions born\e es\hh:\: $\K\to\R$\vvb

\Def{\num}On note\, $\RR\PK$\, la fermeture de\, $\EE\PK$\, dans\, $\FF\PK$\, pour la convergence uniforme\pt
Les fonctions de\, $\RR\PK$\, sont appel\e es les fonctions\, \underline{r\e gl\e es}\, dans\, K\vbc

\Th{\num*}Soit\, $f\in\FF\PK$\pv alors\: $f\in\RR\PK$\: ssi\; $\all(\hw u\,,\ha v\hw)\in\K$\; les\ha\
\emph{huit}\ha\ limites\hb\ suivantes\ha\ existent\, :\vvbc

\hs{25mm}$\lim\pos_{x\,\to\,u^+}\,f\hb(\hw x\,,\ha v\hw)
\hhh\lim\pos_{x\,\to\,u^-}\,f\hb(\hw x\,,\ha v\hw)
\hhh\lim\pos_{y\,\to\,v^+}\,f\hb(\hw u\,,\ha y\hw)
\hhh\lim\pos_{y\,\to\,v^+}\,f\hb(\hw u\,,\ha y\hw)$\vva

\hs{25mm}$\lim\pos_{\sbs{x\,\to\,u^+\\[1mm]y\,\to\,v^+}}\,f\hb(\hw x\,,\ha y\hw)
\hhh\lim\pos_{\sbs{x\,\to\,u^-\\[1mm]y\,\to\,v^+}}\,f\hb(\hw x\,,\ha y\hw)
\hhh\lim\pos_{\sbs{x\,\to\,u^+\\[1mm]y\,\to\,v^-}}\,f\hb(\hw x\,,\ha y\hw)
\hhh\lim\pos_{\sbs{x\,\to\,u^-\\[1mm]y\,\to\,v^-}}\,f\hb(\hw x\,,\ha y\hw)$\vvb

\begin{picture}(,)
\put(20.5,4.5){\frm(124,25.5)}
\end{picture}\vvk

\newpage
\oui Si on trace un carr\e\ de centre\hb\ $P=(\hw u\,,\ha v\hw)$\, et de c\^ot\e s parall\ee les aux axes\vg
ainsi que ses deux m\e dianes\vg les limites ci-dessus correspondent aux limites en\hb\ $P$\hb\ sur les quatre
demi-m\e dianes et aux limites en\hb\ $P$\hb\ dans les quatres carr\e s \emph{ouverts} d\e limit\e s par les
\mbox{m\e dianes\hb.}\vvab

\begin{picture}(,)
\put(1,-2.5){On d\e finit\hh$\PM\PK\vv\,\LL^1\PK\vv\,\LL^2\PK\vv\BB\PK\vv\BA\PK\vv\ha\PR\PK\vv\WW\PK$\hh
comme sur \ab\ha.}
\put(1,-9.5){Les d\e finitions\, et \theo s relatifs \ac\ ces espaces sont analogues au cas de \ab\:.}
\put(-1.5,-13){\frm(168,16.5)}
\end{picture}\vk17.5mm

\emph{\underline{Notation\ha\ int\e grale}}\dd$\all\tdf\in\PM\PK\,\hh\all g\in\RR\PK$\,\hh on note
$$\boxed{\,\tdf\ha(g)\,=\Int\hs{-2mm}\int_{\,\K}g\:\tdf\ha
=\Int\hs{-2mm}\int_{\,\K}g\,(x\ha,\hb y)\;\tdf\ha(x\ha,\hb y)\,}\pt$$\vva

$\all\tdf\in\LL^1\PK$\hh on \e crit\,\hh$\Int\hs{-2mm}\int_{\,\K}\tdf\ha(x\ha,\hb y)
\;d\hb x\ d\hb y$\,\hh au lieu de\hh$\Int\hs{-2mm}\int_{\,\K}\tdf\ha(x\ha,\hb y)$\pv
on a donc\vvb\hs{25mm}\fbox{\ha\,$\Int\hs{-2mm}\int_{\,\K}\tdf\ha(x\ha,\hb y)\;
d\hb x\ d\hb y\,=\Int\hs{-2mm}\int_{\,\K}\tdf\ha(x\ha,\hb y)\ha
=\ha\tdf\ha(\un)$\quad(\ha avec\, $\un=1_{\,\K}\ha)$\,}\ha\vcd

\Def{\num}$\tdf\in\PM\PK$\, est une\, \underline{mesure}\, sur K\, ssi\\
\begin{picture}(,)
\put(13,-6){$\al\;u\in\I\,\hh\lim\pos_{\sbs{\\\alf\ha\to\ha u^\pm}}\,\vtdf\,
\big(\hw1_{\hb{]\,u\vv\alf\,[}\ha\tim\ha\JJ}\ha\big)=\hb0$\,\hhh
\underline{et}$\hhh\al\;v\in\J\,\hh\lim\pos_{\sbs{\\\beta\ha\to\ha v^\pm}}\,\vtdf\,
\big(\hw1_{\hb\II\,\tim\ha{]\,v\vv\beta\,[}}\ha\big)=\hb0$\hb\hh.}
\put(11,-10){\frm(142,10)}
\end{picture}\\\vvbc

\oui Cette propri\e t\e\ha\ s'appelle l'\underline{hypercontinuit\e}\, des mesures\pv elle signifie que si\vg
dans un rectangle \emph{ouvert} de c\^ot\e s parall\ee les aux axes\vg un c\^ot\e\ tend vers le
c\^ot\e\ oppos\e\vg alors la mesure du rectangle tend vers\, $0$\vab

On note\, $\MM\PK$\, l'espace de Riesz des mesures sur K\vb

La mesure\,\hh$\EE\PK\to\R\ :\ g\,\tto\Int\hs{-2mm}\int_{\,\K}g\hb(x\ha,\hb y)\;d\hb x\;d\hb y$\,\hh
est la\, \underline{mesure de Lebesgue}\, sur K\vc

\Th{\num*}\fbox{\,$\LL^1\PK\,\sub\,\MM\PK\,\sub\,\PM\PK$\,}\vc

\Th{\num*}$\MM\PK$\, est un\, sous-espace coh\e rent\vg \fbox{int\e gral}\, et\, \underline{ferm\e}\,
de\, $\PM\PK$\vbc

\num\ha\ \underline{\Theo\ha\ de Lebesgue dans\, $\WW\PK$}\vvc
Soit\; $f\in\WW\PK$\hh et soit une suite\; $f_n\in\WW\PK$\hh telle que\hh$f_n\bb f$\pv alors\ha\hh$\all\tmu\in\MM\PK$\vvww on a\,\hh$f_n\;\tmu\fin f\hb\tmu$\ha\pv
en cons\e quence\,\hh$\tmu\,(f\ha_n)\to\tmu\,(f)$\\
\begin{picture}(,)
\put(-2.75,3.5){\frm(160.5,17.5)}
\end{picture}\vvwa

\Dem M\^eme succession d'\e tapes que pour\: \ab\vbc

\Cor{\num}\underline{Hypercontinuit\e\ha\ g\e n\e ralis\e e}\vvwa
Soit une suite d\e croissante d'ouverts\ha\hh$\Oma\ha\sub\hb\K$\ha\hh tels que\hb\hh
$\under{n}{\bigcap}\ha\ \Om\ha_n=\hb\emptyset$\ha\pv\ha alors\ha\hh$\all\tmu\in\MM\PK$\,\hh
on a\,\hh\fbox{\,$\tmu\,\big(\un\ha_{\Oma}\big)\to\ha0$\,}\vb

\Dem On a\ha\hh$\nn\ha\hh\un\ha_{\Oma}\in\ha\PR\PK$\pv de plus on a clairement\hh$\un\ha_{\Oma}\bb\ha0$\pv\vvww
il suffit d\ee s lors d'appliquer le \theo\ha\ de Lebesgue\vbc

\textsf{\underline{Remarque}\, :\, Il n'existe pas de g\e n\e ralisation naturelle de la notion de mesure\ha\
\emph{diffuse}\vvww en  dimension\hb\ $\ge\hb2$}\vd

\initnum
\quad\textbf{\Num\underline{Produit tensoriel de fonctions}}\vvc

\Def{\num}$\all f\in\FF\PI\,\hh\all g\in\FF\PJ$\hh on note\vva
\fbox{\,$f\ten g\ :\ \K\to\R\ :\ (x\hb,\hb y)\tto f\ha(x)\:g\hb(y)$\hb}\ha\pv\ha
on a donc\hh\fbox{\hb$f\ten g\in\FF\PK$\hb}\vbc

\Th{\num*}\fbox{\,$|\hb f\ten g\ha|=|\ha f\hw|\ten|\hb g\ha|$\,}\quad et\quad
\fbox{\,$\|\hb f\ten g\ha\| =\|\ha f\ha\|\,\|\ha g\ha\|$\hb}\vc

\Th{\num*}On a\,\hh$\all f,\,f_1\vv f_2\in\FF\PI\,\hh\all g\vv g\hw_1\vv g\ha_2\in\FF\PJ$\vva
1)\hh$f_1\ten g+f_2\ten g=(f_1+f_2)\ten g$\vva
2\ha)\hh$f\ten g\ha_1+f\ten g\ha_2=f\ten(g\hw_1+g\ha_2)$\vva
3\ha)\hh$\all\lam\in\R\,\hh(\lam\,f)\ten g=f\ten(\lam\,g)=\lam\,(f\ten g)$\vva
4\ha)\hh$(f_1\ten g\hw_1)\,(f_2\ten g\ha_2)=(f_1\,f_2)\ten(\ha g\hw_1\:g\ha_2)$\vbc

\Not Soient\, V\, un\, sous-espace de \,$\FF\PI$\, et\, W\, un\, sous-espace de\, $\FF\PJ$\pv\vvwa on note\:
\fbox{\hb$\V\ten\W$\hb}\; le\, \underline{sous-espace vectoriel}\, de\, $\FF\PK$\, engendr\e\ par les
\e l\e ments\vva de la forme\hh$f\ten g$\, avec\, $f\in\V$\, et\, $g\in\W$\pv
on peut donc \e crire\,\hh\fbox{\,$\FF\PI\ten\FF\PJ\ha\sub\hb\FF\PK\ph{\big|}$\hs{-.85mm}}\ha\vbc

\Th{\num*}\fbox{\:$\EE\PI\ten\EE\PJ\ \mbof{=}\ha\ \EE\PK\ph{\dot{l}}$\hs{-.85mm}}\ha\vab
Cette propri\e t\e\vg \ac\ contraster avec la suivante\vg fournit une raison suppl\e mentaire de b\^atir
la th\e orie de l'int\e gration sur les fonctions \e tag\e es plut\^ot que sur les fonctions continues\vbc

\Th{\num*}$\CC\PI\ten\CC\PJ\sub\hb\CC\PK$\quad et\quad$\RR\PI\ten\RR\PJ\sub\hb\RR\PK$\vb

\Th{\num*}$\BA\PI\hb\ten\BA\PJ\sub\,\BA\PK$\quad et\quad$\PR\PI\ten\PR\PJ\sub\hb\PR\PK$\vcd

\initnum
\quad\textbf{\Num\underline{Pseudo-mesures marginales}}\vvc

\Def{\num}$\all\tdf\in\PM\PK$\,\hh on pose\vvab
\hs{23mm}$\tdfI\::\ \EE\PI\to\R\::\:h\tto\tdf\,(\ha h\ten\unJ\hw)\ha
=\Int\hs{-2mm}\int_{\,\K}\hs{-.5mm}h\hb(x)\:\tdf\ha(x\vvv y)$\vva
\hs{15.25mm}et\quad$\tdfJ\::\ \EE\PJ\to\R\ :\ h\tto\tdf\,(\hw\unI\ten h\ha)\ha
=\Int\hs{-2mm}\int_{\,\K}\hs{-.5mm}h\hb(y)\:\tdf\ha(x\vvv y)$\vbc

\Th{\num}\vvwa\oui$\tdfI\ha\in\ha\PM\PI$\,\hh et\hh$\tdfJ\ha\in\ha\PM\PJ$\pv
de plus\,\hh$\Big\|\tdfI\,\Big\|_\star\le\ha\bnst{\ha\tdf\hb}$\hh
et\hh$\Big\|\tdfJ\,\Big\|_\star\le\ha\bnst{\ha\tdf\hb}$\ha\vab

\Dem\vvwa On a\,\hh$\all h\in\ha\EE\PI\,\hh\Big|\hb\Big(\tdfI\Big)\hb(h)\ha\Big|
=\big|\ha\tdf\,(\ha h\ten\unJ\hw)\ha\big|\le\bnst{\ha\tdf\hb}\:\big\|\ha|\ha h\ha
|\ten\unJ\ha\big\|=\bnst{\ha\tdf\hb}\,\|\ha h\|$\va

$\tdfI$\; et $\tdfJ$\, s'appellent les\, \underline{pseudo-mesures marginales}\, de\, $\tdf$,\,
respectivement sur\, $\II$\, et\ha\ $\J$\ha.\vvcd

\Cor{\num}\vvwa Soient\hh$\tdf_n\vvv\tdf\in\PM\PK$\hh tels que\hh$\tdf_n\hb\st\tdf$\pv
alors\hh$\tdfI_n\hb\st\tdfI$\;\hh et\ha\hh$\tdfJ_n\hb\st\tdfJ$\vbc

\Lem{\num}\vvwa$\all h\in\ha\EE\PI^+\,\hh\Big|\tdfI\,\Big|\hb(h)\le\vtdf\hb(h\ten\unJ)$\quad
et\quad$\all k\in\ha\EE\PJ^+\,\hh\Big|\tdfJ\,\Big|\hb(k)\le\vtdf\hb(\unI\ten k)$\vb

\Dem$\Big|\tdfI\,\Big|\hb(h)\,=\dis\sup_{\sbs{\\k\,\in\:\EE\PI\\[.5mm]|\ha k\ha|\,\le\:h}}
\Big|\ha\Big(\tdfI\Big)\hb(k)\hb\Big|\ =\sup_{\sbs{\\k\,\in\:\EE\PI\\[.5mm]|\ha k\ha|\,\le\:h}}
\big|\hb\tdf\hb(k\ten\unJ)\ha\big|$\vvab\hs{60mm}$\le\dis\sup_{\sbs{\\k\,\in\:\EE\PI\\[.5mm]
|\ha k\ha|\,\le\:h}}\vtdf\hb\big(|\hb k\ha|\ten\unJ\big)\,=\,\vtdf\,(\ha h\ten\unJ)$\vc

\Th{\num}\fbox{\,$\tdf\in\MM\PK\,\imp\,\Big[\,\tdfI\in\MM\PI$\,\hh et\,\hh$\tdfJ\in\MM\PJ\,\Big]$\,}\vbc

\Dem On a\,\hh$\al\;u\in\I$\vvab$\Big|\ha\tdfI\,\Big|\,\big(\hw1\ha_{]\,u\vv\alf\,[}
\hw\big)\ha\le\ha\vtdf\hb\big(\hw1_{\ha]\,u\vv\alf\,[}\ten\unJ\hw\big)=\ha\vtdf\hb
\big(\hw1\ha_{{]\,u\vv\alf\,[}\ha\times\hw\JJ}\hw\big)\to0$\,\hh quand\,\hh$\alf\to\ha u\ha^\pm.$\vvb

\Not Si\, $\tdf\in\LL^1\PK$\, on notera les mesures marginales\, $\tdfI\:d\hb y$\ha\hh et\, $\tdfJ\:d\hb x$\pv\vvw
compte tenu du \theo\ suivant nous les appelerons\, \underline{fonctionnelles marginales}\ha\vg\vvww
respectivement sur\hb\ $\II$\, et\, $\JJ$\vbc

\Th{\num}\fbox{\,$\tdf\in\LL^1\PK\,\imp\,\Big[\,\tdfI\:d\hb y\ha\in\ha\LL^1\PI$\,\hh
et\,\hh$\tdfJ\:d\hb x\ha\in\ha\LL^1\PJ\,\Big]$\,}\vbc

\Dem Il existe une suite\: $f_n\in\ha\EE\PK$\; et une suite\; $\eps_n>\ha0$\; telles que\;
$\eps_n\to0$\vvab et telles que\hb\hh$\all g\in\ha\EE\PK\,\hh
\Big|\,\tdf\hb(g)-\Int\hs{-2mm}\int_{\,\K}\hs{-1mm}f_n\hb(x\ha,\ha y)\:
g\hb(x\ha,\ha y)\:d\hb x\;d\ha y\:\Big|\le\hb\eps_n\:\|\hb g\hb\|$\pv\vvwa
donc\,\hh$\all h\in\ha\EE\PI\,\hh\Big|\,\tdf\ha(h\ten\unJ)
-\Int\hs{-2mm}\int_{\,\K}\hs{-1mm}f_n\hb(x\ha,\ha y)\:h\hb(x)\ d\hb x\;d\hb y\:\Big|
\le\hb\eps_n\;\|\hb h\ten\unJ\ha\|$\vg\vvwa \cad\,\hh
$\Big|\,\Big(\tdfI\:d\hb y\Big)\hb(h)-\Int_{\,\I}\hs{-.3mm}\Big[\hs{-.3mm}\int_{\ha\J}\hs{-.25mm}f_n\hb
(x\ha,\ha y)\;d\hb y\,\Big]\,h\hb(x)\;d\hb x\:\Big|\le\hb\eps_n\:\|\ha h\ha\|$\pv\vva
or\,\hh$\nn\,\ \Int_{\ha\J}\hs{-.25mm}f_n\;d\hb y\ha\in\ha\EE\PI$\ha\vg
donc\: $\tdfI\:d\hb y\ha\in\ha\LL^1\PI$\vbc

\Th{\num}\fbox{\,$\tdf\in\LL^2\PK\,\imp\,\Big[\,\tdfI\:d\hb y\ha\in\ha\LL^2\PI$\,\hh et\,\hh
$\tdfJ\:d\hb x\ha\in\ha\LL^2\PJ\,\Big]$\,}\pv\vvab de plus\quad$\Big\|\tdfI\:d\hb y\;\Big\|_2\,
\le\,|\ha\JJ\hw|\ha^{1/\ha2}\ \bntwo{\ha\tdf\hb}$\,\quad\low{et}\,\quad
$\Big\|\tdfJ\:d\hb x\;\Big\|_2\,\le\,|\ha\II\ha|\ha^{1/\ha2}\ \bntwo{\ha\tdf\hb}$\vb

\Dem On a\,\hh$\all h\in\ha\EE\PI\,\hh\Big|\ha\Big(\tdfI\Big)\hb(h)\hb\Big|\,=\,|\ha\tdf\ha(h\ten\unJ)\ha|\,
\le\,\bntwo{\ha\tdf\hb}\;\ntwo{\hb h\ten\unJ\ha}$\vva$=\,\bntwo{\ha\tdf\hb}\
|\ha\JJ\hw|\ha^{1/\ha2}\ \ntwo{\ha h\ha}$\pv donc\,\hh$\tdfI\:d\hb y\ha\in\ha\LL^2\PI$\,\hh
et\,\hh$\Big\|\tdfI\:d\hb y\;\Big\|_2\,\le\,|\ha\JJ\hw|\ha^{1/\ha2}\ \bntwo{\ha\tdf\hb}$\vc

\Th{\num}\fbox{\,$\tdf\in\ha\BB\PK\,\imp\,\Big[\,\tdfI\:d\hb y\in\ha\BB\PI$\,\hh et\,\hh
$\tdfJ\:d\hb x\in\ha\BB\PJ\,\Big]$\,}\pv\vvab de plus\,\hh$\Big\|\tdfI\:d\hb y\;\Big\|_\B\,\le\,|\ha\JJ\hw|\:
\bnbb{\ha\tdf\hb}$\,\hh et\,\hh$\Big\|\tdfJ\:d\hb x\;\Big\|_\B\,\le\,|\ha\II\ha|\:\bnbb{\ha\tdf\hb}$\vb

\Dem On a\,\hh$\all h\in\ha\EE\PI\,\hh\Big|\ha\Big(\tdfI\Big)\hb(h)\hb\Big|\,=\,\big|\ha\tdf(h\ten\unJ)\ha\big|\,
\le\,\bnbb{\ha\tdf\hb}\;\none{\hb h\ten\unJ\ha}$\vva$=\,b\nbb{\ha\tdf\hb}\ |\ha\JJ\hw|\,\none{\ha h\ha}$\pv
donc\,\hh$\tdfI\:d\hb y\in\ha\BB\PI$\,\hh et\,\hh$\Big\|\tdfI\:d\hb y\;\Big\|_\B\,
\le\,|\ha\JJ\hw|\,\bnbb{\ha\tdf\hb}$\vc

\Th{\num}
\begin{picture}(,)
\put(2,0){$\WW\PI\ten\WW\PJ\sub\hb\WW\PK$\; .}
\put(.5,-2){\frm(45,6.75)}
\end{picture}\vva

\Dem\vvwa Soient\: $f_1\in\WW\PI^+$\, et\: $f_2\in\WW\PJ^+$\pv soit\; $M\in\R^+$\: tel que\hh$f_1\le M$\hh
et\hh$f_2\le M$\pv\vvw soient\hh$\eps>0$\hh et\hh$\tmu\in\MM\PK^+$\pv soient\hh$g\hw_1\vv h\hw_1\in\PR\PI$\hh
tels que\hh$0\ha\le\ha g\hw_1\ha\le\ha f_1\ha\le\ha h\hw_1\ha\le\ha M$\vvwa
et\hh$\Big(\Int_{\,\J}\tmu\Big)\hb(h\hw_1-g\hw_1)\ha\le\ha\eps$\pv
de m\^eme soient\hh$g\ha_2\vv h\ha_2\in\PR\PJ$\hh tels que\\[-.5mm]$0\le\ha g\ha_2\ha\le\ha f_2\ha
\le\ha h\ha_2\ha\le\ha M$\hh et\hh$\Big(\Int_{\,\I}\tmu\Big)\hb(h\ha_2-g\ha_2)\ha\le\ha\eps$\pv
alors on a\vvaab$g\hw_1\ten g\ha_2\in\PR\PK\vg h\hw_1\ten h\ha_2\in\PR\PK$\,\hh et\,\hh
$g\hw_1\ten g\ha_2\ha\le\ha f_1\ten f_2\ha\le\ha h\hw_1\ten h\ha_2$\pv\vvw de plus on peut \e crire\vvw
$\tmu\,\big(\ha h\hw_1\ten h\ha_2\ha-\ha g\hw_1\ten g\ha_2\hw\big)\ha
=\ha\hb\tmu\,\big(\hb h\hw_1\ten h\ha_2\ha-\ha g\hw_1\ten h\ha_2\hb+\hb g\hw_1\ten h\ha_2\ha
-\ha g\hw_1\ten g\ha_2\ha\big)$\vvw$=\,\tmu\,\big[\hb(h\hw_1- g\hw_1)\ten h\ha_2\hb
+\hb g\hw_1\ten(h\ha_2-g\ha_2)\hb\big]\ha\le\,\tmu\,\big[\hb(h\hw_1-g\hw_1)\ten M\ha
+\hb M\ten(h\ha_2-g\ha_2)\hb\big]$\vvwa$\le\hb M\,\Big[\hb\Big(\Int_{\,\J}\tmu\Big)\hb
(h\hw_1-g\hw_1)\ha+\hb\Big(\Int_{\,\I}\tmu\Big)\hb(h\ha_2-g\ha_2)\hb\Big]\ha\le\,2\;\eps\,M$\pv
donc\hh$ f_1\ten f_2\in\WW\PK$\vaab Le cas g\e n\e ral s'obtient en raisonnant sur les parties positives
et n\e gatives de\ha\ $f_1$\ha\ et\ha\ $f_2$\vcd


\initnum
\quad\textbf{\Num\underline{Produit tensoriel de pseudo-mesures}}\vvc

\Def{\num}On pose\,\hh$\all\tmu\in\PM\PI\,\hh\all\tnu\in\PM\PJ\,\hh\all f\in\ha\EE\PI\,\hh\all g\in\ha\EE\PJ$\\
\begin{picture}(,)
\put(50,-4){$(\hb\tmu\ten\tnu\hb)\,(\ha f\ten g\ha)=\ha\tmu\hb(f)\;\tnu\hb(g)$\, .}
\put(48.5,-6.5){\frm(55.5,7.5)}
\end{picture}\vvcd

Comme\hh$\EE\PI\ten\EE\PJ=\EE\PK$\vg cette relation permet par lin\e arit\e\ de faire agir\: $\tmu\ten\tnu$\vvw
sur\, $\EE\PK$\pv on en conclut imm\e diatement que l'on a\,\hh
\fbox{\,$\tmu\ten\tnu\ha\in\PM\PK\ph{\dot{t}}$\hs{-1.1mm}}\vc

\Not Soient\, V\, un\, sous-espace de \,$\PM\PI$\, et\, W\, un\, sous-espace de\, $\PM\PJ$\ha;\vvwwa on note\:
\fbox{\hb$\V\ten\W$\hb}\: le\, \underline{sous-espace vectoriel}\, de\, $\PM\PK$\, engendr\e\ par les
\e l\e ments\vva de la forme\, $\tmu\ten\tnu$\, avec\: $\tmu\in\V$\: et\: $\tnu\in\W$\vab
On peut donc \e crire\,\hh\fbox{\,$\PM\PI\ten\PM\PJ\ha\sub\ha\PM\PK\ph{\dot{l}}$\hs{-.85mm}}\vc

\Th{\num*}On a\,\hh$\all\tmu\vv\tmu\ha_1\vv\tmu\hb_2\in\PM\PI\,\hh\all\tnu\vv\tnu\ha_1\vv\tnu\ha_2\in\PM\PJ$\vva
1)\hh$\tmu\ha_1\ten\tnu+\tmu_2\ten\tnu=(\tmu\ha_1+\tmu\hb_2)\ten\tnu$\vva
2\ha)\hh$\tmu\ten\tnu\ha_1+\tmu\ten\tnu\ha_2=\tmu\ten(\tnu\ha_1+\tnu\ha_2)$\vva
3\ha)\hh$\all\lam\in\R\hh\,(\lam\;\tmu)\ten\tnu=f\ten(\lam\;\tnu)=\lam\;(\ha\tmu\ten\tnu\hw)$\vc

\Th{\num*}$\all\tmu\ha_1\vv\tmu\hb_2\in\PM\PI^+\,\hh\all\tnu\ha_1\vv\tnu\ha_2\in\PM\PJ^+$\,\hh on a\vvab
\hs{35mm}\fbox{\,$\big(\,\tmu\ha_1\le\ha\tmu\hb_2$\,\hh et\,\hh$\tnu\ha_1\le\ha\tnu\ha_2\,\big)
\,\imp\;\tmu_1\ten\tnu\ha_1\le\ha\tmu\hb_2\ten\tnu\ha_2$\,}\vc

\Th{\num}$\all\tmu\in\PM\PI\quad\all\tnu\in\PM\PJ$\quad on a\\
\begin{picture}(,)
\put(38,-3.5){$|\hb\tmu\ten\tnu\hb|=\vtmu\ten\vtnu$\,\quad et\hb\quad
$\nst{\hb\tmu\ten\tnu\hb}=\nst{\ha\tmu\ha}\:\nst{\ha\tnu\ha}$\hh.}
\put(35,-6){\frm(93.5,7.75)}
\end{picture}\vvcd

\newpage
\Dem Toute fonction\, $h\in\ha\EE\PK$\, peut s'\e crire\hh$h=\sum\pos_{\ha i\ha j}\,
\alf\ha_{ij}\,p\ha_i\,q\hw_j$\hh\ou\, $p\ha_i$\, est la fonction\vvww caract\e ristique d'un intervalle
$\sub$ I\, et \ou\, $q\hw_j$\, est la fonction caract\e ristique d'un intervalle\vvw$\sub$ J\pv on peut
supposer de plus\,\hh$\all i\ne j\,\hh p\ha_i\,p\hw_j=\ha0$\hh et\hh$q\ha_i\,q\hw_j=\ha0$\pv on a d\ee s
lors\dd\vvwa$\all h\in\ha\EE\PK\quad\big|\ha(\hb\tmu\ten\tnu\hb)\,(h)\ha\big|\ha
=\ha\big|\ha(\hb\tmu\ten\tnu\hb)\,\big(\sum\pos_{\ha i\ha j}\:\alf\ha_{ij}\:p\ha_i\:q\hw_j\ha\big)\ha\big|\ha
=\ha\big|\ha\sum\pos_{\ha i\ha j}\,\alf\ha_{ij}\;\tmu\,(\ha p\ha_i)\;
\tnu\,(q\hw_j)\ha\big|$\vvwa$\le\hb\sum\pos_{\ha i\ha j}\,|\ha\alf\ha_{ij}\ha|\:
\big|\ha\tmu\,(\ha p\ha_i)\ha\big|\,\big|\ha\tnu\,(q\hw_j)\ha\big|\ha
\le\hb\sum\pos_{\ha i\ha j}\,|\ha\alf\ha_{ij}|\;\vtmu\,(\ha p\ha_i)\ \vtnu\,(q\hw_j)\ha
=\ha\big(\hb|\ha\tmu\ha|\ten|\ha\tnu\ha|\hb\big)\,\big(\hw|h|\hw\big)$\pv\vvwa
on en d\e duit\,\hh$|\hb\tmu\ten\tnu\hb|\le\vtmu\ten\vtnu$\va
Par ailleurs soit\; $\eps>0$\pv soit\; $p\in\ha\EE\PI$\; tel que\; $\|\ha p\ha\|\le1$\;
et\; $\nst{\hb\tmu\hb}\le\ha\tmu\,(p)+\eps$\pv\vvwwa de m\^eme soit\; $q\in\ha\EE\PJ$\; tel que\;
$\|\ha p\ha\|\le1$\; et\; $\nst{\ha\tnu\ha}\le\ha\tmu\,(\ha q\ha)+\eps$\pv
on  a\; $\|\hb p\:q\hb\|\le1$\vg\vvwwa
donc\,\hh$\nst{\hb\tmu\ten\tnu\ha}\ha\ge\ha(\hb\tmu\ten\tnu\hb)\,(p\ten q)
=\,\tmu\,(p)\;\tnu\,(q)\ha\ge\ha\big(\ha\nst{\ha\tmu\ha}-\eps\ha\big)\,\big(\ha\nst{\ha\tnu\ha}-\eps\ha\big)$\vvwwa
$\ge\hb\nst{\ha\tmu\ha}\,\nst{\ha\tnu\ha}-\ha\eps\,\big(\ha\nst{\ha\tmu\ha}+\nst{\tnu}\ha\big)$\pv
comme\, $\eps$\, est arbitraire on obtient\vvwa$\nst{\hb\tmu\ten\tnu\ha}\ha\ge\ha\nst{\ha\tmu\ha}\,\nst{\tnu}$\va
Posons\,\hh$\tchi\ha=\ha\vtmu\ten\vtnu-|\hb\tmu\ten\tnu\hb|\in\PM\PK^+$\pv
on a donc\,\hh$\nst{\ha\tchi\ha}=\tchi\,(\un_{\II}\ten\un_{\!\JJ})$\vvw
$=\,(\hb\vtmu\ten\vtnu\hb)\,(\un_{\II}\ten\un_{\!\JJ})-|\hb\tmu\ten\tnu\hb|\,(\un_{\II}\ten\un_{\!\JJ})
=\,\vtmu\,(\un_{\II})\;\vtnu\,(\un_{\!\JJ})-\nst{\hb\tmu\ten\tnu\hb}$\vvw
$=\,\nst{\ha\tmu\ha}\,\nst{\tnu}-\nst{\hb\tmu\ten\tnu\ha}\le\hb0$\pv donc\,\hh
$\nst{\,\tchi\,}=\ha0$\vg\cad\,\hh$\nst{\hb\tmu\ten\tnu\hb}=\nst{\ha\tmu\ha}\,
\nst{\tnu}$\vg\vvw et donc aussi\,\hh$\tchi=\ha0$\vg \cad\,\hh$|\hb\tmu\ten\tnu\hb|=\vtmu\ten\vtnu$\vc

\Th{\num}\fbox{\,$\MM\PI\ten\MM\PJ\ha\sub\,\MM\PK$\ha}\vb

\Dem On a\quad$\all u\in\I\,\hh\lim\pos_{\alf\ha\to\ha u^\pm}\,\big|\hb\tdf\ten\tdg\hb\big|\:
\big(\hw1_{\hb{]\,u\vv\alf\,[}\ha\tim\hw\JJ}\ha\big)\hb
=\lim\pos_{\alf\ha\to\ha u^\pm}\,\big(\ha\vtdf\ten\vtdg\ha\big)\:
\big(\hw1_{\hb]\,u\vv\alf\,[}\ten1_\JJ\ha\big)$\vva$=\lim\pos_{\alf\ha\to\ha u^\pm}
\,\vtdf\hb(1_{\hb]\,u\vv\alf\,[})\,\vtdg\ha(1_{\JJ})\hb=\hb0$\vc

\Th{\num}\fbox{\,$\LL^1\PI\ten\LL^1\PJ\ha\sub\hb\LL^1\PK\ph{\big|\dot{l}}$\hs{-2.25mm}}\quad et\quad
$\all\tdf\in\LL^1\PI\,\hh\all\tdg\in\LL^1\PJ$\,\hh on a
$$\boxed{\,\bnone{\hb\tdf\ten\tdg\,}\,=\,\bnone{\ha\tdf\hb}\:\none{\hb\tdg\hb}\,}\pt$$\vvab

\Dem Soit\,\hh$f_n\in\ha\EE\PI$\,\hh tel que\,\hh$f_n\one\tdf$\,\hh et\,\hh$g\ha_n\in\ha\EE\PJ$\,\hh
tel que\,\hh$g\ha_n\one\tdg$\pv\vvwa alors on a\,\hh$\nn\,\hh\bnst{\hb\tdf_n\ten\tdg\ha_n-\tdf\ten\tdg\hb}\ha
=\hb\bnst{\hb\tdf_n\ten\tdg\ha_n-\tdf_n\ten\tdg\ha+\ha\tdf_n\ten\tdg-\tdf\ten\tdg\hb}$\vvwa
$\le\ha\bnst{\ha\tdf_n\ten(\tdg\ha_n-\tdg)\ha}\ha+\ha\bnst{\ha(\tdf_n-\tdf)\ten\tdg\hb}
\ha=\hb\bnone{\ha\tdf_n\ha}\,\bnone{\ha\tdg\ha_n-\tdg\ha}\ha+\ha\bnone{\ha\tdf_n-\tdf\ha}\,\none{\ha\tdg\ha}$\pv\vvwa
donc\, $\tdf_n\ten\tdg\ha_n\one\tdf\ten\tdg$\,\hh et\,\hh$\tdf\ten\tdg\in\LL^1\PK$\vcd

\Th{\num}\fbox{\,$\LL^2\PI\ten\LL^2\PJ\ha\sub\hb\LL^2\PK\ph{\big|\dot{l}}$\hs{-2.25mm}}\quad et\quad
$\all\tdf\in\LL^2\PI\,\hh\all\tdg\in\LL^2\PJ$\,\hh on a
$$\boxed{\,\bntwo{\hb\tdf\ten\tdg\,}\,=\,\bntwo{\ha\tdf\hb}\:\ntwo{\hb\tdg\hb}\,}\pt$$\vvab

\Dem Soit\,\hh$f_n\in\ha\EE\PI$\,\hh tel que\,\hh$f_n\two\tdf$\,\hh et\,\hh$g\ha_n\in\ha\EE\PJ$\,\hh tel que\,\hh
$g\ha_n\two\tdg$\pv\vvwa on sait d\e j\ac\ que\,\hh$\tdf_n\ten\tdg\ha_n\one\tdf\ten\tdg\in\LL^1\PK$\pv
de plus on a\, $\all m\vv n\in\N$\vvwa$\bntwo{\hb\tdf_m\ten\tdg\ha_m-\tdf_n\ten\tdg_n\hb}\,
=\,\bntwo{\hb\tdf_m\ten\tdg\ha_m-\tdf_m\ten\tdg\ha_n+\tdf_m\ten\tdg\ha_n-\tdf_n\ten\tdg_n\hb}$\vva
$\le\ha\bntwo{\hb\tdf_m\ten(\tdg\ha_m-\tdg\ha_n)\hb}\hb
+\hb\bntwo{\hb(\tdf_m-\tdf_n)\ten\tdg\ha_n\hb}\,=\,\none{\hb\tdf_m^{\ha2}\ten
(\tdg\ha_m-\tdg\ha_n)^2\hb}\hb+\hb\bnone{\hb(\tdf_m-\tdf_n)^2\ten\tdg\ha_n^{\hw2}\hb}$\vva
$=\,\bnone{\ha\tdf_m^{\ha2}\ha}\,\bnone{\hb(\tdg\ha_m-\tdg\ha_n)^2\hb}\hb
+\hb\bnone{\hb(\tdf_m-\tdf_n)^2\hb}\,\none{\hb\tdg\ha_n^{\hw2}\ha}$\vva
$=\,\bntwo{\ha\tdf_m\ha}^2\;\ntwo{\hb\tdg\ha_m-\tdg\ha_n\ha}^2
+\bntwo{\hb\tdf_m-\tdf_n\ha}^{\ha2}\;\ntwo{\hb\tdg\ha_n\ha}^2$\pv\vva
donc\, $\tdf_n\ten\tdg\ha_n$\, est une suite de Cauchy dans\, $\LL^2\PK$\vg
donc\, $\tdf_n\ten\tdg\ha_n\two\tdf\ten\tdg\in\LL^2\PK$\pv\vvwa
par ailleurs on a\,\hh$\bntwo{\hb\tdf\ten\tdg\hb}=\bnone{\hb\tdf^{\ha2}\ten\tdg^{\ha2}\hb}
=\bnone{\ha\tdf^{\ha2}\ha}\,\bnone{\ha\tdg^{\hb2}\ha}=\bntwo{\ha\tdf\ha}\:\ntwo{\hb\tdg\hb}$\vc

\Th{\num*}$\all\tdf,\,\tdg\in\LL^2\PI\,\hh\all\tdh\vv\tdk\in\LL^2\PJ$\,\hh on a\vvab
\hs{50mm}\fbox{\,$(\tdf\ten\tdg)\,(\tdh\ten\tdk)\ha=\hb(\tdf\,\tdh)\ten(\ha\tdg\:\tdk)$\hb}\ha\vc

\Th{\num}\fbox{$\BB\PI\,\ten\BB\PJ\ha\sub\,\BB\PK\ph{\big|\dot{l}}$\hs{-2.25mm}}\quad
et\quad$\all\tdf\in\BB\PI\,\hh\all\tdg\in\BB\PJ$\,\hh on a\vvb\hs{55mm}
\fbox{\,$\bnbb{\hb\tdf\ten\tdg\,}\,=\,\bnbb{\ha\tdf\hb}\;\nbb{\hb\tdg\hb}$\,}\vbc

\Dem On a\,\hh$\vtdf\le\bnbb{\ha\tdf\hb}$\,\hh et\,\hh$\vtdg\le\nbb{\hb\tdg\hb}$\vg donc\,\hh
$\big|\hb\tdf\ten\tdg\hb\big|=\vtdf\ten\vtdg\le\bnbb{\ha\tdf\hb}\:\nbb{\hb\tdg\hb}$\vg\vvwwa
donc\,\hh$\bnbb{\hb\tdf\ten\tdg\,}\le\bnbb{\ha\tdf\hb}\;\nbb{\hb\tdg\hb}$\ha\pv
par ailleurs soit\; $\eps>0$\pv soit\hh$p\in\ha\EE\PI$\hh tel que\hh$\none{\hb p\hb}\le1$\vvwwa
et\hh$\tdf\ha(\ha p)\ge\bnbb{\ha\tdf\hb}-\ha\eps$\pv de m\^eme soit\hh$q\in\ha\EE\PJ$\hh
tel que\hh$\none{\ha q\ha}\le1$\hh et\hh$\tdf\ha(q)\ge\nbb{\hb\tdg\hb}
-\ha\eps$\ha\pv\vvwwa on a\,\hh$\none{\hb p\ten q\hb}=\hb\none{\ha p\ha}\,\none{\ha q\ha}\le1$\,\hh
et\,\hh$(\ha\tdf\ten\tdg\ha)\,(\ha p\ten q\ha)\hb
=\hb\tdf\ha(\ha p)\;\tdg\hb(q)$\vvwwa$\ge\ha\bnbb{\ha\tdf\hb}\:\nbb{\hb\tdg\hb}
-\ha\eps\;\big(\ha\bnbb{\ha\tdf\hb}+\nbb{\hb\tdg\hb}\ha\big)$\vg
donc\,\hh$\bnbb{\hb\tdf\ten\tdg\,}\ha\ge\ha\bnbb{\ha\tdf\hb}\:
\nbb{\hb\tdg\hb}\ha-\hb\eps\;\big(\ha\bnbb{\ha\tdf\hb}+\nbb{\hb\tdg\hb}\ha\big)$\pv\vvwwa
donc\,\hh$\bnbb{\hb\tdf\ten\tdg\,}\ha\ge\ha\bnbb{\ha\tdf\hb}\;\nbb{\hb\tdg\hb}$\vd

\initnum
\quad\textbf{\Num\underline{Points de Lebesgue}}\vvc

\oui Soit\, $\II$\, un intervalle compact de\, $\R$\pt Quelle signification donner \ac\: $\tdf\ha(x+u)$\, quand\vvx
$\tdf\in\LL^1(\hw\II\hw)$\hh?\, Si\, $\tdf$\, est la limite en norme\, $\none{\ }$\, de la suite\,
$f_n\in\ha\EE\hw(\hw\II\hw)$\,,\hb\ il semble naturel\vvx de d\e finir\, $\tdf\ha(x+u)$\, comme la limite
en norme\, $\none{\ }$\, de la suite\, $f_n\ha(x+u)$\vbc

\Th{\num}Soit\, $\tdf\in\LL^1(\hw\II\hw)$\, et soit une suite\, $f_n\in\ha\EE\hw(\hw\II\hw)$\,
telle que\, $f_n\one\tdf$\hb;\\on convient de prolonger\ha\ $\tdf$\ha\ et\ha\ $\tdf_n$\ha\ par\, 0\,
en dehors de\hb\ $\II$\pv alors\, $f_n\ha(x\ha+u)$\, converge en norme\, $\none{\ }$\, sur\hh$\II\tim\R$\:
vers une fonctionnelle\: $\tdf\ha(x+u)\in\ha\LL^1\ha\big(\hw\II\tim\R\big)$\vg  ind\e pendante de la suite particuli\ee re\ha\ $f_n$\vab

\Dem On a\,\hh$\all p>n\,\ \Int\hs{-2mm}\int_{\ha\II\,\tim\ha\R}\hs{-5mm}|\hb f_p\ha(x+u)-f_n\ha(x+u)\hb|\;d\hb x\ d\hb u$\vvwa
$=\Int_{\ha\II}\Big[\int_{\R}\hs{-1mm}|\hb f_p\ha(x+u)-f_n\ha(x+u)\hb|\:d\hb u\,\Big]\;d\hb x
=\int_{\ha\II}\Big[\int_{\R}\hs{-1mm}|\hb f_p\ha(v)-f_n\ha(v)\hb|\;d\hb v\,\Big]\:d\hb x$\vvab
$=\hb|\ha\II\ha|\:\none{\hb f_p-f_n\hb}$\pv donc\hh$f_n\ha(x+u)$\hh est bien une suite de Cauchy
pour la norme\, $\none{\ }$\vvw  sur\, $\II\ha\tim\R$\vg dont nous noterons la limite\, $\tdf\ha(x+u)$\pt
On d\e montre de fa\c con analogue que\vvw cette limite est effectivement ind\e pendante de la suite particuli\ee re\ha\ $f_n$\vb

\underline{Attention}\, !\hh$\tdf\ha(x+u)$\, n'est pas une \emph{fonction} des ``\,variables\," $x$\,
et\, $u$\vg mais bien une\, \emph{fonctionnelle} agissant sur\, $\EO\ha\big(\hw\II\tim\R\big)$\pv
la meilleure fa\c con d'interpr\e ter les  symboles\, $x$\, et\, $u$\, revient donc \ac\ les consid\e rer
comme des \emph{variables potentielles d'int\e gration}\vx
\oui Il en va de m\^eme pour l'expression\, $\tdf\ha(x)$\vg d\e pourvue d'int\e gration explicite sur\, $x$\vg \mbox{(\ha\e criture que nous avons
soigneusement} \e vit\e e jusqu'\ac\ pr\e sent\ha)\, :\, dans ce qui suit\, $\tdf\ha(x)$\, repr\e sente en r\e alit\e\ la fonctionnelle\:
$\tdf\hs{-.15mm}\ten\hs{-.05mm}\un_{\R}$\vg elle aussi d\e finie sur\, $\EO\ha\big(\hw\II\tim\R\big)$\hw.\vvb

\Def{\num}Soit\: $\tdf\in\LL^1(\hw\II\hw)$\pv on pose\,\hh\underline{$\all s\vv t>\ha0$}\vvb
\hs{40mm}\fbox{\,$\Psist\hb(\tdf)\ha=\ha\Inv{s+t}\,
\int_{-\ha s}^{\,t}\hs{-1mm}\big|\hb\tdf\ha(x+u)-\tdf\ha(x)\hb\big|\;d\hb u$\,}\,.\vvbc

$(s\ha+\ha t)\,\Psist\hb(\tdf)$\, n'est rien d'autre que la fonctionnelle marginale sur\hb\ $\II$\hb\ de\ha\
$\big|\ha\tdf\ha(x\ha+\ha u)\ha-\hw\tdf\ha(x)\ha\big|$\vvww restreint \ac\: $\II\tim[\ha-\,s\vv t\,]$\pv
en cons\e quence\hh$\all s\vvv t>0\,\hh\Psist\hb(\tdf)\in\hw\LL^1(\hw\II\hw)$\vc

\Th{\num}Soit\, $\tdf\in\LL^1(\hw\II\hw)$\pv alors\,\hh\fbox{\,$\Psist\hb(\tdf)\ha\pp\ha0$\,}\,\hh
quand\ha\hh$s+t\to0$\vb

\Dem Soit\hh$\eps>0$\hh et\ha\ soit\hh$g\in\ha\EE\hw(\hw\II\hw)$\; tel que\hh$\bnone{\ha\tdf-g\hb}
\le\ha\eps\hb^2$\pv alors on a\,\hh$\all s\vvv t>\ha0$\vvaab$\Psist\hb(\tdf)\ha
\le\ha\Inv{s+t}\,\int_{-\ha s}^{\,t}\hs{-1mm}\big|\hb\tdf\ha(x+u)-g\hb(x+u)\hb\big|\ d\hb u$\vvw
\hs{55mm}$+\ \Inv{s+t}\,\int_{-\ha s}^{\,t}\hs{-1mm}|\,g\hb(x+u)-g\hb(x)\hb|\ d\hb u\,
+\,\big|\hb\tdf\ha(x)-g\hb(x)\ha\big|$\vva
$=\ha\Inv{s+t}\,\int_{x\ha-\ha s}^{\,x\hb+\hb t}\hs{-1mm}\big|\hb\tdf\ha(u)-g\hb(u)\hb\big|\ d\hb u\,
+\,\Inv{s+t}\,\int_{-\ha s}^{\,t}\hs{-1mm}|\,g\hb(x+u)-g\hb(x)\hb|\ d\hb u\,+\,\big|\hb\tdf-g\hb\big|$\pv\vvab

donc\quad$\Big\{\ha\Psist\hb(\tdf)\ha>\ha\eps\ha\Big\}\ha\le\ha\Big\{\ha\Inv{s+t}\,\int_{x\ha-\ha s}
^{\,x\hb+\hb t}\hs{-1mm}\big|\hb\tdf\ha(u)-g\hb(u)\hb\big|\ d\hb u\ha>\ha\frac{\eps}{3}\hb\Big\}$\vva\hs{30mm}
$+\ha\ \Big\{\ha\Inv{s+t}\,\int_{-\ha s}^{\,t}\hs{-1mm}|\,g\hb(x+u)-g\hb(x)\hb|\ d\hb u\ha
>\ha\frac{\eps}{3}\hb\Big\}\hb+\hb\Big\{\ha\big|\hb\tdf-g\hb\big|\ha>\ha\frac{\eps}{3}\hb\Big\}$\vb
On continue ensuite comme au\ha\ \Theo\, X\ha\ 4\ha.\:4\vvbc

\newpage
\Cor{\num*}\vvw Soit\, $\tdf\in\LL^1(\hw\II\hw)$\pv alors\hb\hh\fbox{\:$\Inv{t}\int_{-\ha t}^{\:t}\hs{-.5mm}
\big|\hb\tdf\ha(x+u)-\tdf\ha(x)\hb\big|\ d\hb u\ha\pp\ha0$\:}\hb\hh quand\ha\hh$t\to0$\vc

\oui Ce corollaire correspond \ac\ l'affirmation classique de l'existence des ``\,points de\\Lebesgue\,"
presque partout sur\, $\II$\vd

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\initnum
\quad\textbf{\Num\underline{\Theo\ha\ de Fubini}}\vvc

\num\ \underline{\Theo\ha\ de projection}\, (\ha voir notation au\ha\ \Theo\ha\ XIII\ha\ B\ha.\ha)\vva

Soit\, $f\in\ha\EE\PK$\pv alors\quad$\all x\ze\in\I$\,\hh\fbox{\,$f\,(x\ze\hb,\hb*)\in\ha\EE\PJ$\,}\vbc

\begin{picture}(,)
\put(5,0){Idem en rempla\c{c}ant\, $\EE$\, par\hh$\CC\vv\,\RR\vv\,\PR$\, ou\, $\WW$\,\hh.}
\put(3,-2.25){\frm(89.5,7)}
\end{picture}\\

\Dem Le \theo\ha\ est \e vident pour\,  $\EE\vv\,\CC\vv\,\RR\vv\,\PR$\pt D\e montrons\ha-\ha le pour\, $\WW$\vww
Soient\hh$\eps>0$\vv\hh$x\ze\in\I$\vv\hh$\tnu\in\MM^+\PJ$\pv soient\: $g\vv h\ha\in\PR\PK$\hh telles que\vvww
$g\le f\le\ha h$\,\hh et\,\hh$(\hb\del\ha_{x\ze}\ten\tnu\hb)\,(h-g)\le\hb\eps$\pv on a\,\hh$g\hb(x\ze\hb,\hb*)
\in\PR\PJ\,,\ h\hb(x\ze\hb,\hb*)\in\PR\PJ$\vvww et\,\hh$g\hb(x\ze\hb,\hb*)\le\ha
f\ha(x\ze\hb,\hb*)\le\ha h\hb(x\ze\hb,\hb*)$\pv de plus gr\^ace au \theo\ha\ de Fubini dans\, $\PR\PK$\vvww
(que nous d\e montrons plus loin)\, on peut \e crire\vvw
$\tnu\,\big[\hb h\hb(x\ze\hb,\hb*)-g\,(x\ze\hb,\hb*)\ha\big]
=(\hb\del\ha_{x\ze}\ten\tnu\hb)\,(h-g)\ha\le\hb\eps$\pv donc\,\hh$f\ha(x\ze\hb,\hb*)\in\WW\PJ$\vc

\num\ \underline{\Theo\ha\ de transfert}\vva

Soient\; $f\in\ha\EE\PK$\hh et\hh$\tnu\in\MM\PJ$\pv on pose\,\hh$\all x\in\I$\vva
$F\ha(x)=\,\tnu\,\big[\hb f\ha(x\hb,\hb*)\ha\big]=\Int_{\ha\J}f\ha(x\hb,\hb y)\:\tnu\,(y)$\pv alors\,\hh
\fbox{\hb$F\in\ha\EE\PI$\ha}\vab

\begin{picture}(,)
\put(5,-2){Idem en rempla\c{c}ant\, $\EE$\, par\hh$\CC\vv\,\RR\vv\,\PR$\, ou\, $\WW$\,\hh.}
\put(3,-4.25){\frm(89.5,7)}
\end{picture}\vvc

\Dem\vva

\hhh a)\hh$f\in\ha\EE\PK$\dd Evident\va

\hhh b)\hh$f\in\CC\PK$\dd
Soit\, $\eps>0$\pv par compacit\e\ de\, K\, il existe\hh$\eta>0$\hh tel que\hh
$\al\;x\hw_1\vv x\ha_2\in\I$\vvw$\all y\hw_1\vv y\ha_2\in\J\quad
|\hb f\ha(x\hw_1\ha,\hb y\hw_1)-f\ha(x\hw_2\ha,\hb y\ha_2)\hb|\le\hb\eps$\pv
en particulier\, $\al\;x\hw_1\vv x\ha_2\in\I\quad\all y\in\J$\vvwa
$|\hb f\ha(x\hw_1\ha,\hb y)-f\ha(x\ha_2\ha,\hb y)\hb|\le\hb\eps$\pv donc\hh$\al\;x\hw_1\vv x\ha_2\in\I$\hh
on a\vvwa$|\,F\hb(x\hw_1)-F\hb(x\ha_2)\ha|\ha\le\Int_{\ha\J}\big|\hb f\ha(x\hw_1\ha,\hb y)
-f\ha(x\ha_2\ha,\hb y)\hb\big|\;\tnu\,(y)\ha\le\hb\eps\:\nst{\hb\tnu\hb}$\pv donc\hh$F\in\CC\PI$\vab

\newpage
\hhh c)\hh$f\in\RR\PK$\dd Soit une suite\, $f_n\in\ha\EE\PK$\; telle que\: $f_n\uu f$\pv\vvw
posons\,\hh$\nn\,\hh\all x\ze\in\I\,\hh F_n\hb(x\ze)=\tnu\,\big[\hb f_n\hb(x\ze\hb,\hb*)\ha\big]$\pv
on a\,\hh$\all x\ze\in\I$\vvw$|\hb F\hb(x\ze)-F_n\hb(x\ze)\ha|\ha
\le\Int_{\ha\J}\big|\hb f\ha(x\ze\ha,\hb y)-f_n\ha(x\ze\ha,\hb y)\hb\big|\;
\tnu\,(y)\ha\le\hb\|\ha f-f_n\ha\|\,\nst{\hb\tnu\hb}$\pv\vvw donc\hh$F_n\uu F$\pv
or\hb\hh$\nn\,\hh F_{\ha n}\in\ha\EE\PI$\vg donc\hh$F\in\RR\PI$\vab

\hhh d)\hh$f\in\PR\PK$\dd Soit une suite\, $f_n\in\ha\EE\PK$\; telle que\: $f_n\bb f$\pv
on a en particulier\vvwa$\all x\ze\in\I\quad f_n\ha(x\ze\hb,\hb*)\bb
f\ha(x\ze\hb,\hb*)$\vg donc par le \theo\ha\ de Lebesgue\ha\ sur\, J\vva
$\all x\ze\in\I\quad F_{\ha n}\hb(x\ze)=\!\Int_{\ha\J}f_n\ha(x\ze\ha,\hb y)\
\tnu\,(y)\to\!\Int_{\ha\J}f_n\ha(x\ze\ha,\hb y)\ \tnu\,(y)=F\hb(x\ze)$\pv
de plus\hh$\nn$\vvwa$\|F_{\ha n}\|\le\|\ha f_n\ha\|\,\nst{\ha\tnu\ha}$\vg donc\hh$F_{\ha n}\bb F$\pv
or\hb\hh$\nn\,\hh F_{\ha n}\in\ha\EE\PI$\vg donc\hh$F\in\PR\PI$\vab

\hhh e)\hh$f\in\WW\PK$\dd Soient\hh$\eps>0$\vv\hh$\tmu\in\MM^+\PI$\vv\hh$\tnu\in\MM^+\PJ$\pv
soient\: $g\vv h\ha\in\PR\PK$\vvwa telles que\hh$g\le f\le\ha h$\hh et\hh
$(\hb\tmu\ten\tnu\hb)\,(h\ha-\ha g)\le\hb\eps$\pv posons\,\hh$\all x\ze\in\I$\vva
$G\hb(x\ze)=\!\Int_{\ha\J}\hb g\hb(x\ze\hb,\hb y)\;\tnu\,(y)$\hh et\hh$H\hb(x\ze)
=\!\Int_{\ha\J}\hb h\hb(x\ze\hb,\hb y)\:\tnu\,(y)$\pv\vva on a\,\hh$G\vv H\in\PR\PI$\hh\hb
et\,\hh$G\le F\le H$\pv\vvwa de plus gr\^ace au \theo\ha\ de Fubini dans\, $\PR\PK$\, (que nous d\e montrons
plus loin)\vvwa on peut \e crire\,\hh$\tmu\,(H\!-G)=(\hb\tmu\ten\tnu\hb)\,(h-g)\le\hb\eps$\pv
donc\,\hh$F\in\WW\PI$\vwa Le \theo\ha\ est donc d\e montr\e\ pour tout\; $\tnu\in\MM^+\PJ$\pv soit alors\hh
$\tnu\in\MM\PJ$\pv\vvwa on a\,\hh$\all x\ze\in\I\quad F\hb(x\ze)
=\!\Int_{\ha\J}\hb f\ha(x\ze\hb,\hb y)\;\tnu\,(y)=\!\int_{\ha\J}\hb f\ha(x\ze\hb,\hb y)\;\tnu^{\hb+}\,(y)
-\!\int_{\ha\J}\hb f\ha(x\ze\hb,\hb y)\;\tnu^{\hb-}\,(y)$\vg\vvwa donc\hh$F\in\WW\PI$\vc

\num\ \underline{\Theo\ha\ de Fubini}\vvb

Soient\hh$f\in\WW\PK\vv\ \tmu\in\MM\PI\vv\ \tnu\in\MM\PJ$\pv alors on a\vvb

\hs{17.5mm}\fbox{\,$(\hb\tmu\ten\tnu\hb)\ha(f)
=\Int_{\,\I}\Big[\int_{\ha\J}f\ha(x\ha,\hb y)\:\tnu\hb(y)\ha\Big]\hb\tmu\hb(x)\ha
=\int_{\ha\J}\Big[\int_{\,\I}f\ha(x\ha,\hb y)\:\tmu\hb(x)\ha\Big]\hb\tnu\hb(y)$\,}\vc

\Dem\vva

\hhh a)\hh$f\in\ha\EE\PK$\dd Evident\vab

\hhh b)\hh$f\in\PR\PK$\dd Soit une suite\, $f_n\in\ha\EE\PK$\; telle que\: $f_n\bb f$\pv\vva
par d\e finition on a\:\hh$(\hb\tmu\ten\tnu\hb)\,(f)=\hb\lim\pos_n\;\big[\ha(\hb\tmu\ten\tnu\hb)\,
(f_n)\ha\big]$\pv or on peut \e crire\hh$\nn$\vvwa$(\hb\tmu\ten\tnu\hb)\,(f_n)=\Int_{\,\I}
\Big[\hb\int_{\ha\J}\,f_n\hb(x\ha,\hb y)\:\tnu\hb(x)\ha\Big]\hb
\tmu\hb(y)$\;;\; posons\,\hh$\all x\ze\in\I$\vvwa$F\hb(x\ze)=\!\Int_{\ha\J}
f\ha(x\ze\hb,\hb y)\:\tnu\,(y)$\,\hh et\,\hh$\nn\,\hh F_n\hb(x\ze)=\!\Int_{\ha\J}
f_n\ha(x\ze\hb,\hb y)\:\tnu\,(y)$\pv\vva on a\hh$F\in\PR\PI$\hh\ha et\ha\hh$\nn\hh\ha F_n\in\ha\EE\PI$\pv
comme\,\hh$\all x\ze\in\I\,\hh f_n\ha(x\ze\hb,\hb*)\bb f\ha(x\ze\hb,\hb*)$\,,\vva
le \theo\ha\ de Lebesgue sur\, J\, nous donne\,\hh$\all x\ze\in\I\,\hh F_n\hb(x\ze)\to F\hb(x\ze)$\pv\vva
on en d\e duit\,\hh$F_n\bb F$\pv le \theo\ha\ de Lebesgue sur\, I\, nous donne donc\vva
$(\hb\tmu\ten\tnu\hb)\,(f_n)=\!\Int_{\,\I}F_n\hb(x)\,\tmu\hb(x)\to\!
\int_{\,\I}F\hb(x)\,\tmu\hb(x)$\pv on en d\e duit\vvwa$(\hb\tmu\ten\tnu\hb)\,(f)
=\lim\pos_n\;\big[\ha(\hb\tmu\ten\tnu\hb)\,(f_n)\ha\big]=\!\Int_{\,\I}F\hb(x)\,\tmu\hb(x)$\vbc

\hhh c)\hh$f\in\WW\PK$\dd Soient\, $\tmu\in\MM^+\PI$\, et\, $\tnu\in\MM^+\PI$\pv
soit\, $\eps>0$\vva et soient\, $g\,,\, h\in\PR\PK$\hh tels que\hh$g\le f\le\hb h$\hh
et\hh$(\hb\tmu\ten\tnu\hb)\,(h-g)\le\hb\eps$\pv\vvab on a\,\hh$(\hb\tmu\ten\tnu\hb)\,(g)
=\Int_{\,\I}\hb\Big[\hb\int_{\ha\J}g\hb(x\ha,\hb y)\:\tnu\hb(y)\hb\Big]\hb\tmu\hb(x)$\vva
et\quad$(\hb\tmu\ten\tnu\hb)\,(h)=\Int_{\,\I}\hb\Big[\hb
\int_{\ha\J}h\ha(x\ha,\hb y)\:\tnu\hb(y)\hb\Big]\hb\tmu\hb(x)$\pv\vvb
on en d\e duit\quad$(\hb\tmu\ten\tnu\hb)\,(g)\le(\hb\tmu\ten\tnu\hb)\,(f)
\le(\hb\tmu\ten\tnu\hb)\,(h)$\vvab et\quad$(\hb\tmu\ten\tnu\hb)\,(g)
\le\Int_{\,\I}\hb\Big[\hb\int_{\ha\J}f\ha(x\ha,\hb y)\:\tnu\hb(y)\hb\Big]\hb\tmu\hb(x)
\le(\hb\tmu\ten\tnu\hb)\,(h)$\pv\vvab de plus\,\hh$(\hb\tmu\ten\tnu\hb)\,(h)
-(\hb\tmu\ten\tnu\hb)\,(g)\ha\le\,\eps$\pv\vvab donc\quad
$\Big|\,(\hb\tmu\ten\tnu\hb)\,(f)-\Int_{\,\I}\hb\Big[\hb\int_{\ha\J}
f\ha(x\ha,\hb y)\:\tnu\hb(y)\hb\Big]\hb\tmu\hb(x)\,\Big|\le\hb\eps$\,;\vvw
comme\, $\eps$\, est arbitraire on en d\e duit\:\hh$(\hb\tmu\ten\tnu\hb)\,(f)
=\Int_{\,\I}\hb\Big[\hb\int_{\ha\J}f\ha(x\ha,\hb y)\:\tnu\hb(y)\hb\Big]\hb\tmu\hb(x)$\vaab
Le cas g\e n\e ral s'obtient en d\e composant\: $\tmu$\: et\: $\tnu$\: en parties positives et n\e gatives\vd

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\initnum
\quad\textbf{\Num\underline{G\e n\e ralisation \ac\hb\ $\R^2$}}\vvc

\oui Les deux \theo s suivants se d\e duisent facilement des \theo s correspondants pour les rectangles\vc

A.\, \underline{\Theo\ha\ de Lebesgue sur\hb\ $\R^2$}\vvbc

\hs{5mm}Soit une suite\, $f_n\in\WB\ha(\R^2)$\hh telle que\hh$f_n\bb f\in\WB\ha(\R^2)$\pv
alors\,\hh$\all\tmu\in\MX(\R^2)$\vvw\hs{5mm}on a\,\hh$f_n\;\tmu\fin f\hb\tmu$\ha\pv
en cons\e quence\,\hh$\tmu\,(f_n)\to\tmu\,(f)$\vpt

\begin{picture}(,)
\put(2,3.5){\frm(154.75,17.75)}
\end{picture}\vvb

\newpage
B.\, \underline{\Theo\ha\ de Fubini sur\hb\ $\R^2$}\vva

Soient\, $f\in\WW_B\ha(\R^2)$\hh et\hh$\tmu\vv\tnu\in\MX\hw(\R)$\pv alors\,
$\Int_{\ha\R}\hs{-1.5mm}f\ha(*\ha,\ha y)\:\tnu\hb(y)$\hb\hh
et\, $\Int_{\ha\R}\hs{-1.5mm}f\ha(x\ha,\ha *)\:\tmu\hb(x)$\vvwa
appartiennent \ac\, $\WB\ha(\R)$\,  et on a\vvb\hs{15mm}\fbox{\,$(\hb\tmu\ten\tnu\hb)\ha\ha(f)
=\Int_{\,\R}\hs{-.75mm}\Big[\int_{\ha\R}\hs{-1.5mm}f\ha(x\ha,\ha y)\:\tnu\hb(y)\ha\Big]\hb\tmu\hb(x)\ha
=\int_{\,\R}\hs{-.75mm}\Big[\int_{\ha\R}\hs{-1.5mm}f\ha(x\ha,\ha y)\:\tmu\hb(x)\ha\Big]\hb\tnu\hb(y)$\,}\vf

C.\, \underline{\Theo\ha\ de Fubini sur\hb\ $\R^2$\, :\, \tsf{Int\e grales g\e n\e ralis\e es}}\vvbc

Soit\hh$f\in\WW\ha(\R^2)$\hh et supposons\,\hh
\fbox{\:$\dis\sup_{k}\ha\int_{-\hw k}^{\hb k}\int_{-\hw k}^{\hb k}\hs{-1mm}|\hb f\ha(x\ha,\ha y)\ha|\;
d\hb x\;d\hb y\ha<\pin$\:}\pv\vvaab alors\, $\Int_{\ha\R}\hs{-1.5mm}f\ha(*\ha,\ha y)\:d\hb y$\hb\hh
et\, $\Int_{\ha\R}\hs{-1.5mm}f\ha(x\ha,\ha *)\:d\hb x$\hh appartiennent \ac\, $\LL^1\ha(\R)$\;
et on a\vvab\hs{35mm}\fbox{\:$\Int_{\,\R}\hs{-.75mm}\Big[\int_{\ha\R}\hs{-1.5mm}f\ha(x\ha,\ha y)\:
d\hb y\hb\Big]\ha d\hb x\ha=\int_{\,\R}\hs{-.75mm}\Big[\int_{\ha\R}\hs{-1.5mm}f\ha(x\ha,\ha y)\:
d\hb x\hb\Big]\ha d\hb y$\;}\ha\vcd

\Dem Posons\,\hh$\all k\in\N\,\hh\alf\ha_k\ha=\Int_{-\hw k}^{\hb k}\int_{-\hw k}^{\hb k}\hs{-1mm}
|\hb f\ha(x\ha,\ha y)\ha|\;d\hb x\;d\hb y$\,\hh et soit\,\hh$\alf=\hb\dis\sup_{k}\,\alf\ha_k<\pin$\pv\vva
on a\quad$\all\el\ge k\quad\alf\ha_{\el}-\alf\ha_k\ha
=\Int_{-\hw\el}^{\hb\el}\int_{-\hw\el}^{\hb\el}\hs{-1mm}|\hb f\ha(x\ha,\ha y)\ha|\;d\hb x\;d\hb y
-\int_{-\hw k}^{\hb k}\int_{-\hw k}^{\hb k}\hs{-1mm}|\hb f\ha(x\ha,\ha y)\ha|\;d\hb x\;d\hb y$\vva
$=\Int_{-\hw\el}^{\hb\el}\int_{-\hw\el}^{\hb\el}\hs{-1mm}|\hb f\ha(x\ha,\ha y)\ha|\;d\hb x\;d\hb y
-\int_{-\hw\el}^{\hb\el}\int_{-\hw k}^{\hb k}\hs{-1mm}|\hb f\ha(x\ha,\ha y)\ha|\;d\hb x\;d\hb y$\vva
\hs{50mm}$+\Int_{-\hw\el}^{\hb-\hw k}\int_{-\hw k}^{\hb k}\hs{-1mm}|\hb f\ha(x\ha,\ha y)\ha|\;d\hb x\;
d\hb y\,+\int_k^{\hb\el}\int_{-\hw k}^{\hb k}\hs{-1mm}|\hb f\ha(x\ha,\ha y)\ha|\;d\hb x\;d\hb y$\vva

$=\Int_{-\hw\el}^{\hb\el}\int_{-\hw\el}^{-\hw k}\hs{-1mm}|\hb f\ha(x\ha,\ha y)\ha|\;d\hb x\;d\hb y\,
+\int_{-\hw\el}^{\hb\el}\int_k^{\hb\el}\hs{-1mm}|\hb f\ha(x\ha,\ha y)\ha|\;d\hb x\;d\hb y$\vva
\hs{50mm}$+\Int_{-\hw\el}^{\hb-\hw k}\int_{-\hw k}^{\hb k}\hs{-1mm}|\hb f\ha(x\ha,\ha y)\ha|\;d\hb x\;
d\hb y\,+\int_k^{\hb\el}\int_{-\hw k}^{\hb k}\hs{-1mm}|\hb f\ha(x\ha,\ha y)\ha|\;d\hb x\;d\hb y$\ha\va

Par ailleurs posons\,\hh$\all k\in\N\,\hh F_{\hw k}=\Int_{-\hw k}^{\hb k}\hs{-1.5mm}f\ha(*\ha,\ha y)\:
d\hb y\ha\in\ha\WW\ha(\R)$\pv\vvaab$F_{\hw k}$\, est une fonction sommable\pv en effet on a\hh
$\all n\ge k$\vvab$\Int_{-\hw n}^{\hb n}|\hb F_{\hw k}\ha(x)\hb|\;d\hb x
=\int_{-\hw n}^{\hb n}\Big|\int_{-\hw k}^{\hb k}\hs{-1.5mm}f\ha(x\ha,\ha y)\hb\Big|\;d\hb x\;d\hb y\ha
\le\int_{-\hw n}^{\hb n}\int_{-\hw k}^{\hb k}\hs{-1.5mm}|\hb f\ha(x\ha,\ha y)\ha|\;d\hb x\;d\hb y\hb
\le\hb\alf\ha_n\hb\le\hb\alf$\pv\vva donc\,\hh$\all k\in\N\,\hh\none{\ha F_{\hw k}\hb}
=\hb\dis\sup_n\int_{-\hw n}^{\hb n}|\hb F_{\hw k}\ha(x)\hb|\;d\hb x\ha\le\hb\alf$\vb
\newpage
Montrons que la suite\, $F_{\hw k}$\, converge sur\, $\R$\, en norme\, $\none{\ }$\pv
on a\hh$\all\ha n\ge\el>k$\vva
$\Int_{-\hw n}^{\hb n}|\ha F_{\hw\el}\ha(x)-F_{\hw k}\ha(x)\ha|\ d\hb x
=\Int_{-\hw n}^{\hb n}\Big|\Int_{-\hw\el}^{\hb\el}\hs{-1.5mm}f\ha(x\ha,\ha y)\:d\hb y
-\int_{-\hw k}^{\hb k}\hs{-1.5mm}f\ha(x\ha,\ha y)\:d\hb y\,\Big|\ d\hb x$\vva
$=\Int_{-\hw n}^{\hb n}\Big|\int_{-\hw\el}^{\hb-\hw k}\hs{-1.5mm}f\ha(x\ha,\ha y)\:d\hb y\,
+\int_k^{\hb\el}\hs{-1.5mm}f\ha(x\ha,\ha y)\:d\hb y\,\Big|\;d\hb x$\vva
$\le\Int_{-\hw n}^{\hb n}\int_{-\hw\el}^{\hb-\hw k}\hs{-1.5mm}|\hb f\ha(x\ha,\ha y)\ha|\ d\hb x\;d\hb y\,
+\int_{-\hw n}^{\hb n}\int_k^{\hb\el}\hs{-1.5mm}|\hb f\ha(x\ha,\ha y)\ha|\ d\hb x\;d\hb y$\vva
$\le\Int_{-\hw n}^{\hb n}\int_{-\hw n}^{\hb-\hw k}\hs{-1.5mm}|\hb f\ha(x\ha,\ha y)\ha|\ d\hb x\;d\hb y\,
+\int_{-\hw n}^{\hb n}\int_k^{\hb n}\hs{-1.5mm}|\hb f\ha(x\ha,\ha y)\ha|\ d\hb x\;d\hb y\hb
\le\hb\alf\ha_n-\alf\ha_k\hb\le\hb\alf-\alf\ha_k$\pv\vva
on en d\e duit\,\hh$\all\el>k\,\hh\none{\hb F_{\hw\el}-F_{\hw k}\hb}\hb=\,\dis\lim_n\ha\int_{-\hw n}^{\hb n}
|\ha F_{\hw\el}\ha(x)-F_{\hw k}\ha(x)\ha|\ d\hb x\hb\le\hb\alf-\alf\ha_k$\vab
Par cons\e quent la suite\, $F_{\hw k}$\, est de Cauchy dans\, $\LL^1(\R)$\, et converge vers une
fonctionnelle\vvaab de\, $\LL^1(\R)$\vg que nous notons\, $\Int_{\ha\R}\hs{-1.5mm}f\ha(*\ha,\ha y)\:d\hb y$\ha\va
De m\^eme\,\hh$\all k\in\N\,\hh G_{\hw k}=\Int_{-\hw k}^{\hb k}\hs{-1.5mm}f\ha(x\ha,\ha*)\:d\hb x\ha\in\WW\ha(\R)$\hh
est une fonction sommable\vg et la suite\, $G_{\hw k}$\, converge sur\, $\R$\, en norme\: $\none{\ }$\: vers\,
$\Int_{\ha\R}\hs{-1.5mm}f\ha(x\ha,\ha *)\:d\hb x\ha\in\ha\LL^1(\R)$\vab

Montrons que l'on a\hh$\Int_{\,\R}\hs{-.75mm}\Big[\int_{\ha\R}\hs{-1.5mm}f\ha(x\ha,\ha y)\:
d\hb y\hb\Big]\, d\hb x\ha=\int_{\,\R}\hs{-.75mm}\Big[\int_{\ha\R}\hs{-1.5mm}f\ha(x\ha,\ha y)\:
d\hb x\hb\Big]\,d\hb y$\ha\vg\vva\cad\hhh$\dis\lim_k\int_{\,\R}\hs{-1.5mm}F_{\hw k}\ha(x)\;d\hb x\ha
=\,\lim_k\int_{\,\R}\hs{-1.25mm}G_{\hw k}\ha(x)\;d\hb x$\vg\vva
\cad\hhh$\dis\lim_k\int_{\,\R}\Big[\int_{-\hw k}^{\hb k}\hs{-1.5mm}f\ha(x\ha,\ha y)\:
d\hb y\ha\Big]\,d\hb x\ha=\,\lim_k\int_{\,\R}\Big[\int_{-\hw k}^{\hb k}\hs{-1.5mm}f\ha(x\ha,\ha y)\:d\hb x\ha\Big]\,d\hb y$\vva
\cad\hhh$\dis\lim_k\:\lim_j\ha\int_{-\hw j}^{\hb j}\Big[\int_{-\hw k}^{\hb k}\hs{-1.5mm}f\ha(x\ha,\ha y)\:d\hb y\ha\Big]\,d\hb x\ha
=\,\lim_k\:\lim_j\ha\int_{-\hw j}^{\hb j}\Big[\int_{-\hw k}^{\hb k}\hs{-1.5mm}f\ha(x\ha,\ha y)\:d\hb x\ha\Big]\,d\hb y$\vva
\cad\hhh$\dis\lim_k\:\lim_j\ha\int_{-\hw k}^{\hb k}\int_{-\hw j}^{\hb j}\hs{-1.5mm}f\ha(x\ha,\ha y)\
d\hb x\;d\hb y\ha=\,\lim_k\:\lim_j\ha\int_{-\hw j}^{\hb j}\int_{-\hw k}^{\hb k}\hs{-1.5mm}f\ha(x\ha,\ha y)\ d\hb x\;d\hb y$\va

On a\quad$\all k>j\hhh\Big|\Int_{-\hw k}^{\hb k}\int_{-\hw j}^{\,j}\hs{-1.5mm}f\ha(x\ha,\ha y)\ d\hb x\;d\hb y\,
-\int_{-\hw j}^{\,j}\int_{-\hw k}^{\hb k}\hs{-1.5mm}f\ha(x\ha,\ha y)\ d\hb x\;d\hb y\:\Big|$\vva
$=\,\Big|\Int_{-\hw k}^{\hb k}\int_{-\hw j}^{\,j}\hs{-1.5mm}f\ha(x\ha,\ha y)\ d\hb x\;d\hb y\,
-\int_{-\hw j}^{\,j}\int_{-\hw j}^{\,j}\hs{-1.5mm}f\ha(x\ha,\ha y)\ d\hb x\;d\hb y$\vva
\hs{50mm}$-\Int_{-\hw j}^{\,j}\int_{-\hw k}^{\hb-\hw j}\hs{-1.5mm}f\ha(x\ha,\ha y)\ d\hb x\;d\hb y\,
-\int_{-\hw j}^{\,j}\int_j^{\hb k}\hs{-1.5mm}f\ha(x\ha,\ha y)\ d\hb x\;d\hb y\:\Big|$\vva
$=\,\Big|\Int_{-\hw k}^{-\hw j}\int_{-\hw j}^{\,j}\hs{-1.5mm}f\ha(x\ha,\ha y)\ d\hb x\;d\hb y\,
+\int_j^{\hb k}\hs{-1mm}\int_{-\hw j}^{\,j}\hs{-1.5mm}f\ha(x\ha,\ha y)\ d\hb x\;d\hb y$\vva
\hs{50mm}$-\Int_{-\hw j}^{\,j}\int_{-\hw k}^{\hb-\hw j}\hs{-1.5mm}f\ha(x\ha,\ha y)\ d\hb x\;d\hb y\,
-\int_{-\hw j}^{\,j}\int_j^{\hb k}\hs{-1.5mm}f\ha(x\ha,\ha y)\ d\hb x\;d\hb y\:\Big|$\vva

\newpage
$\le\Int_{-\hw k}^{-\hw j}\int_{-\hw j}^{\,j}\hs{-1.5mm}|\hb f\ha(x\ha,\ha y)\ha|\ d\hb x\;d\hb y\,
+\int_j^{\hb k}\hs{-1mm}\int_{-\hw j}^{\,j}\hs{-1.5mm}|\hb f\ha(x\ha,\ha y)\ha|\ d\hb x\;d\hb y$\vva
\hs{50mm}$+\Int_{-\hw j}^{\,j}\int_{-\hw k}^{\hb-\hw j}\hs{-1.5mm}|\hb f\ha(x\ha,\ha y)\ha|\ d\hb x\;d\hb y\,
+\int_{-\hw j}^{\,j}\int_j^{\hb k}\hs{-1.5mm}|\hb f\ha(x\ha,\ha y)\ha|\ d\hb x\;d\hb y$\vva
$\le\Int_{-\hw k}^{-\hw j}\int_{-\hw j}^{\,j}\hs{-1.5mm}|\hb f\ha(x\ha,\ha y)\ha|\ d\hb x\;d\hb y\,
+\int_j^{\hb k}\hs{-1mm}\int_{-\hw j}^{\,j}\hs{-1.5mm}|\hb f\ha(x\ha,\ha y)\ha|\ d\hb x\;d\hb y$\vva
\hs{20mm}$+\Int_{-\hw k}^{\hb k}\int_{-\hw k}^{\hb-\hw j}\hs{-1.5mm}|\hb f\ha(x\ha,\ha y)\ha|\ d\hb x\;d\hb y\,
+\int_{-\hw k}^{\hb k}\int_j^{\hb k}\hs{-1.5mm}|\hb f\ha(x\ha,\ha y)\ha|\ d\hb x\;d\hb y\ha=\hb\alf\ha_k-\alf\ha_j$\pv\vvab
donc\quad$\dis\lim_k\:\lim_j\:\Big|\int_{-\hw k}^{\hb k}\int_{-\hw j}^{\,j}\hs{-1.5mm}f\ha(x\ha,\ha y)\ d\hb x\;d\hb y\,
-\int_{-\hw j}^{\,j}\int_{-\hw k}^{\hb k}\hs{-1.5mm}f\ha(x\ha,\ha y)\ d\hb x\;d\hb y\:\Big|\hb\le\hb\alf-\alf\hb=\hb0$\ha\vf

\initnum
\quad\textbf{\Num\underline{G\e n\e ralisation \ac\hb\ $\R^n$}}\vvc

\begin{picture}(,)
\put(30,0){\textbf{Tous les r\e sultats\ha\ de ce chapitre\hb\ho se g\e n\e ralisent}}
\put(30,-7){\textbf{sans difficult\e\ majeure aux espaces\, $\R^n$ (\ha$n\ge2$\ha)}\pt}
\put(27,-10.5){\frm(104.5,16.75)}
\end{picture}\vk22.5mm

\oui\textbf{\underline{R\e capitulatif}}\dd (\ha les fl\ee ches simples repr\e sentent des inclusions\ha)\vvaab

\begin{picture}(,)
\put(-1,-44){\frm(159.5,45)}
\end{picture}

\hs{10mm}$\WW\ha\Rn\ \under{surj.}\fun\ \BB\Loc\ha\Rn\ \to\ \LL^2\Loc\Rn\to
\LL^1\Loc\Rn\,\to\,\MM\ha\Rn\;\to\ \PM\ha\Rn$\vvw
\hs{15mm}$\upa\hs{25mm}\upa\hs{29mm}\upa\hs{21mm}\upa\hs{21mm}\upa\hs{20mm}\upa$\vvw
\hs{9.5mm}$\WB\ha\Rn\ha\ \under{surj.}\fun\,\ \BB\ha\Rn\hs{14.5mm}\LL^2\hw\Rn\hs{10mm}\LL^1\Rn\
\to\ \MX\Rn\,\to\,\PMX\Rn$\vvw
\hs{12mm}$\nearrow\hs{26.5mm}\upa\hs{30mm}\nwarrow\hs{10mm}\nearrow$\vvw
\hs{2mm}$\WO\ha\Rn\,\ \fun\,\LL^1\Rn\:\cap\ha\BB\ha\Rn\:\to\:\LL^1\Rn\cap\LL^2\hw\Rn$\vvd

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage\null\thispagestyle{empty}\newpage
\initNum

\centerline{\textbf{CHAPITRE\, XVI}}\vva
\centerline{\textbf{APPLICATIONS}}\vvcd

\oui D\e rivation des primitives\pt Convolution\pt Transform\e e de Laplace\pt \Theo\ha\ de Titchmarsh\pt
Transform\e e de Fourier\vcd

\initnum
\quad\textbf{\Num\underline{D\e rivation des primitives}}\vvc

\Def{\num}Soit\, $\tmu\in\MXD\ha(\R)$\pv on pose\:\hh$\all s>0\,\hh\all t>0\:\hh\all x\in\R$\vvb
\hs{30mm}\fbox{\:$\Phist\hb(\ha\tmu)\hb(x)\ha=\hb\Inv{s+t}\ha\ \tmu\,
\big(\ha\un\ha_{[\hb x\ha-\ha s\vv\ha x\hb+\hb t\hb]}\ha\big)\ha
=\hb\inv{s+t}\int_{\ha x\hw-\hw s}^{\hb x\ha+\ha t}\hs{-3mm}\tmu\hb(u)$\,}\ha\vcd

\Th{\num}$\al\;\tmu\in\MXD\ha(\R)$\hh on a\hh$\Phist\hb(\ha\tmu)\in\CB\ha(\R)$\hh
et\hh$\big\|\hb\Phist\hb(\ha\tmu)\hb\big\|\ha\le\hb\Inv{s+t}\ \nst{\ha\tmu\ha}$\vab

\Dem Si\, $F$\, est une primitive de\, $\tmu$\, on a\,\hh$\all x\in\R$\vvaab
$\Phist\hb(\ha\tmu)\hb(x)\ha=\hb\Inv{s+t}\,\big[\hw F\ha(x+\ha t)-F\ha(x-s)\ha\big]$\vg
donc\: $\Phist\hb(\ha\tmu)$\: est une fonction continue\pv\vva
de plus\ha\hh$\all x\in\R\,\hh\big|\ha\Phist\hb(\ha\tmu)\,(x)\ha\big|\ha\le\:\Inv{s+t}
\int_{\ha x\hw-\hw s}^{\hb x\ha+\ha t}\hs{-2mm}|\ha\tmu\ha|\,(x)\:\le\,\inv{s+t}\ \nst{\ha\tmu\hb}$\vc

\Th{\num}$\al\;\tmu\in\MXD\ha(\R)$\,\hh on a\:\hh$\Phist\hb(\ha\tmu)\in\LL^1(\R)$\:\hh
et\:\hh\fbox{\:$\bnone{\hb\Phist\hb(\ha\tmu)\hb}\le\ha\nst{\ha\tmu\ha}$\hb}\ha\vbc

\Dem$\bnone{\hb\Phist\hb(\ha\tmu)\hb}=\Int_{\R}\big|\ha\Phist\ha(\ha\tmu)\hb(x)
\ha\big|\ d\,x\,\le\,\inv{s+t}\int_{\R}\Big[\int_{\R}\un\ha_{[\,x\ha-\ha s\vv\ha x\hb+\hb t\ha]}
\hb(u)\:|\ha\tmu\ha|\hb(u)\ha\Big]\,d\,x$\vvb
or\,\hh$\all x\vv s\vv t\in\R$\,\hh on a\,\hh$\un\ha_{[\,x\ha-\ha s\vv\ha x\hb+\hb t\,]}\hb(u)\ha
=\ha\un\ha_{[\,u\hb-\hb t\vv\ha u\hb+\ha s\,]}\hb(x)$\hb\vg\ha donc\vvb
$\bnone{\hb\Phist\hb(\ha\tmu)}\le\;\Inv{s+t}\int_{\R}\Big[\int_{\R}
\un\ha_{[\,u\hb-\hb t\vv\ha u\hb+\ha s\,]}\hb(x)\ d\,x\,\Big]\,|\ha\tmu\ha|\hb(u)\:
=\:\inv{s+t}\int_{\R}\Big[\int_{\ha u\ha-\ha t}^{\hb u\ha+s}\hs{-1.5mm}d\,x\,\Big]
\,|\ha\tmu\ha|\hb(t)$\vvab\hs{92mm}$=\Int_{\R}|\ha\tmu\ha|\hb(t)\,=\,\nst{\ha\tmu\hb}$\vc

\Th{\num*}$\al\;\tmu\in\MXD\ha(\R)$\,\hh on a\quad
\fbox{\,$\Int_{\R}\Phist\hb(\ha\tmu)\,(x)\ d\, x=\!\int_{\R}\hs{-.5mm}\tmu$ }\ha\vcd

\Th{\num}$\all f\in\CB\ha(\R)$\ha\hh on a\,\hh$\big\|\hb\Phist\hb(f)\hb\big\|\ha\le\ha\|\ha f\ha\|$\pv\vvwa
de plus\hb\hh$\Phist\hb(f)\,\to f$\ha\hh uniform\e ment sur tout compact\, quand\hb\hh$s+t\to0$\vb

\Dem On a \,\hh$\all x\in\R\,\hh\big|\hb\Phist\hb(f)\hb(x)\ha\big|\hb\le\:\Inv{s+t}
\int_{\ha x\ha-\ha s}^{\hb x\hb+\hb t}\hs{-2mm}|\ha f\ha|\hb(u)\ d\,u\,\le\ha\|\ha f\ha\|$\pv\vvaab
d'autre part\ha\ soit\, $\ab$\, un intervalle compact de\, $\R$\, et soit\, $\eps>0$\hb\pv\vvww
comme\hb\ $f$\hb\ est uniform\e ment continue dans\, $\ab$\vg il existe\: $\alf>\ha0$\; tel que\vvwa
$\al\;u\vv v\in\ab\quad\Big[\ |\hb u-v\hb|\le\hb\alf\ \imp\;|\ha f\ha(u)-f\ha(v)\ha|\le\hb\eps\ \Big]$\ha\pv\ha
on a donc\,\hh$\all s\vv t\le\ha\alf$\vvw$\all x\in\ab\quad\big|\hb\Phist\hb\ha(f)\ha(x)-f\ha(x)\ha\big|
=\hb\Big|\Int_{\ha0}^{\,1}\Big[\,f\ha\big[\ha x+\hb(s+t)\:u-s\hb\big]-f\ha(x)\,\Big]\;d\,u\;\Big|$\vvw
$\,\le\Int_{\ha0}^{\,1}\,\Big|\hb f\ha\big[\ha x+\hb(s+t)\:u-s\hb\big]-f\ha(x)\hb\Big|\;d\,u\hb\le\hb\eps$\vc

\Th{\num}$\all\tdf\in\LL^1(\R)$\,\hh on a\:\hh$\Phist\hb(\tdf)\,\one\tdf$\:\hh quand\hb\hh$s+t\to0$\vb

\Dem On applique le\, \emph{LFAF}\, aux op\e rateurs lin\e aires\,\hh
$\Phist-\:\I\ :\ \LL^1(\R)\ha\to\ha\LL^1(\R)$\vbc

\Th{\num}$\all\tdf\in\LL^2(\R)$\,\hh on a\:\hh$\Phist\hb(\tdf)\,\two\tdf$\:\hh quand\hb\hh$s+t\to0$\vb

\Dem Analogue \ac\ la pr\e c\e dente\vbc

\Lem{\num}$\all f\in\EB\ha(\R)$\hh on a\hh$\big\|\hb\Phist\hb(f)\hb\big\|\ha\le\ha\|\ha f\ha\|$\pv
de plus\vg si\, $x\in\R$\, est un point\vvw de continuit\e\ de\, $f$\, on a\hh$\Phist\hb(f)\ha(x)\,\to
f\ha(x)$\ha\hh quand\hb\hh$s+t\to0$\vab

\Dem Il suffit de faire un dessin\hh!\vvc

\Th{\num}$\al\;\tmu\in\MXD\ha(\R)$\,\hh on a\:\hh$\Phist\hb(\ha\tmu)\,\fd\,\tmu$\:\hh
quand\ha\hh$s+t\to0$\vb

\Dem Comme\,\hh$\all s\vv t>0$\,\hh on a\,\hh$\bnone{\hb\Phist\hb(\ha\tmu)\hb}
\le\ha\nst{\ha\tmu\ha}$\ha\vg\ha il suffit de montrer que\vvab$\all h\in\EB\ha(\R)$\,\hh
$\Int_{\R}\Phist\hb(\ha\tmu)\,(x)\;h\hb(x)\ d\,x\,\to\int_{\R}h\hb(u)\,\tmu\,(u)$\,\hh
quand\hb\hh$s+t\to0$\ha\va On a\hh$\Int_{\R}\Phist\hb(\ha\tmu)\,(x)\:h\hb(x)\ d\,x\,
=\:\inv{s+t}\int_{\R}\Big[\int_{\R}\un\ha_{[\,x\hb-\hb s\vv\ha x\hb+\hb t\,]}\hb(u)\:
|\ha\tmu\ha|\hb(u)\ha\Big]\,h\hb(x)\ d\,x$\vvab
$=\ \Inv{s+t}\int_{\R}\Big[\int_{\R}\un\ha_{[\,u\hb-\hb t\vv\ha u\hb+\ha s\,]}\hb(x)\;
h\hb(x)\;d\,x\hb\Big]\,\tmu\hb(u)\,=\int_{\R}\Phist\hb(\ha h)\,(u)\;\tmu\hb(u)$\pv\vvab
or\ha\hh$\Phist\hb(\ha h)\,\bb\,h$\hb\vg
donc\hh$\Int_{\R}\Phist\hb(\ha\tmu)\,(x)\;h\hb(x)\ d\,x\,\to\int_{\R}h\hb(u)\;\tmu\,(u)$\vd

\newpage
\quad\textbf{\Num\underline{Convolution sur\, $\R$}}\vvc
\initnum

\Lem{\num}Soit\, $f\in\EO\ha(\R)$\pv posons\,\hh$F\,:\:\R^2\to\R\::\:(\ha r,\ha s\hw)\tto f\ha(r+s)$\pv\vvww alors\ha\ on a\,\hh$F\in\ha\PRB\ha(\hw\R^2)$\vab

\Dem Il suffit de faire un dessin\vbc

\Def{\num}La\, \underline{convol\e e}\, de\hh$\tmu\vv\tnu\ha\in\MX(\R)$\hh est d\e finie par\vvbc
\hs{35mm}\fbox{\,$\tmu*\tnu\ :\ \EO\ha(\R)\to\R\ :\ f\tto\Int\hs{-2mm}\int_{\,\R^2}\hs{-2.5mm}f\ha(r+s)\:\tmu\hb(r)\:\tnu\ha(s)$\,}\ha\vcd

\Th{\num}$\al\;\tmu\vv\tnu\ha\in\MX(\R)$\,\hh on a\,\hh$\tmu*\tnu\ha\in\MX(\R)$\hh et\hh$\nst{\hb\tmu*\tnu\hb}\le\nst{\ha\tmu\ha}\,\nst{\hw\tnu\hw}$\vb

\Dem On a\,\hh$\all h\in\EO\ha(\R)\,\hh\big|\ha(\tmu*\tnu)(h)\ha\big|\le\|\ha h\ha\|\Int\hs{-2mm}\int_{\R^2}|\ha\tmu\ha|\ha(r)\,|\ha\tnu\ha|\ha(s)
=\nst{\ha\tmu\ha}\,\nst{\hw\tnu\hw}\,\|h\|$\pv\vva donc\,\hh$\tmu*\tnu\ha\in\PM(\R)$\,\hh et\,\hh$\nst{\hb\tmu*\tnu\hb}\le\nst{\ha\tmu\ha}\,\nst{\hw\tnu\hw}$\vwa
Montrons que\,\hh$\tmu*\tnu\ha\in\MM(\R)$\pv soit\, $\alf\in\R$\pv posons\,\hh$\all\eps\ne0\quad g\hb_{\eps}=\hb\un\ha_{]\,\alf\hb,\,\alf\,+\hb\eps\hb[}$\pv\vvwa
fixons\hh$s\in\R$\, et posons\,\hh$h\hb_{\eps}\,:\ \R\to\R\ :\ r\tto g\hb_{\eps}\ha(r+s)$\pv on a\,\hh
$h\hb_{\eps}\hb=\hb\un\ha_{]\,\alf\hb-\ha s\hb,\,\alf\hb-\ha s\,+\hb\eps\hb[}$\pv\vvwa
donc\,\hh$h\hb_{\eps}\sm0$\,\hh quand\,\hh$\eps\to0$\pv de plus\,\hh$\all\eps\ne0\,\hh\|\ha h\hb_{\eps}\|=1$\pv donc\,\hh$\all s\in\R$\,\hh on a\vvwa
$k\hb_{\eps}\ha(s)\,=\Int_{\R}g\hb_{\eps}\ha(r+s)\:\tmu\hb(r)\,=\int_{\R}h\hb_{\eps}\ha(r)\:\tmu\hb(r)\to\ha0$\,\hh quand\,\hh$\eps\to0$\pv
de plus\,\hh$\all\eps\ne0$\vvwa on a\,\hh$\|\ha k\hb_{\eps}\ha\|\le\nst{\ha\tmu\ha}$\vg
donc\,\hh$\Int_{\R}k\hb_{\eps}\ha(s)\;\tnu\,(s)\, =\,(\hb\tmu*\tnu\ha)\,(g\hb_{\eps})\ha\to\ha0$\,\hh
quand\,\hh$\eps\to0$\pv\vva donc\,\hh$\tmu*\tnu\ha\in\MM(\R)$\vc

\Th{\num*}$\al\;\tmu\vv\tnu\ha\in\MX(\R)$\quad on a\quad$(\tmu*\tnu)\,(\un)
=\hb\tmu\ha(\un)\:\tnu\ha(\un)$\pv de plus\vvw si\quad$\tmu\vv\tnu\ha\in\MX(\R)^+$\vg
alors\quad$\tmu*\tnu\ha\in\MX(\R)^+$\,\hh et\quad$\nst{\hb\tmu*\tnu\hb}
=\nst{\ha\tmu\ha}\,\nst{\hw\tnu\hw}$\vc

\Th{\num}$(\MX(\R)\vv*\,)$\, est une alg\ee bre de Banach associative\ha, commutative et\vvw unitaire\pt
$\LL^1(\R)$\, est une sous-alg\ee bre de Banach de\, $\MX(\R)$\vab

\Dem Classique\vc

\Th{\num}$\al\;\tmu\vv\tnu\ha\in\MX(\R)\quad\all f\in\PRB\ha(\R)$\quad on a\vvb

\hs{45mm}\fbox{\,($\tmu*\tnu)\,(f)=\Int\hs{-2mm}\int_{\,\R^2}\hs{-2.5mm}f\ha(r+s)\:\tmu\hb(r)\:\tnu\ha(s)$\,}\hb\vc

\newpage
\Dem Soit\hh$f_n\in\EO(\R^2)$\hh avec\hh$f_n\bb f$\,;\hh on a donc\hh$(\tmu*\tnu)\,(f_n)\to(\tmu*\tnu)\,(f)$\pv\vva
posons\hh$F\ha(r,s)=f\ha(r+s)$\hh et\hh$\nn\hh F_n\ha(r,s)=f_n\ha(r+s)$\pv on a\vva$\nn\hh F_n\in\PRB\ha(\R^2)$\hh
et\hh$F_n\bb F$\vg donc\hh$F\in\WB\ha(\R^2)$\vva et\hh$(\tmu\ten\tnu)\,(F_n)\to(\tmu\ten\tnu)\,(F)$\pv
or on a\hh$\nn\,\hh(\tmu*\tnu)\,(f_n)=(\tmu\ten\tnu)\,(F_n)$\vg\vva
donc\hh$(\tmu*\tnu)\,(f)=(\tmu\ten\tnu)\,(F)$\,\hh quand\hh$n\to\pin$\vc

\Th{\num}$\all\tmu\vv\tnu\ha\in\MX(\R)\quad\all f\in\WB\ha(\R)$\quad on a\vvb\hs{40mm}
\fbox{\,$(\tmu*\tnu)\,(f)=\Int\hs{-2mm}\int_{\,\R^2}\hs{-2.5mm}f\ha(r+s)\:\tmu\hb(r)\:\tnu\ha(s)$\,}\ha\vc

\Dem Par linarit\e\, on peut supposer\hh$\tmu\vv\tnu\ha\in\MX(\R)^+$\vg donc\hh$\tmu*\tnu\ha\in\MX(\R)^+$\vvwa
et\hh$\tmu\ten\tnu\in\MX(\R^2)^+$.\; Soit\hh$\eps>0$\hh et soient\hh$g\vv h\in\PRB\ha(\R)$\hh
tels que\hh$g\le f\le h$\vvwa et\hh$(\tmu*\tnu)\,(h-g)\le\eps$\pt Posons\hh$G\hb(r,s)=\ha g\hb(r+s)$\hh
et\hh$H\ha(r,s)=\ha h\hb(r+s)$\va On a\quad$(\tmu*\tnu)\,(g)=\ha(\tmu\ten\tnu)\,(G)\le(\tmu\ten\tnu)\,(F)
\le(\tmu\ten\tnu)\,(H)=\ha(\tmu*\tnu)\,(h)$\vvwa et\quad$(\tmu*\tnu)\,(g)\le(\tmu*\tnu)\,(f)\le(\tmu*\tnu)\,(h)$\vg
donc\vva$|\hb(\tmu\ten\tnu)\,(f)-(\tmu*\tnu)\,(F)\hb|\,\le\,|\hb(\tmu*\tnu)\,(h)-(\tmu*\tnu)\,(g)\hb|=\ha(\tmu*\tnu)\,(h-g)\le\,\eps$\pv\vvwa
on en d\e duit\,\hh$(\tmu*\tnu)\,(f)=\ha(\tmu\ten\tnu)\,(F)$\vd

\quad\textbf{\Num\underline{Convolution sur\, $\R^+$}}\vvb
\initnum
(\ha D\e monstrations analogues \ac\ celles du paragraphe pr\e c\e dent\ha.\ha)\vvbc

\Lem{\num}Soit\, $f\in\EO\ha(\R^+)$\pv posons\,\hh$F\,:\:(\R^+)^2\to\R\::\:(\ha r,\ha s\hw)\tto f\ha(r+s)$\,;\vvw
alors on a\,\hh$F\in\hb\PRO\hb[\ha(\hw\R^+)^2\hb]$\vb

\Def{\num}La\, \underline{convol\e e}\, de\hh$\tmu\vv\tnu\ha\in\MM(\R^+)$\hh est d\e finie par\vvb
\hs{35mm}\fbox{\,$\tmu*\tnu\ :\ \EO\ha(\R^+)\to\R\ :\ f\tto\Int\hs{-2mm}\int_{\,(\R^+)^2}\hs{-6mm}f\ha(r+s)\:\tmu\hb(r)\:\tnu\ha(s)$\,}\ha\vc

\Th{\num}$\al\;\tmu\vv\tnu\ha\in\MM(\R^+)$\quad on a\,\quad$\tmu*\tnu\ha\in\MM(\R^+)$\vb

\Th{\num}\vvwa$\al\;\tmu\vv\tnu\ha\in\MX(\R^+)$\quad on a\ha\quad$\tmu*\tnu\ha\in\MX(\R^+)$\quad
et\quad\,$\nst{\hb\tmu*\tnu\hb}\le\nst{\ha\tmu\ha}\,\nst{\hw\tnu\hw}$\vbc

\Th{\num}$\al\;\tmu\vv\tnu\ha\in\MX(\R^+)$\quad on a\quad$(\tmu*\tnu)\,(\un)\ha
=\,\tmu\,(\un)\:\tnu\,(\un)$\pv de plus\vvwa si\quad$\tmu\vv\tnu\in\MX(\R^+)^+$\vg
alors\quad$\tmu*\tnu\in\MX(\R^+)^+$\,\hh et\quad$\nst{\hb\tmu*\tnu\hb}
=\ha\nst{\ha\tmu\ha}\,\nst{\hw\tnu\hw}$\vbc

\Th{\num}$(\MM(\R^+)\vv*\,)$\, est une alg\ee bre de Banach associative\, commutative et\vvw unitaire\pt
$\MX(\R^+)$\, et\, $\LL^1(\R^+)$\, sont des sous-alg\ee bres de Banach de\, $\MM(\R^+)$\vbc

\Th{\num}$\al\;\tmu\vv\tnu\ha\in\MM(\R^+)\quad\all f\in\WO\ha(\R^+)$\quad on a\vvb
\hs{45mm}\fbox{\,$(\tmu*\tnu)\,(f)=\Int\hs{-2mm}
\int_{\,(\R^+)^2}\hs{-6mm}f\ha(r+s)\,\tmu\hb(r)\;\tnu\ha(s)$\,}\,.\\[12.5mm]

\quad\textbf{\Num\underline{Transform\e e de Laplace}}\vvc
\initnum

\Def{\num}\vva On pose\ha\hh$\HZ=\ens{\ha z\in\hb\C\ ||\;R\ha e\hb(z)>0\hb}$\pv
soit\hh$\tmu\in\MX(\R^+)$\pv on d\e finit\hh $\all z\in\HZ$\vvab\hs{55mm}
\fbox{\,$(\LZ\,\tmu)\,(z)=\Int_{\R^+}\hs{-1mm}e^{\ha-\ha z\ha t}\:\tmu\hb(t)$\,}\vbc

$\LZ\,\tmu$\: est la\, \underline{transform\e e de Laplace}\, de\: $\tmu$\vbc

\Th{\num}$\LZ\,\tmu$\hh est une fonction analytique dans\, $\HZ$\,;\: de plus\hh
$\|\ha\LZ\,\tmu\ha\|\le\nst{\ha\tmu\hw}$\vab

\Dem On applique le \theo\ha\ d'holomorphie\vbc

\Lem{\num}\vvwa Soit\hh$h\in\CO\ha(\R^+)\hh\mrom{et}\hh\eps>0$\pv alors il existe un polyn\^ome\, $P\in\ha\R\hw[\hw X]$\: tel que\vvb
\begin{picture}(,)
\put(55,0){$\all t\ge0$\hh\hh$\big|\,h\ha(t)-P\ha(e^{\ha-\hb t})\,\big|\le\eps$\hh.}
\put(53,-3.5){\frm(59,9)}
\end{picture}\vvc

\Dem On d\e finit la fonction\hh$k\ :\ [\hb0\hb,\ha1\ha]\to\R$\,\hh de la mani\ee re suivante\, :\;
$k\ha(0)=\ha0$\\[1.25mm] et\ha\hh$k\ha(u)=h\hb($--\;$\ln\,u)$\,\hh si\,\hh$0<u\le1$\pv la fonction\,$k$\,
\e tant continue sur\, $[\hb0\hb,\ha1\ha]$\vg il existe un\\[1.5mm] polyn\^ome\hh$P\in\ha\R\hw[\hw X]$\,\hh
tel que\,\hh$\all u\in[\hb0\hb,\ha1\ha]\,\hh\big|\hb k\ha(u)-P\ha(u)\hb\big|\le\ha\eps\ha$\pv
en particulier on a\vvwa$\all u\in\;]\hb0\hb,\ha1\ha]\,\hh
\big|\hb h\hb($--\;$\ln\,u)-P\ha(u)\hb\big|\le\ha\eps\ha$\,;\; on en d\e duit\,\hh
$\all t\in\R^+$\,\hh$\big|\,h\ha(t)-P\ha(e^{\ha-\hb t})\,\big|\le\ha\eps$\vbc

\Th{\num}$\all\tmu\in\MX(\R^+)$\quad on a\quad$\big[\,\LZ\,\tmu=\ha0$\hh dans\,\ $\HZ\;\sam\;\tmu=\ha0\,\big]$\vb

\Dem Supposons\,\hh$\LZ\,\tmu=\ha0$\hh dans\,\ $\HZ$\pv alors on a\,\hh$\nn\hh
\Int_{\R^+\hs{-2mm}}e^{-n\hb t}\,\tmu\hb(t)=\ha0$\vg donc\vvw aussi\,\hh
$\all P\in\R\hw[\hw X]\hh\Int_{\R^+}\hs{-2mm}P\ha(e^{\ha-\hb t})\:\tmu\hb(t)=\ha0$\pv
en vertu du lemme pr\e c\e dent on en d\e duit\vvwa$\all h\in\CO(\R^+)\,\
\Int_{\R^+}\hs{-2mm}h\ha(t)\:\tmu\hb(t)=\ha0$\vg\hb\cad\,\hh$\tmu=\ha0$\vc

\Th{\num}$\al\;\tmu\vv\tnu\ha\in\MX(\R^+)$\quad on a\quad\fbox{\:$\LZ\,(\ha\tmu*\tnu)$\ho$
=(\LZ\,\tmu)\,(\LZ\,\tnu)$\hb}\vb

\Dem On a\,\hh$\all z\in\HZ$\vvab$\LZ\,(\ha\tmu*\tnu)\ha(z)\ha=\Int\hs{-2mm}
\int_{(\R^+)^2}\hs{-5mm}e^{-z\ha(\ha r+s\ha)}\,\tmu\hb(r)\:\tnu\,(s)\ha=\int_{\R} e^{-z\ha r}\hb\tmu\hb(r)
\int_{\R}e^{-z\ha s}\hb\tnu\,(s)\ha=\ha(\LZ\,\tmu)\hb(z)\ (\LZ\,\tnu)\hb(z)\hb.$\\\vvcd

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\initnum
\quad\textbf{\Num\underline{\Theo\ha\ de Titchmarsh}}\vvc

\num\ \underline{\Theo\ha\ de Titchmarsh dans\, $\MX(\R^+)$}\vvbc
\hs{27.5mm}\fbox{$\ \al\;\tmu\vv\tnu\ha\in\MX(\R^+)$\,\hh on a\ha\hh$\Big[\;\tmu*\tnu
=\ha0\;\sam\;(\hb\tmu=\ha0\hh\mathrm{ou}\hh\tnu=\ha0\ha)\,\Big]$\,}\ha\vc

\Dem Supposons\,\hh$\tmu*\tnu=\ha0$\pv on en d\e duit\,\hh$(\LZ\,\tmu)\,(\LZ\,\tnu)=\ha0$\pv $\LZ\,\tmu$\;
et\, $\LZ\,\tnu$\; sont des fonctions analytiques dans\, H\pv supposons\,\hh$\LZ\,\tmu\ne0$\pv alors\,
les z\e ros de\, $\LZ\,\tmu$\, sont isol\e s\pv donc les z\e ros de\, $\LZ\,\tnu$\; sont denses\vg
donc\; $\LZ\,\tnu=\ha0$\vg donc\; $\tnu=\ha0$\vc

\Def{\num}$\all a>0$\,\hh on pose\,\hh\fbox{\,$\Ya\hb=\hb\un\ha_{[\:0\ha\vv\hw a\ha]}\ph{\dot{t}}$\hs{-1mm}}\,
$\in\,\EO\ha(\R^+)$\pv\vvaab$\all\tmu\in\MM(\R^+)$\hh on pose\,\hh
\fbox{\:$\tmu\,_a\hb=\,\Ya\,.\,\tmu\ph{\big|}$\hs{-.75mm}}\, $\in\hb\MX(\R^+)$\vc

\Lem{\num}$\all\tmu\in\MX(\R^+)\quad\all a>0$\quad on a\vvb
\hs{40mm}$\Big[\,\Int_0^{\ha a}e\hb^{z\,(\hw a\ha-\ha t\hw)}\,\tmu\ha(t)$\, born\e\, dans\, H\;$\Big]\:
\imp\;\tmu\,_a=\ha0$\vab

\Dem\vvwa Posons\,\hh$\all z\in\,\C\quad G\,(z)=\Int_0^{\ha a}e\hb^{z\,(\hw a\ha-\ha t\hw)}\,
\tmu\hb(t)$\,\hh et supposons\, $G$\, born\e\ dans\, H\,;\: $G$\, est par\vvwa ailleurs born\e\
dans\,\hh$\C-\HZ$\pv or\, $G$\, est anlytique dans\hb\hh$\C$\vg donc\hh$G=\ha0$\hh dans\hb\hh$\C$\va
On en d\e duit\,\hh$\nn\,\hh G\,^{(\hw n\hw)}\ha(0)\ha=\Int_0^{\ha a}(a-t)\hw^n\,\tmu\hb(t)=\ha0$\vg
donc\,\hh$\all P\in\R\ha[X]$\\$\Int_0^{\ha a}P\hb(t)\:\tmu\hb(t)=\ha0$\vg
donc aussi\,\hh$\all h\in\CC\ha\big(\hb[\,0\,,\hb a\hb]\hb,\hb\R\hb\big)\hh
\Int_0^{\ha a}h(t)\,\tmu\ha(t)=\ha0$\vg\vva donc\hh$\tmu=\ha0$\hh sur\: $[\hb0\,,\hb a\hb]$\vbc

\newpage
\Th{\num}$\all\tmu\in\MX(\R^+)\quad\all a>0$\quad on a\vvab
\hs{40mm}$\Big[\ e\hb^{a\ha z}\,(\LZ\,\tmu)\hb(z)$\, born\e\, dans\, H\:$\Big]\ \sam\ \tmu\,_a=\ha0$\va

\Dem\vva
 a)\, $\pmi$\dd On a\quad$e\hb^{a\ha z}\,(\LZ\,\tmu)\hb(z)=\Int_a^{\ha\infty}
\hs{-2mm}e\hb^{-\ha z\,(\hw t\ha-\ha a\hw)}\,\tmu\hb(t)$\,\hh qui est bien born\e\ sur\, H\vab

b)\, $\imp$\dd On a\,\hh$\all z\in\HZ\,\hh\Int_0^{\ha a}e\hb^{z\,(\hw a\ha-\ha t\hw)}\,
\tmu\hb(t)=\hb e\hb^{a\ha z}\,(\LZ\,\tmu)\hb(z)\ha-\Int_a^{\ha\infty}\hs{-2mm}
e\hb^{-\ha z\,(\hw t\ha-\ha a\hw)}\,\tmu\hb(t)$\vvwa qui est born\e\ dans\, H\vg donc\,\hh$\tmu\,_a=\ha0$\vc

\Lem{\num}\vvwa Soit\hh\,$\tmu\in\MM(\R^+)$\,\hh tel que\,\hh$\tmu*\tmu\ha=\ha0$\,\hh
et soit\hh$a>0$\ha\pv alors\,\hh$\big(\ha\tmu_{\,a}*\tmu\,_a\hw\big)\ha_a=\ha0$\vb

\Dem On a\,\hh$\all\ha h\in\ha\EE\ha\big(\hb[\,0\,,\hb a\hb]\hb,\hb\R\hb\big)$\vvab
$(\ha\tmu\,_a*\tmu\,_a\ha)\,(h)\ha
=\Int_0^{\ha\infty}\hs{-1.75mm}\int_0^{\ha\infty}\hs{-1.5mm}h\hb(r+s)\:\tmu\,_a\ha(r)\:\tmu\,_a\ha(s)\ha
=\int_0^{\ha a}\hs{-1mm}\int_0^{\ha a}h\hb(r+s)\:\tmu\hb(r)\:\tmu\hb(s)\ha$\vvwa
$=\Int_0^{\ha \infty}\hs{-1.75mm}\int_0^{\ha \infty}\hs{-1.5mm}h\hb(r+s)\:
\tmu\hb(r)\:\tmu\hb(s)=(\ha\tmu*\tmu\ha)\,(h)=\ha0$\vcd

\Th{\num}$\all\tmu\in\MM(\R^+)$\,\hh on a\,\hh$\big[\;\tmu*\tmu\ha=\ha0\;\imp\:\tmu\ha=\ha0\;\big]$\vab

\Dem Soit\, $a>0$\pv alors\,\hh$(\tmu_{\,a}*\tmu_{\,a})\ha_a=\ha0$\vg
donc\,\hh$e\ha^{a\ha z}\,\LZ\hb(\ha\tmu_{\,a}*\tmu_{\,a}\ha)=\,e\ha^{a\ha z}\,[\hb\LZ\hb(\ha\tmu_{\,a})\ha]\ha^2$\vvwa
$=\hb[\hb e\ha^{a\ha z/2}\,\LZ\hb(\ha\tmu_{\,a})\hb]\ha^2$\,\hh est born\e\ dans\, H\vg
donc \e galement\hh$e\ha^{a\ha z/2}\,\LZ\hb(\ha\tmu_{\,a})$\pv
donc\,\hh$(\tmu\,_a)\ha_{a\hw/2}\ha=\ha0$\vg\vvw\cad\hh\,$\tmu\,_{a\hw/2}\ha=\ha0$\pv comme\, $a$\, est arbitraire\vg$\tmu\ha=\ha0$\vc

\Lem{\num}Soient\hh\,$\tmu\vv\tnu\ha\in\MM(\R^+)$\hh\,tels que\,\hh$\tmu*\tnu\ha=\ha0$\pv alors\hh\,$(t\,\tmu)*\tnu=\ha0$\vb

\Dem On a\,\hh$\all\ha h\in\ha\EO\ha\big(\hb\R^+\!,\hb\R\hb\big)$\vvab$0=(\ha\tmu*\tnu\ha)\,[\,t\ha.\,h\ha(t)\hb]
=\Int_0^{\ha\infty}\hs{-1.75mm}\int_0^{\ha\infty}\hs{-1.5mm}(r+s)\:h\hb(r+s)\:\tmu\hb(r)\:\tnu\hb(s)$\vvwa
$=\Int_0^{\ha\infty}\hs{-1.75mm}\int_0^{\ha\infty}\hs{-1.5mm}h\hb(r+s)\:r\:\tmu\hb(r)\:\tnu\hb(s)
+\int_0^{\ha\infty}\hs{-1.75mm}\int_0^{\ha\infty}\hs{-1.5mm}h\hb(r+s)\:\tmu\hb(r)\:s\:\tnu\hb(s)
=\big[\ha(t\,\tmu)*\tnu\ha+\ha\tmu*(t\,\tnu)\ha\big](h)$\,;\vvab posons\quad$\tmu\ha'=\,t\;\tmu$\,\hh
et\,\hh$\tnu\hb'=\hb t\;\tnu$\hb\pv on a\quad$\tmu\ha'*\tnu\ha+\hw\tmu*\tnu\hb'=\ha0$\vg donc\vvwa
$0=\ha(\tmu\ha'*\tnu)*(\ha\tmu\ha'*\tnu\ha+\hw\tmu*\tnu\hb'\ha)=(\tmu\ha'*\tnu)*(\tmu\ha'*\tnu)$\vg
donc\,\hh$\tmu\ha'*\tnu\ha=\ha0$\vc

\Th{\num}\vva Soient\,\hh$\tmu\vv\tnu\ha\in\MM(\R^+)$\,\hh tels que\,\hh$\tmu*\tnu\ha=\ha0$\pv
alors\,\hh$\al\;i\,,\ha j\in\N\,\hh\big(t\hb^i\,\tmu\big)*\big(t\hb^j\,\tnu\big)=\ha0$\vb

\Dem On fait une r\e currence sur\, $i$\, et\, $j$\ha.\vvc

\Lem{\num}\vvwa Soient\,\hh$\tmu\vv\tnu\ha\in\MM(\R^+)$\hh\,tels que\,\hh$\tmu*\tnu=\ha0$\pv soit\, $a>0$\,
et soit\, $\T_a$\, le triangle dont les\vvww sommets sont\hh$(0\ha,\ha0)\,,(a\ha,\ha0)\,,(0\ha,\ha a)$\pv
alors\hh$\all P\in\R\hb[\hb X\ha,\ha Y\ha]$\hh on a\hh$(\ha\tmu\ten\tnu\ha)\,(\ha\un\ha_{\T_a}.\hb P\ha)=\ha0$\vbc

\Dem$(\ha\tmu\ten\tnu\ha)\,(\ha\un\ha_{\T_a}.\hb P\ha)=\Int_0^{\ha\infty}\hs{-1.75mm}\int_0^{\ha\infty}\Ya\hb(r+s)\,P\hb(r,s)\
\tmu\hb(r)\:\tnu\,(s)$\vvwa$=\dis\sum\pos_{i,\ha j}\alf\hb_{i,\ha j}\int_0^{\ha\infty}\hs{-1.75mm}\int_0^{\ha\infty}\Ya\hb(r+s)\ r\ha^i\,s\ha^j\
\tmu\hb(r)\:\tnu\,(s)=\sum\pos_{i,\ha j}\alf\hb_{i,\ha j}\,\big(\ha t\hb^i\ha\tmu*t\ha^j\ha\tnu\ha\big)\hb\big(\Ya\big)=\ha0$\vcd

\num\ \underline{\Theo\ha\ de Titchmarsh dans\, $\MM(\R^+)$}\vvbc
\hs{27.5mm}\fbox{$\ \al\;\tmu\vv\tnu\ha\in\MM(\R^+)$\,\hh on a\,\hh$\Big[\;\tmu*\tnu
=\ha0\;\sam\;(\hb\tmu=\ha0\hh\mathrm{ou}\hh\tnu=\ha0\ha)\,\Big]$\,}\ha\vc

\Dem On suppose\,\hh$\tmu*\tnu\ha=\ha0$\pt Soit\, $g\in\CO\ha(\ha\R^+\hs{-.75mm}\tim\R^+,\,\R\ha)$\,
et choisissons\, $a>0$\, tel\vvww que\hh$g=\ha0$\hh en dehors du triangle\, $\T_a$\pv soit\, $\eps>0$\pv
d'apr\ee s le \theo\ha\ de Weierstrass\vvww pour un compact du plan\vg il existe un polyn\^ome\;
$P\in\R\hb[\hb X\ha,\ha Y\ha]$\hh tel que\hh$\|\,g-P\,\|\le\hb\eps$\vvw dans\: $\T_a$\pv
on peut d\ee s lors \e crire\,\hh$\big|\hb(\ha\tmu\ten\tnu\ha)\,(\ha g\ha)\ha\big|=\big|\hb(\ha\tmu\ten\tnu\ha)\,(\ha\un\ha_{\T_a}.\hb g\ha)\hb\big|
\le\big|\hb(\ha\tmu\ten\tnu\ha)\,(\ha\un\ha_{\T_a}.\hb P\ha)\hb\big|$\vvw
$+\ \big|\hb(\ha\tmu\ten\tnu\ha)\,[\hb\un\ha_{\T_a}.\hb(g-P)\ha]\hb\big|\ha\le\,\eps\,|\,\tmu\ten\tnu\hb|\,(\ha\un\ha_{\T_a}\ha)$\pv
comme\, $\eps$\, est arbitraire\vg on a\vvw
$(\ha\tmu\ten\tnu\ha)\,(\ha g\ha)=\ha0$\pv on en d\e duit\,\hh$\tmu\ten\tnu\ha=\ha0$\ha\vg donc\,\hh$\tmu=\ha0$\,\hh ou\,\hh$\tnu=\ha0$\vd

\quad\textbf{\Num\underline{Transform\e e r\e elle de Fourier}}\vvc
\initnum

\Def{\num}Soit\hh$\tmu\in\MX(\R)$\pv on pose\hh$\all x\in\R$\vvw
\begin{picture}(,)
\put(18,-7){$(\AZ\,\tmu)\,(x)=\Int_{\R}\cos\,(x\,t)\,\tmu\hb(t)\qquad\mathrm{et}\qquad(\BZ\,\tmu)\,(x)=\int_{\R}\,\sin\,(x\,t)\,\tmu\hb(t)$}
\put(16,-13){\frm(54,14)}
\put(85,-13){\frm(54,14)}
\end{picture}\vk17.5mm

$\AZ\,\tmu$\: et\, $\B\,\tmu$\: s'appellent la\, \underline{transform\e e cosinuso\"idale}\, et la\, \underline{transform\e e sinuso\"idale}\, de\hb\ $\tmu$\vab

\Not\vvw On note\hh$\CBU(\R)$\hh l'alg\ee bre des fonctions\hh$\R\to\R$\hh born\e es uniform\e ment continues\vbc

\Th{\num*}$\big(\hb\CBU(\R)\vv\|\ \|\hb\big)$\hh est une alg\ee bre de Riesz\hb-Banach\vbc

\Th{\num}$\all\tmu\in\MX(\R)$\quad on a\vva
\hs{30mm}1)\hh$\AZ\,\tmu\in\CBU(\R)$\,\hh et\,\hh$\BZ\,\tmu\in\CBU(\R)$\vva
\hs{30mm}2\ha)\hh$\|\AZ\,\tmu\|\le\nst{\tmu}$\,\hh et\,\hh$\|\BZ\,\tmu\|\le\nst{\tmu}$\vva
\hs{30mm}3\ha)\hh$\AZ\,\tmu$\hh est une fonction paire\hh et\hh$\BZ\,\tmu$\hh est une fonction impaire\vvwa
\hs{30mm}4\ha)\hh\big(\,$\tmu$\; impaire\, $\imp\ \AZ\,\tmu\ha=\ha0$\,\big)\quad et\:\hh\big(\,$\tmu$\; paire\, $\imp\ \BZ\,\tmu\ha=\ha0$\,\big)\vb

\Dem Le reste \e tant trivial, montrons que\, $\AZ\,\tmu$\, et\, $\BZ\,\tmu$\, sont uniform\e ment continues\va
Soit\hh$\eps>0$\,\hh et soit\hh$N>0$\,\hh tel que\hh$\Int_{|\ha t\ha|\hb\ge\hb N}\!|\hb\tmu\hb(t)|\le\ha\eps$\ha\pv\ha
on a\,\hh$\all u\vv v\in\R$ \\ $\big|\hb(\AZ\,\tmu)\hb(u)-(\AZ\,\tmu)\hb(v)\hb\big|
=\Big|\Int_{-N}^{\hb N}\big[\ha\cos\,(u\,t)-\cos\,(v\,t)\ha\big]\,\tmu\hb(t)
+\int_{|\ha t\ha|\hb\ge\hb N}\!\big[\ha\cos\,(u\,t)-\cos\,(v\,t)\ha\big]\,\tmu\hb(t)
\,\Big|$ \\ $\le\,|u-v|\underbrace{\Int_{-N}^{\hb N}|\hb t\ha|\,|\hb\tmu\hb(t)|}_{K}\,+\;2\;\eps$\pv\vva
on en d\e duit\,\hh$\Big[\;|\ha u-v\ha|\le\ha\eps\ha/K\,\imp\,\big|\hb(\AZ\,\tmu)\hb(u)
-(\AZ\,\tmu)\hb(v)\hb\big|\le\hb3\;\eps\;\Big]$\vc

\Not On note\: $\ROD$\: l'alg\ee bre des fonctions\hh$\R\to\R$\hh\ac\ support born\e\ \ha et \ac\ d\e riv\e e seconde r\e gl\e e\vc

\Th{\num}Soit\hh$f\in\ROD$\pv alors on a\quad$\AZ f\in\LL^1(\R)\cap\LL^2(\R)$\vg\vvwa
$\BZ\ha f\in\LL^1(\R)\cap\LL^2(\R)$\:\hh et\:\hh\fbox{\,$(\ha\AZ^2+\BZ^{\,2}\ha)\hb f=\,2\,\pi\,f$\,}\ha\vb

\Dem On a\,\hh$\all x\in\R\,\hh(\AZ f)\hb(x)=\Int_{\R}\cos\,(x\,t)\,f(t)\;d\hb t$\pv
donc\,\hh$\all x\ne0$\vvwa$(\AZ f)\hb(x)=-\hb\Inv{x}\int_{\R}\sin\,(x\,t)\,f\ha'(t)\;d\hb t
=\hb\inv{x\ha^2}\int_{\R}\cos\,(x\,t)\,f\ha''(t)\;d\hb t$\pv on en d\e duit\vva
si\,\hh$|x|\le1\quad|\ha(\AZ f)\hb(x)\ha|\le\none{\ha f\ha}$\quad et si\,\hh
$|\ha x\ha|\ge1\quad|\ha(\AZ f)\hb(x)\ha|\le\Inv{x\ha^2}\;\none{\ha f\ha''\ha}$\pv\vvab
donc\,\hh$\AZ f\in\LL^1(\R)\cap\LL^2(\R)$\pv idem pour\, $\BZ\ha f$;\:
$\AZ^2+\BZ^{\,2}$\; est donc bien d\e fini sur\, $\RR^2_O(\R)$\vab

On a\,\hh$\all x\in\R\quad\big[\,(\ha\AZ^2+\BZ^{\,2}\ha)\hb f\,\big]\hb(x)$\vva
$=\Int_{\R}\cos\,(x\,u)\,\Big[\int_{\R}\cos\,(u\,t)\,f(t)\;d\hb t\,\Big]\,d\hb u
\ +\hb\int_{\R}\sin\,(x\,u)\,\Big[\int_{\R}\sin\,(u\,t)\,f(t)\;d\hb t\,\Big]\,d\hb u$\vva
$=\Int_{\R}\Big[\int_{\R}\big[\cos\,(x\,u)\cos\,(u\,t)\hb+\;\sin\,(x\,u)\sin\,(u\,t)\,\big]\,f(t)\;d\hb t\hb\Big]\hb d\hb u$\va
$=\Int_{\R}\Big[\int_{\R}\cos\big[\hb u\hb(t-x)\ha\big]\,f(t)\;d\hb t\hb\Big]\hb
d\hb u=2\int_0^{\ha\infty}\Big[\int_{\R}\cos\big[\hb u\hb(t-x)\ha\big]\,f(t)\;d\hb t\hb\Big]\hb d\hb u$\vva

Fixons\hh$x\in\R$\hh et posons\,\hh$\all u\ge0\quad\Phi\,(u)\hb=\Int_{\R}\cos\big[\hb u\hb(t-x)\ha\big]\,f(t)\;d\hb t$\pv\vvwa
soit\hh$M>0$\hh tel que\, $\Big[\:|\ha t\ha|\ge M\imp f(t)=\ha0\:\Big]$\ha;\vvwa
on a\quad$\Phi\,(u)\hb=\Int_{-M}^{\ha M}\cos\big[\hb u\hb(t-x)\ha\big]\,f(t)\;d\hb t$\ha\vg\hb
donc\:\hh$\all u>0$\vva$\Phi\,(u)\hb=\dis-\,\inv{u}\int_{-M}^{\ha M}
\sin\big[\hb u\hb(t-x)\ha\big]\,f\ha'(t)\;d\hb t\,=\,\frac{2}{u\ha^2}\int_{-M}^{\ha M}\sin^{\ha2}
\big[\hb u\hb(t-x)/\hw2\hb\big]\,f\ha''(t)\;d\hb t$\pv\vva

posons\,\hh$\all u>0\,\hh\all t\in[\ha-\ha M\hw,\ha M\hb]\quad H\ha(u\,,\ha t\hw)
=\,\Frac{1+\,u\hb^{3\ha/\hw2}}{u\ha^2}\ \sin^{\ha2}\big[\hb u\hb(t-x)/\hw2\ha\big]$\pv\vva

$\all u\in\ ]\,0\hb,\hw1\hb]\,\hh\all t\in[\ha-\ha M\hw,\ha M\hb]$\,\hh
on a\,\hh$0\le H\ha(u\,,\ha t\hw)\le\ha\Inv{2}\ (t-x)\ha^2$\,\hh
car\,\hh$\all a\in\R\,\hh|\sin a\ha|\le|\hb a\ha|$\vg\vvwa
donc\,\hh$0\le H\ha(u\,,\ha t\hw)\le\ha\Inv{2}\ (M+\ha|x|\ha)\ha^2$\ha\pv
de m\^eme\,\hh$\all u>1\,\hh\all t\in[\ha-\ha M\hw,\ha M\hb]$\,\hh on a\vvwa$0\le H\ha(u\,,\ha t\hw)\le2$\ha\pv
donc\, $H$\, est born\e\, dans\,\hh$\R_{\;*}^{\,+}\tim[\ha-\ha M\hw,\ha M\hb]$\va

On en d\e duit\vva$\big[\,(\ha\AZ^2+\BZ^{\,2}\ha)\hb f\,\big]\hb(x)\hb
=\,2\Int_0^{\ha\infty}\hs{-1.5mm}\Phi\,(u)\;d\,u\,
=\,4\int_0^{\ha\infty}\Big[\int_{-M}^{\ha M}H\ha(u\,,\ha t\hw)\,f\ha''(t)\;d\hb t
\hb\Big]\,\frac{\hb d\hb u}{1+\,u\hb^{3\ha/\hw2}}$\vvwa
$=\,4\Int_{-M}^{\ha M}\Big[\int_0^{\ha\infty}\hs{-1.5mm}H\ha(u\,,\ha t\hw)\
\frac{\hb d\hb u}{1+\,u\hb^{3\ha/\hw2}}\hb\Big]\,f\ha''(t)\;d\hb t
=\,4\Int_{-M}^{\ha M}\Big[\int_0^{\ha\infty}\:\frac{\sin^{\ha2}
\big[\hb u\hb(t-x)/\hw2\hb\big]}{u\ha^2}\hb\Big]\,f\ha''(t)\;d\hb t$\pv\vva

or un r\e sultat classique nous dit que\,\hh$\all a\in\R$\,\hh
$\Int_0^{\ha\infty}\:\frac{\sin^{\ha2}\ha(\ha a\,u)}{u\ha^2}\hb=\hb\frac{\pi}{2}\;|\ha a\ha|$\pv\vvwa on a donc\quad
$\Inv{\pi}\:\big[\,(\ha\AZ^2+\BZ^{\,2}\ha)\hb f\,\big]\hb(x)=\int_{-M}^{\ha M}|\hb t-x\hb|\;f\ha''(t)\;d\hb t
=\int_{-\ha\infty}^{\ha+\ha\infty}\!|\hb t-x\hb|\;f\ha''(t)\;d\hb t$\vvwa
$=\Int_{-\ha\infty}^{\ha x}\!(x-t\ha)\;f\ha''(t)\;d\hb t\;+\int_x^{\ha+\ha\infty}\!(\ha t-x)\;f\ha''(t)\;d\hb t$
$=\Int_{-\ha\infty}^{\ha x}\hs{-1.5mm}f\ha'(t)\;d\hb t\,-\int_x^{\ha+\ha\infty}\hs{-2mm}f\ha'(t)\;d\hb t=\ha2\:f\ha(x)$\vcd

\Th{\num}$\all f\in\ROD$\,\hh on a\quad
\fbox{\,$\ntwo{\hb\AZ f\ha}^2\,+\,\ntwo{\hb\BZ\ha f\ha}^2\,=\,2\:\pi\,\ntwo{\ha f\ha}^2$\,}\ha\vb

\Dem$\ntwo{\ha f\ha}^2=\Int_{\R}f\ha^2\hw(x)\:d\,x=\int_{-M}^{\ha M}f\ha^2\hw(x)\:d\,x=\inv{2\,\pi}\int_{-M}^{\ha M}
\big[\,(\ha\AZ^2+\BZ^{\,2}\ha)\hb f\hb\big]\hw(x)\,f(x)\,d\hb x$\vva
$=\Inv{\pi}\int_{-M}^{\ha M}\Big[\int_0^{\ha\infty}\Big\{\int_{-M}^{\ha M}
\cos\big[\hb u\hb(t-x)\ha\big]\,f(t)\;d\hb t\hb\Big\}\ d\hb u\:\Big]\,f(x)\,d\hb x$\vva
$=\Frac{2}{\pi}\int_{-M}^{\ha M}\Big[\int_0^{\ha\infty}\Big\{\int_{-M}^{\ha M}
\sin^{\ha2}\big[\hb u\hb(t-x)/\hw2\hb\big]\,f\ha''(t)\;d\hb t\hb\Big\}\ \frac{d\hb u}{u\ha^2}\:\Big]\,f(x)\,d\hb x$\vvbc

Posons\quad$\all u>0\quad\all x\in[\ha-\ha M\hw,\ha M\hb]$\vvab
$K\hb(\hw u\hb,\ha x\hw)\hb=\,\Frac{1+\,u\hb^{3\ha/\hw2}}{u\ha^2}\int_{-M}^{\ha M}
\sin^{\ha2}\big[\hb u\hb(t-x)/\hw2\hb\big]\,f\ha''(t)\;d\hb t$\pv\vvw
$\all u\in\ ]\,0\hb,\hw1\hb]\,\hh\all x\in[\ha-\ha M\hw,\ha M\hb]$\,\hh on a\,\hh
$|\ha K\ha(u\,,\ha x\hw)\ha|\ha\le\,\Inv{2}\,\int_{-M}^{\ha M}(\ha t-x)\ha^2\,
|\hb f\ha''(t)\ha|\;d\hb t$\vvwa$\le\hb2\;M\ha^2\:\none{\ha f\ha''\ha}$\pv
de m\^eme\,\hh$\all u>1\,\hh\all x\in[\ha-\ha M\hw,\ha M\hb]$\,\hh on a\,\hh
$|\ha K\ha(u\,,\ha x\hw)\ha|\le\hb2\;\none{\ha f\ha''\ha}$\ha\pv\vvwa
donc\, $K$\, est born\e\, dans\,\hh$\R_{\;*}^{\,+}\tim[\ha-\ha M\hw,\ha M\hb]$\vb

On en d\e duit\quad$\ntwo{f}^2\,=\,\Frac{2}{\pi}\int_{-M}^{\ha M}
\Big[\int_0^{\ha\infty}K\hb(\hw u\hb,\ha x\hw)\ \frac{d\hb u}{1+\,u\hb^{3\ha/\hw2}}\:\Big]\,f(x)\,d\hb x$\vva
$=\,\Frac{2}{\pi}\int_0^{\ha\infty}\Big[\int_{-M}^{\ha M}K\hb(\hw u\hb,\ha x\hw)\,f(x)\,d\hb x\,\Big]\,\frac{d\hb u}{1+\,u\hb^{3\ha/\hw2}}$\vva
$=\,\Frac{2}{\pi}\int_0^{\ha\infty}\Big[\int_{-M}^{\ha M}\Big\{\int_{-M}^{\ha M}
\sin^{\ha2}\big[\hb u\hb(t-x)/\hw2\hb\big]\,f\ha''(t)\;d\hb t\,\Big\}\,f(x)\:d\hb x\,\Big]\,\frac{d\hb u}{u\ha^2}$\vva
$=\,\Inv{\pi}\int_0^{\ha\infty}\Big[\int_{-M}^{\ha M}\Big\{\int_{-M}^{\ha M}
\cos\big[\hb u\hb(t-x)\ha\big]\,f(t)\;d\hb t\,\Big\}\,f(x)\:d\hb x\,\Big]\,d\hb u$\vva
$=\,\Inv{\pi}\int_0^{\ha\infty}\Big[\,\int_{-M}^{\ha M}\cos\,(u\,t)\,f(t)\,d\hb t\int_{-M}^{\ha M}\cos\,(u\,x)\,f(x)\,d\hb x$\vvwa
\hs{60mm}$+\Int_{-M}^{\ha M}\sin\,(u\,t)\,f(t)\,d\hb t\int_{-M}^{\ha M}\sin\,(u\,x)\,f(x)\,d\hb x\,\Big]\,d\hb u$\vva
$=\,\Inv{\pi}\int_0^{\ha\infty}\Big[\hb(\ha\AZ f\ha)\ha^2\ha(u)+(\ha\BZ\ha f\ha)\ha^2\ha(u)\hb\Big]\,d\hb u
=\,\inv{2\;\pi}\:\Big(\hb\ntwo{\hb\AZ f\ha}^2\:+\ha\ntwo{\hb\BZ\ha f\ha}^2\,\Big)$\ha.\vvc

\Th{\num}\vvw Soit\hh$\tdf\in\LL^2(\R)$\pv soit\hh$f_n\in\ROD$\hh avec\hh$f_n\two\tdf$\pv alors\hh
$\AZ\hb f_n$\hh et\hh $\BZ\,f_n$\hh convergent\vvww dans\hh$\LL^2(\R)$\hh vers une limite ind\e pendante de la suite particuli\ee re\, $f_n$\vab

\Dem On applique le \theo\ pr\e c\e dent\vbc

\emph{\underline{Notation\ha\ provisoire}}\dd$\all\tdf\in\LL^2(\R)$\quad on pose\vvb
\hs{15mm}\fbox{\,$\AZ'\tdf=\ha\limtwo{\hh n}\ (\AZ\hb f_n)$\,} $\in\LL^2(\R)$\hhh
et\hhh\fbox{\,$\BZ\,'\tdf=\ha\limtwo{\hh n}\ (\BZ\hb f_n)$\,} $\in\LL^2(\R)$\vcd

\Def{\num}$\all h\in\,\CBU(\R)\cap\LL^2(\R)$\quad
on pose\quad$\|\ha h\ha\|_{\hb u\ha,\ha2}=\|\ha h\ha\|+\ntwo{\ha h\ha}$\vbc

\Th{\num}\big(\,$\CBU(\R)\cap\LL^2(\R)\vv\|\ \|_{\hb u\ha,\ha2}\,\big)$\hh est un espace de Riesz\ha-Banach\vb

\Dem Il suffit de montrer que\: $\CBU(\R)\cap\LL^2(\R)$\: est complet pour la norme\, $\|\ \|_{\hb u\ha,\hw2}$\pt\vvwa
Soit\, $f_n\in\CBU(\R)\cap\LL^2(\R)$\, une suite de Cauchy pour\, $\|\ \|_{\hb u\ha,\ha2}$\pv comme\hh $\|\ \|\le\|\ \|_{\hb u\ha,\ha2}$\hh
et\vvwa $\ntwo{\ }\le\|\ \|_{\hb u\ha,\ha2}$\vg $f_n$\, est aussi une suite de Cauchy pour\, $\|\ \|$\,
et\, $\ntwo{\ }$\pv donc\,\hh$\tdf_n\uu g\in\CBU(\R)$\vvwa
et\,\hh$f_n\two\tdh\in\LL^2(\R)$\pt Soit\, $k\in\EO\hw(\R)$\pv on a\,\hh$\nn\,\hh|\hb f_n\hw(k)-g\ha(k)\ha|\le\|\ha f_n-\tdg\ha\|\,\none{\ha k\ha}$\vvwa
et\,\hh$\big|\hb f_n\hw(k)-\tdh\ha(k)\ha\big|\le\bntwo{\ha f_n-\tdh\ha}\:\ntwo{\hb k\ha}$\pv
donc\,\hh$f_n\hw(k)\to g\ha(k)$\,\hh et\,\hh$f_n\hw(k)\to\tdh\ha(k)$\pv\vvwa
on en d\e duit\,\hh$g=\ha\tdh$\,\hh et\,\hh$\|\ha f_n-g\ha\|_{\hb u\ha,\ha2}
=\hb\|\ha f_n-g\ha\|+\ntwo{\ha f_n-g\ha}\,\to\,0$\ha\vg donc\,\hh$\tdf_n\,\stack{u\ha,\ha2}{\to}\,g$\vbc

\Th{\num}$\all\tdf\in\LL^1(\R)\cap\LL^2(\R)$\quad on a\quad$\AZ'\tdf=\AZ\tdf\in\,\CBU(\R)\cap\LL^2(\R)$\vvw
et\quad$\BZ\,'\tdf=\BZ\ha\tdf\in\,\CBU(\R)\cap\LL^2(\R)$\vab

\Dem\vvwa Soit\, $\eps>0$\pv soit\, $N\in\N^*$\, tel que\hh$\Int_{|\ha t\ha|\,\ge\,N}\hs{-2.5mm}\big|\ha\tdf\ha(t)\ha\big|\,d\hb t\hb\le\hb\frac{\eps}{2}$\quad
et\hh$\Int_{|\ha t\ha|\,\ge\,N}\hs{-2.5mm}\tdf\ha(t)^{\hb2}\,d\hb t\hb\le\hb\frac{\eps}{2}$\ha\pv\vvw
soit\, $g\in\RR^{(\hw2\hw)}_O(\R)$\, nulle en dehors de\, $[-N,N\ha]$\, telle que\hh
$\Int_{-N}^{\ha N}|\ha\tdf-g\hb|^{\hb2}\:d\hb t\hb\le\hb\frac{\eps^{\hb2}}{8\,N}$\pv\vvw
on a donc\,\hh$\Int_{-N}^{\ha N}\big|\ha\tdf-g\hb\big|\:d\hb t\le\hb\sqrt{2\,N}\,\frac{\eps}{\sqrt{8\,N}}\,=\,\frac{\eps}{2}$\pv\vvwwa on en d\e duit\,\hh
$\bnone{\ha\tdf-g\hb}\hb=\Int_{-N}^{\ha N}\big|\ha\tdf-g\hb\big|\:d\hb t\:
+\int_{|\ha t\ha|\,\ge\,N}\hs{-2.5mm}\big|\ha\tdf\ha(t)\ha\big|\,d\hb t\hb\le\hb\eps$\vvwwa
et\,\hh$\bntwo{\ha\tdf-g\hb}^{\ha2}\,=\Int_{-N}^{\ha N}\big|\ha\tdf-g\hb\big|^{\ha2}\:d\hb t\:
+\int_{|\ha t\ha|\,\ge\,N}\hs{-2.5mm}\tdf\ha(t)^{\hb2}\,d\hb t\hb
\le\hb\frac{\eps^{\hb2}}{8\,N}\hb+\hb\frac{\eps^{\hb2}}{2}\hb\le\hb\eps\ha^2$\pv\vva
donc\,\hh$\ntwo{\ha\tdf-g\hb}\ha\le\ha\eps$\vab On peut donc construire une suite\hh$g\ha_n\in\RR^{(\hw2\hw)}_O(\R)$\hh
telle que\,\hh$g\ha_n\one\tdf$\,\hh et\,\hh$g\ha_n\two\tdf$\pv\vva
on a donc\,\hh $\AZ\,g\ha_n\uu\AZ\tdf$\,\hh et\,\hh$\AZ\,g\ha_n\two\AZ'\tdf$\pv donc\,\hh$\AZ\tdf=\AZ'\tdf$.\vvbc

\emph{\underline{Notation\ha\ d\e finitive}}\dd$\all\tdf\in\LL^2(\R)$\,\hh on peut donc poser\quad
$\AZ\tdf=\ha\AZ'\tdf$\quad et\quad$\BZ\ha\tdf=\ha\BZ\,'\tdf$.\vvbc

\Th{\num*}$\all\tdf\in\LL^2(\R)$\,\hh on a\quad
\fbox{\,$\bntwo{\hb\AZ\tdf\hb}^{\ha2}\,+\,\bntwo{\hb\BZ\ha\tdf\hb}^{\ha2}\,=\,2\,\pi\,\bntwo{\ha\tdf\hb}^{\ha2}$\,}\ha\vbc

\Cor{\num}$\ntwo{\ha\AZ\ha}=\ntwo{\ha\BZ\ha}=\sqrt{2\,\pi}$\vb

\Dem Si\, $\tdf\in\LL^2(\R)$\, est paire on a\hh$\BZ\ha\tdf=\ha0$\vg donc\,\hh$\bntwo{\hb\AZ\tdf\hb}^{\ha2}\,=\,2\,\pi\,\bntwo{\ha\tdf\hb}^{\ha2}$\vbc

\Def{\num}On pose\,\hh\fbox{\,$\T\ph{^t}\hs{-1.25mm}=\ha\AZ+\BZ$\,}\pv c'est un endomorphisme de\, $\LL^2(\R)$\pv\vva$\all\tdf\in\LL^2(\R)$\:
on appelle\; $\T\tdf$\, la\, \underline{transform\e e r\e elle de Fourier}\, de\hb\ $\tdf$\hw.\vvbc

\Th{\num*}1)\hh$\all\tdf\in\LL^2(\R)$\:\hh\fbox{\,$\bntwo{\hb\T\tdf\hb}=\sqrt{2\,\pi}\ \bntwo{\ha\tdf\hb}$\hb}\vvb
\hs{38mm}2\ha)\quad$\AZ\hb\BZ=\BZ\,\AZ=\ha0$\\
\begin{picture}(,)
\put(44.5,5.75){\frm(30,5.5)}
\end{picture}\vk-2.5mm
\hs{38mm}3\ha)\hh\fbox{\,$\T^{\ha2}=\hb\AZ^2+\BZ^{\ha2}=\hb2\,\pi\,\II$\,}\ha\vbc

\Th{\num*}$\AZ\vv\BZ\vv\T$\hh sont des endomorphismes sym\e triques de\, $\LL^2(\R)$\vbc

\Def{\num}\fbox{\,$\T\ph{^t}\hs{-1.25mm}=\ha\AZ+\BZ$\,}\,\hh est aussi un morphisme lin\e aire\hh$\MX(\R)\to\CBU(\R)$\pv\vva
$\T\ha\tmu$\hh est la\, \underline{transform\e e r\e elle de Fourier}\, de\, $\tmu$\vbc

On a donc\,\hh$\al\hb\tmu\in\ha\MX(\R)$\,\hh
\fbox{\,$\big(\hw\T\ha\tmu\hw\big)\ha(x)=\hs{-.5mm}\Int_{\R}\big[\cos\,(x\,t)+\ha\sin\,(x\,t)\hb\big]\,\tmu\hb(t)$\,}\ha\vc

\newpage
\Th{\num}Soient\hh$\tmu\vv\tnu\in\MX(\R)$\pv alors on a\vvab
\hs{15mm}\fbox{\,$\tmu\,\hpo\AZ\:\tnu\hw\hpu=\hb\tnu\,\hpo\AZ\:\tmu\hw\hpu$\ha}\qquad
\fbox{\,$\tmu\,\hpo\hw\BZ\:\tnu\hw\hpu=\hb\tnu\,\hpo\hw\BZ\:\tmu\hw\hpu$\ha}\qquad
\fbox{\,$\tmu\,\hpo\hw\T\hb\tnu\hw\hpu=\hb\tnu\,\hpo\hw\T\hb\tmu\hw\hpu$\ha}\ha\vc

\Dem Comme\ha\hh$\AZ\,\tmu\vv\BZ\,\tmu\ \ldots\,\in\ha\CBU(\R)$\vg les expressions ci-dessus ont un sens\pv
les \e galit\e s r\e sultent d'une interversion des signes int\e graux\vbc

\Th{\num}Soient\hh$\tmu\in\MX(\R)$\hh et\hh$f\in\ROD$\pv alors on a\vvab
\quad\fbox{\:$\big<\ha\AZ\,\tmu\,,\ha\AZ f\ha\big>\,+\,\big<\ha\BZ\,\tmu\,,\ha\BZ\ha f\ha\big>\ha
=\ha2\:\pi\:\tmu\,(f)$\,}\hhh et\hhh\fbox{\:$\big<\ha\AZ\,\tmu\,,\ha\BZ\ha f\ha\big>\ha
=\ha\big<\ha\BZ\,\tmu\,,\ha\AZ\ha f\ha\big>\ha=\hb0$\:}\ha\pt\vvbc

\Dem\vva
\quad1)\hh$2\:\pi\:\tmu\,(f)=\hb\tmu\,\big[\ha(\ha\AZ^2+\ha\BZ^{\,2}\ha)\hb f\hb\big]
=\hb\tmu\,\big[\hb\AZ\,(\hw\AZ f)\hb\big]+\tmu\,\big[\hb\BZ\,(\hw\BZ\ha f)\hb\big]$\vvwa
\hs{63mm}$=\hb\big<\ha\AZ\,\tmu\,,\ha\AZ f\ha\big>\,+\,\big<\ha\BZ\,\tmu\,,\ha\BZ\ha f\ha\big>$\vb

\quad2\ha)\hh$\AZ\,\tmu$\, est paire\, et\, $\BZ\ha f$\, est impaire\vg donc\,\hh$\big<\ha\AZ\,\tmu\,,\ha\BZ\ha f\ha\big>\ha=\hb0$\vc

\Cor{\num*}$\all\tmu\in\MX(\R)$\hh on a\quad\big[\:$\big(\ha\AZ\,\tmu\ha=\ha0\ha\hh\mrom{et}\hh\BZ\,\tmu\ha=\ha0\hb\big)\:\sam\ \tmu=\ha0\;\big]$\vc

\Th{\num*}$\all\tmu\in\MX(\R)\,\hh\all f\in\ROD$\quad on a\quad
\fbox{\,$\big<\ha\T\,\tmu\,,\T f\ha\big>$\ha\ho$=\ha2\:\pi\:\tmu\,(f)$\hb}\ha\vc

\Cor{\num*}$\all\tmu\in\MX(\R)$\hh on a\,\hh$\big[\:\T\,\tmu=\ha0\ \sam\ \tmu=\ha0\;\big]$\vd

\quad\textbf{\Num\underline{Transform\e e complexe de Fourier}}\vvc
\initnum
\oui On est amen\e\ \ac\ utiliser des espaces de mesures et de fonctions \ac\ valeurs complexes\pv
nous les noterons\hh$\wh{\MX}(\R)$\vg etc\ldots\vvc

\Def{\num}On pose\,\hh\fbox{\,$\ZZ\ph{^t}\hs{-.75mm}=\ha\AZ+\hb i\,\BZ$\,}\,\hh
et\,\hh\fbox{\,$\ZZ\hb^\#=\,\AZ-\hb i\,\BZ$\,}\ha\pv
on a donc\vvab$\all\tmu\in\wh{\MX}(\R)\,\hh\all x\in\R\quad
\ZZ\,(\tmu)\,(x)=\Int_{\R}e\,^{i\,x\,t}\,\tmu\,(t)$\quad et\quad
$\ZZ\hb^\#(\tmu)\,(x)=\Int_{\R}e\,^{-i\,x\,t}\,\tmu\,(t)$\pv\vva

$\ZZ$\, et\, $\ZZ\hb^\#$\, sont naturellement aussi d\e finis en tant qu'op\e rateurs\hh
$\wh{\LL^2}(\R)\to\wh{\LL^2}(\R)$\vab

$\ZZ$\, et\, $\ZZ\hb^\#$\, sont les\, \underline{transform\e es\ha\ complexes\ha\
(\ha directe et r\e ciproque\ha) de Fourier}\vcd

\Th{\num*}Dans\hh$\wh{\LL^2}(\R)$\hh on a\,\hh\fbox{\:$\ZZ\ \ZZ\hb^\#
=\,\ZZ\hb^\#\hb\ZZ\,=\lo2\,\pi\,\I\ph{\dot{l}}$\hs{-.75mm}}\ha\vcd

\Lem{\num}$\all z\in\,\C$\,\hh on a\,\hh\fbox{\:$\Int_{\R}e\,\ha^{z\:t\,-\;t\ha^2/\hw 2}\;d\hb t\hb
=\hb\sqrt{2\,\pi}\ha\ e\:^{z\ha^2\hs{-.25mm}/\hw2}$\:}\hb\vbc

\Dem Posons\, $\all z\in\,\C\quad f(z)=\Int_{\R}e\,\ha^{z\:t\,-\;t\ha^2/\hw 2}\;
d\hb t$\pv en vertu du \theo\ha\ d'holomorphie\vg\vva $f$ est analytique dans\hh$\C$\ha\pv par ailleurs\,\hh
$\all x\in\R$\,\hh on a\vvb$\Int_{\R}e\,\ha^{x\:t\,-\;t\ha^2/\hw 2}\;d\hb t\,
=\int_{\R}e\,^{x\ha^2\hs{-.3mm}/\hw2\,-\,(\ha t-x)\ha^2/\hw 2}\;d\hb t\,
=\ e\,^{x\ha^2\hs{-.3mm}/\hw2}\Int_{\R}e\,^{-\,(\ha t-x)\ha^2/\hw 2}\;d\hb t\,
=\ e\,^{x\ha^2\hs{-.3mm}/\hw2}\Int_{\R}e\,^{-\,\ha t\ha^2/\hw 2}\;d\hb t$\vvab
$=\,\sqrt{2\,\pi}\ha\ e\:^{x\ha^2/\hw2}$\hb\pv
donc\,\hh$\all z\in\,\C\quad f(z)=\,\sqrt{2\,\pi}\ha\ e\:^{z\ha^2/\hs{-.25mm}\hw2}$\vc

\Th{\num}\fbox{\:$\ZZ\,(e\hb^{-X^2\hs{-.3mm}/\hw2})\hb=\hb\AZ\ha(e\hb^{-X^2\hs{-.3mm}/\hw2})\hb
=\,\sqrt{2\,\pi}\,\ e\hb^{-X^2\hs{-.3mm}/\hw2}$\,}\ha\vb

\Dem\vva$\AZ\,(e\hb^{-X^2\hs{-.3mm}/\hw2})\,(x)\hb=\Int_{\R}\cos\,(x\:t)\;e\,^{-\;t\ha^2/\hw 2}\;d\hb t\,
=\,\inv{2}\,\int_{\R}e\,^{\hb i\:x\:t\:-\;t\ha^2/\hw 2}\;d\hb t\,
+\,\inv{2}\,\int_{\R}e\,^{-\hb i\:x\:t\:-\;t\ha^2/\hw 2}\;d\hb t$\vva
$=\,\sqrt{2\,\pi}\,\ e\hb^{-\,x^2\hs{-.3mm}/\hw2}$\vbc

\Def{\num}$\nn$\quad on pose\quad\fbox{\,$\HE\ha_n\,(\ha X)=(-1)^n \;
e\ha^{\,X^2\hs{-.3mm}/\ha2}\;\prt^{\,n}\hi{\big(}e\hb^{-X^2}\hi{\big)}$\,}\vb

\oui Les fonctions\, $\HE\ha_n$\, sont les\, \underline{fonctions de Hermite}\pv elles sont \e gales aux\,
\underline{polyn\^omes de} \underline{Hermite}\, multipli\e s\ha\ par\, $e\hb^{-X^2\hs{-.3mm}/\ha2}$\vbc

\Th{\num*}$\all\ell\in\N\,\hh\HE\hb_{2\,\ell}$\, est une fonction paire\, et\, $\HE\hb_{2\,\ell\ha+\hw1}$\,
une fonction impaire\vbc

\Th{\num}$\nn$\quad on a\quad\fbox{\,$\HE\ha_{n+1}=$\ho$X\hw.\:\HE\ha_n-(\HE\ha_n)\hb'$\ha}\vb

\Dem$X\hw.\,\HE\ha_n-(\HE\ha_n)\hb'\,=\:(-1)^n\;X\;e\ha^{\,X^2\hs{-.3mm}/\hw2}\;
\prt^{\,n}\hi{\big(}e\hb^{-X^2}\hi{\big)}-\,(-1)^n\;X\;e\ha^{\,X^2\hs{-.3mm}/\hw2}\;
\prt^{\,n}\hi{\big(}e\hb^{-X^2}\hi{\big)}$\vva$-\,(-1)^n\;e\ha^{\,X^2\hs{-.3mm}/\hw2}\;
\prt^{\,n+1}\hi{\big(}e\hb^{-X^2}\hi{\big)}\,=\:(-1)^{n+1} \ e\ha^{\,X^2\hs{-.3mm}/\hw2}\;
\prt^{\,n+1}\hi{\big(}e\hb^{-X^2}\hi{\big)}\,=\,\HE\ha_{n+1}$\vc

\Th{\num}$\nn$\quad on a\quad\fbox{\,$\ZZ\,(\HE\ha_n)\ha=\hb i\hb^n\,\sqrt{2\,\pi}\,\ \HE\ha_n$\,}\ha\vb

\Dem Par r\e currence sur\, $n$\vab

1)\hh vrai pour\, $n=\ha0$\vva
2\ha)\hh supposons vrai pour\, $n$\vva
3\ha)\hh d\e montrons vrai pour\, $n+1$\, :\vvab

$\ZZ\,(\HE\ha_{n+1})\,(x)=\Int_{\R}\HE\ha_{n+1}(t)\ e\,^{i\,x\,t}\,d\,t=\int_{\R}t\;\HE\ha_{n+1}\hb(t)\
e\,^{i\,x\,t}\,d\,t\,-\int_{\R}\HE\ha_{n+1}^{\hb'}\hb(t)\ e\,^{i\,x\,t}\,d\,t$\vva
$=\,-\ i\:\Big(\Int_{\R}\HE\ha_{n}(t)\ e\,^{i\,x\,t}\,d\,t\Big)\ha'\hb
+\,\,i\;x\int_{\R}\HE\ha_{n}(t)\ e\,^{i\,x\,t}\,d\,t=\,-\ i\:\big[\,\ZZ\,(\HE\ha_n)\hb(x)\hb\big]\ha'\hb
+\,\,i\;x\ha\;\ZZ\,(\HE\ha_n)\hb(x)$\vvab
$=\;i\hb^{n+1}\ha\sqrt{2\,\pi}\ \big[\hb x\:\HE\ha_n\hb(x)-\hb(\HE\ha_n)\hb'\hb(x)\hb\big]
=\,i\hb^{n+1}\ha\sqrt{2\,\pi}\,\ \HE\ha_{n+1}\hb(x)$\vbc

\Cor{\num*}$\all\el\in\N$\quad on a\quad$\AZ\,(\HE\ha_{2\,\ell})\hb
=\,(-1)^{\hb\ell}\;\sqrt{2\,\pi}\,\ \HE\hb_{2\,\ell}$\vva\hs{58.5mm}et\quad
$\BZ\:(\HE\hb_{2\,\ell\ha+\hw1})\hb=\,(-1)^{\hb\ell}\;\sqrt{2\,\pi}\,\ \HE_{2\,\ell+1}$\vbc

\Cor{\num*}$\nn$\,\hh on a\quad\fbox{\,$\T\,(\HE\ha_n)=(-1)^{\,[\:n\ha/\ha2\:]}\;\sqrt{2\,\pi}\,\
\HE\ha_n$\,}\ha\vf

\initnum
\quad\textbf{\Num\underline{In\e galit\e\ de Jensen dans\hb\ $\R^n$}}\vvc

\oui\Th{}On se donne un\, \underline{convexe}\, \underline{ferm\e}\hh$\AX\sub\ha\R\hw^n$\ha\
(\hw non vide et non r\e duit \ac\ un point\hw)\vg une probabilit\e\, $\tmu$\,  sur\, $\AX$\:
et une fonction\; $\Phi\ha:\,\AX\to\R$\; \underline{convexe}\ha\vg \underline{continue}\, et\,
\underline{sommable}\, pour\: $\tmu$\pv alors on a\vvab
\hs{47.5mm}\fbox{\,$\Phi\hb\Big[\hs{-.2mm}\Int_{\ha\AX}\hs{-.5mm}x\:\tmu\hb(x)\ha\Big]
\le\int_{\ha\AX}\Phi\hb(x)\:\tmu\hb(x)$\ha}\ha\vc

\Dem Nous faisons la d\e monstration dans\, $\R\hw^2$\vg la g\e n\e ralisation \e tant \e vidente\vw
Supposons d'abord\: $\AX$\: born\e\pt Consid\e rons\hw\hh$\nn^*$\hw\, le quadrillage canonique de\vvw
longueur\, $\Inv{n}$\, et soit\, $\SC_{\hw n}$\, la famille de parties de\, $\R\hw^2$\, form\e e des
sommets\vg des ar\^etes\vvw \emph{ouvertes} et des carr\e s \emph{ouverts} du quadrillage\va

Posons\,\hh$u\ha=\Int_{\ha\AX}\hs{-.5mm}x\ \tmu\hb(x)\ha\in\ha\R\hw^2$\hb\hh
et\,\hh$\nn^*\hh x_{\ha n}\ha=\sum\pos_{T\hw\in\:\SC_{\hw n}}x\hb_{T}\ \tmu\,(\AX\cap T\hw)\ha\in\ha\R\hw^2$\vva
avec\ha\hh$\all\hb T\hs{-.5mm}\in\hw\SC_{\hw n}\ha\hh x\hb_T\in T$\ha.\;
On a\hh$\sum\pos_{T\hw\in\:\SC_{\hw n}}\tmu\,(\AX\cap T\hw)\ha=\ha1$\vg donc\hh$x_{\ha n}\ha\in\ha\AX$\vva
et\hh$\Phi\hb(x_{\ha n})\le\sum\pos_{T\hw\in\:\SC_{\hw n}}\Phi\hb(x\hb_T)\;\tmu\,(\AX\cap T\hw)$\pt
Comme la fonction\hh$x\tto x$\hh et la fonction\, $\Phi$\vva sont continues\vg les ``\,s\e ries de Riemann\,"
$\sum\pos_{T\hw\in\:\SC_{\hw n}}x\hb_{T}\ \tmu\,(\AX\cap T\hw)$\,\hh
et\, $\sum\pos_{T\hw\in\:\SC_{\hw n}}\Phi\hb(x\hb_{T})\;\tmu\,(\AX\cap T\hw)$\vvk
convergent respectivement vers\,\hh$u\ha=\Int_{\ha\AX}\hs{-.5mm}x\ \tmu\hb(x)$\,\hh
et\hh$\Int_{\ha\AX}\Phi\hb(x)\;\tmu\hb(x)$\,\hh quand\hh$n\to\pin$\pv\vvwa
on en d\e duit\hh$u\in\AX$\,\hh et\ha\hh$\Phi\hb(u)\le\Int_{\ha\AX}\Phi\hb(x)\;\tmu\hb(x)$\vab

Supposons\, $\AX$\, non born\e\vg on consid\ee re alors la suite\hh$\AX\ha_n=\ha\AX\hb\cap\hb\D\ha_n$\vg
\ou\, $\D\ha_n$\, est le disque\vvwa ferm\e\vg centr\e\ \ac\ l'origine\vg de rayon\, $n$\ha.\: $\AX\ha_n$\,
est un convexe compact et on a\hh$\bigcup\pos_n\,\AX\ha_n=\ha\AX$\pv\vvk on peut alors \e crire\hh
$\nn\quad\Phi\hb\Big[\,\Inv{\tmu\hb(\AX\ha_n)}\int_{\ha\AX\ha_n}\hs{-1.75mm}x\ \tmu\hb(x)\hb\Big]\ha
\le\hb\inv{\tmu\hb(\AX\ha_n)}\int_{\ha\AX\hw_n}\hs{-1.5mm}\Phi\hb(x)\;\tmu\hb(x)$\pv il suffit\vva
ensuite de prendre la limite pour\, $n\to\pin$\vb

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au\ha\ \Theo\, X\ha\ 4\ha.\:4
