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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\centerline{\fbox{\,\textbf{CINQUIEME PARTIE}$\ph{\big|\dot{l}}$\hs{-2mm}}}\vvbc
\centerline{\fbox{\;\Large{\textbf{\tsf{Int\e gration sur $\ZP\ph{\big|}$}}}\hs{-1mm}}}\vs{12.5mm}

\oui Comme application des principes mis en oeuvre dans cet ouvrage nous pr\e sentons dans cette
avant\hw-\hw derni\ee re partie une th\e orie de l'int\e gration des fonctions et des mesures\ha\ \emph{r\e elles}\ha\
ou\ha\ \emph{complexes}\ha\ sur\, $\ZP$\pt En particulier l'existence d'une mesure de Haar\ha\ \emph{r\e elle}\ha\
sur\, $\ZP$\, permet de recycler sans grand changement la th\e orie des mesures sur\hb\ \ab\vw

\oui On remarquera qu'il n'est fait aucune allusion\ha\ aux\ha\ \emph{pseudo\ha-mesures}\, :\, c'est parce que\\
sur\, $\ZP$\, les fonctions \e tag\e es sont continues\vg donc que l'espace des fonctions r\e gl\e es\ha\
\emph{co\"incide}\ha\ avec l'espace des fonctions continues\vg donc que le concept de pseudo\ha-mesure
\emph{s'identifie}\ha\ au concept de mesure\vd

\initNum

\centerline{\textbf{CHAPITRE\, XX}}\vva
\centerline{\textbf{MESURES ET FONCTIONNELLES SUR\, $\ZP$}}\vvcd

\quad\textbf{\Num\underline{D\e finitions et notations}}\vvc
\initnum

$\all a\vv b\in\N$\quad\fbox{\,$a\dv b$\,}\,\hh signifie\hh\fbox{``\,$a$\lo\:divise\, $b$\,"}\ha\vg\vvab
\hs{23mm}\fbox{\,$a\ndv b$\,}\,\hh signifie\hh\fbox{``\,$a$\, ne divise pas\, $b$\,"}\ha\vb

$\CP\,=$\hh alg\ee bre des fonctions\hh:\hh$\ZP\,\to\R$\,\hh\underline{continues}\vvw
$\FP\,=$\hh alg\ee bre des fonctions\hh:\hh$\ZP\,\to\R$\,\hh\underline{born\e es}\vvb

\Not$\all f\in\FP$\,\hh on note\,\hh$\|\ha f\ha\|=\sup\pos_{x\,\in\,\ZP}|\ha f(x)\ha|$\vb

\Def{\num}On note\,\hh$\all u\in\ZP\,\hh\all\el\in\N\,\hh\Bal{u}\ha=\:u+\ha p\ha^\el\:\ZP$\ha\pv\ha
c'est la\, \underline{boule}\, de ``\,centre\," $u$\: et de ``\,rayon\," $1\ha/\ha p\ha^\el$\pv on a\,\hh$\all u\in\ZP\hh\B\,(u\,,\ha0)\ha=\:\ZP$\vb

\Th{\num*}Tout point d'une boule de\, $\ZP$\, est centre de cette boule\vpt
Deux boules de\, $\ZP$\, sont soit disjointes soit incluses l'une dans l'autre\vpt
L'ensemble des boules de\, $\ZP$\, est d\e nombrable\vb

\Th{\num*}\vvwa$\all u\vv v\in\ZP\,\hh\all\el\in\N\,\hh\Big[\;\Bal{u}=\ha\B\,(\hw v\vv\el\ha)\ \sam\ u\in\Bal{v}\:\Big]$\vbc

\Def{\num} On note\hh$\un\ha_{\bal{u}}$\hh la fonction \e gale \ac\: 1\: sur\: $\Bal{u}$\; et \ac\; 0\; sur\hh
$\ZP-\Bal{u}$\pv c'est la\, \underline{fonction indicatrice}\, de la boule\, $\Bal{u}$\vw
\oui Une combinaison lin\e aire\ha\ (\ha r\e elle et finie\ha)\ha\ de fonctions indicatrices de boules de\,
$\ZP$\, s'appelle une \underline{fonction \e tag\e e}\pt On note\, $\EP$\, l'espace vectoriel des fonctions
\e tag\e es\vbc

\Th{\num*}
\begin{picture}(,)
\put(1,0){$\EP\hb\sub\hb\CP\hb\sub\hb\FP$}
\put(-1,-2.5){\frm(30.25,7)}
\end{picture}\vvbc

\oui Compte tenu de ce \theo\ha\ le concept de fonction r\e gl\e e n'est pas significatif
\mbox{sur $\ZP$ :} il co\"incide avec le concept de fonction continue\vb

\Th{\num*}$\EP\vg\CP\vg\FP$\,\hh sont des\, \underline{alg\ee bres de Riesz}\vc

\quad\textbf{\Num\underline{Int\e grale de Haar des fonctions continues sur\, $\ZP$}}\vvc
\initnum

\Lem{\num}\vva$f\in\hb\CP\ \sam\ \Big[\;\all\eps>0$\hh il existe\hh$m\in\N$\hh tel que\,\hh$\all u\vv v\in\ZP\,\hh
\big|\,f\ha(\ha u+v\,p\ha^m)-f\ha(u)\hb\big|\le\hb\eps\;\Big]$\vvab

\Dem\vva
a)\hh$\imp$\dd$\CP$\, est compact\vg donc\, $f$\, est uniform\e ment continue sur\, $\ZP$\pv il existe donc\vvw
$\eta>\ha0$\,\hh tel que\,\hh$\all u\vv w\in\ZP\hh
\Big[\:|\ha w\ha|\ha_p\le\eta\ \imp\,\big|\ha f\ha(\ha u+w)-f\ha(u)\ha\big|
\le\hb\eps\:\Big]$\pv choisissons\vvwa$m\in\N$\hh tel que\hh$1\ha/\ha p\hb^m\le\ha\eta$\pv alors on a bien\,\hh
$\all u\vv v\in\ZP\,\hh\big|\ha f\ha(\ha u+v\,p\ha^m)-f\ha(u)\ha\big|\le\hb\eps$\vab

b)\hh$\pmi$\dd Trivial\vab

\Th{\num}Soit\, $f\in\hb\CP$\pv alors\,\hh\fbox{\,$\dis\lim_{n\to\pin}\
\inv{p\ha^n}\:\sum\pos_{r\,=\,0}^{p\ha^n-1}\,f\ha(r)$\:}\,\hh existe\pv on note cette limite\,\hh
\fbox{\,$\Int_{\;\ZP}\hs{-2mm}f\ha(x)\;\del\ha x$\,}\,
et on l'appelle l'\underline{int\e grale de Haar}\, de\, $f$\vbc

\Dem Soit\hh$\eps>0$\,\hh et soit\hh$m\in\N$\,\hh tel que\,\hh
$\al\;u\vv v\in\ZP\,\hh\big|\,f\ha(\ha u+v\,p\ha^m)-f\ha(u)\,\big|\le\hb\eps$\pv\vvab
on a\quad$\all n>m$\hhh${\Inv{p\ha^m}}\:\sum\pos_{r\,=\,0}^{p\ha^m-1}\,f\ha(r)\,
-{\Inv{p\ha^n}}\:\sum\pos_{r\,=\,0}^{p\ha^n-1}\,f\ha(r)\,
=\,{\Inv{p\ha^n}}\:\Big[\;p^{\hb n-m}\:\sum\pos_{r\,=\,0}^{p\ha^m-1}\,f\ha(r)
-\sum\pos_{r\,=\,0}^{p\ha^n-1}\,f\ha(r)\,\Big]$\vvab
$=\,{\Inv{p\ha^n}}\:\Big[\;p\ha^{n-m}\:\sum\pos_{r\,=\,0}^{p\ha^m-1}f\ha(r)\;
-\sum\pos_{r\,=\,0}^{p\ha^m-1}\ \sum\pos_{s\,=\,0}^{p\ha^{n-m}-1}f\ha(\ha r+s\,p\ha^m)\,\Big]$\vvab
$=\,{\Inv{p\ha^n}}\:\sum\pos_{r\,=\,0}^{p\ha^m-1}\:
\Big[\;p\ha^{n-m}\:f\ha(r)\;-\sum\pos_{s\,=\,0}^{p\ha^{n-m}-1}f\ha(\ha r+s\,p\ha^m)\,\Big]
=\,{\Inv{p\ha^n}}\:\sum\pos_{r\,=\,0}^{p\ha^m-1}\:\sum\pos_{k\,=\,0}^{p\ha^{n-m}-1}
\big[\:f\ha(r)-f\ha(\ha r+k\,p\ha^m)\,\big]$\pv\vvab donc\quad
$\nn\hhh\Big|\;{\Inv{p\ha^m}}\:\sum\pos_{r\,=\,0}^{p\ha^m-1}\,f\ha(r)\,
-\,{\Inv{p\ha^n}}\:\sum\pos_{r\,=\,0}^{p\ha^n-1}\,f\ha(r)\;\Big|\hb\le\,\eps$\vc

\Th{\num*}$\Int_{\;\ZP}\hs{-2.25mm}\un\;\del\ha x\hb=\hb1$\vbc

\Th{\num*}$\all f\in\hb\CP$\,\hh on a\,\hh$\Int_{\;\ZP}\hs{-2mm}|\hb f\ha(x)\ha|\
\del\ha x\ha=\hb0\,\ \sam\ f=\hb0$\vbc

\Th{\num*}$\all f,\,g\ha\in\hb\CP$\,\hh on a\quad\fbox{\:$f\le g\ \imp
\Int_{\;\ZP}\hs{-2mm}f\ha(x)\;\del\ha x\,\le\Int_{\;\ZP}\hs{-2mm}g\ha(x)\;\del\ha x$\:}\vc

On se donne d\e sormais\, $f\in\hb\CP$\vc

\Th{\num*}\fbox{\:$\Big|\Int_{\;\ZP}\hs{-2mm}f\ha(x)\;\del\ha x\,\Big|\,
\le\Int_{\;\ZP}\hs{-2mm}|\ha f\ha(x)\ha|\hb(x)\ \del\ha x\,\le\hb\|\ha f\ha\|$\,}\ha\vc

\Th{\num}$\all u\in\ZP$\quad\fbox{$\,\Int_{\;\ZP}\hs{-2mm}f\ha(\hw x+u\hw)\;\del\ha x\,
=\int_{\;\ZP}\hs{-2mm}f\ha(x)\;\del\ha x$\:}\ha\vc

\Dem Supposons d'abord\,\hh$u=k\in\N$\pv alors on a\,\hh$\nn$\vvab
${\Inv{p\ha^n}}\:\sum\pos_{r\,=\,0}^{p\ha^n-1}\,f\ha(r+k)\,
=\,{\Inv{p\ha^n}}\:\sum\pos_{r\,=\,k}^{p\ha^n+\,k-1}\,f\ha(r)\,
=\,{\Inv{p\ha^n}}\:\Big[\:\sum\pos_{r\,=\,0}^{p\ha^n-1}\,f\ha(r)\,-\,\sum\pos_{r\,=\,0}^{k-1}\,f\ha(r)\
+\sum\pos_{r\,=\,p\ha^n}^{p\ha^n+\,k-1}\,f\ha(r)\;\Big]$\pv\vva
en faisant\hh$n\to\pin$\,\hh on obtient\,\hh$\Int_{\;\ZP}\hs{-2mm}f\ha(\hw x+k\hw)\;\del\ha x\,
=\Int_{\;\ZP}\hs{-2mm}f\ha(x)\;\del\ha x$\vb

Supposons maintenant\hh$u\in\ZP$\pv soit\hh$\eps>0$\,\hh et soit\hh$m\in\N$\,\hh tel que\vva
$\all v\vv w\in\ZP\quad\big|\,f\ha(\ha v+w\,p\ha^m)-f\ha(v)\,\big|\ha
\le\hb\eps$\pv soit\hh$k\in\N$\,\hh tel que\,\hh$u-k\in\ha p\ha^m\:\ZP$\pv\vva
alors on peut \e crire\,\hh$\all r\in\N\,\hh\big|\,f\ha(r+u)-f(r+k)\,\big|\ha\le\hb\eps$\pv\hb
donc\,\hh$\nn$\vvab$\Big|\;{\Inv{p\ha^n}}\:\sum\pos_{r\,=\,0}^{p\ha^n-1}\,f\ha(r+u)
-{\Inv{p\ha^n}}\:\sum\pos_{r\,=\,0}^{p\ha^n-1}\,f\ha(r+k)\;\Big|\hb\le\,\eps$\pv\hb
en faisant\hh$n\to\pin$\,\hh on obtient\vvab
$\Big|\Int_{\;\ZP}\hs{-2mm}f\ha(\hw x+u\hw)\;\del\ha x\,
-\Int_{\;\ZP}\hs{-2mm}f\ha(x)\;\del\ha x\,\Big|\hb\le\,\eps$\vbc

\Th{\num}Soit\hh$u\in\ZP$\hh tel que\hh$|\ha u\ha|\hb_p=1$\pv alors\quad
\fbox{$\,\Int_{\;\ZP}\hs{-2mm}f\ha(\ha u\hb x\ha)\;\del\ha x\,=\int_{\;\ZP}\hs{-2mm}f\ha(x)\;
\del\ha x$\:}\vvbc

\Dem Soit\hh$\eps>0$\hh et soit\hh$m\in\N$\hh tel que\,\hh
$\all v\vv w\in\ZP$\,\hh$\big|\,f\ha(\ha v+w\,p\ha^m)-f\ha(v)\,\big|\ha
\le\hb\eps$\va Supposons d'abord\hh$u=k\in\N^*$\pv on a\,\hh$p\ndv k$\pv soit\hh$n\in\N\vv\hb n\ge2$\ha\pv\ha
effectuons\vva$\all r\ha\in\ha\db{\,1\vv p\hb^n-1\,}$\,\hh la division euclidienne de\, $k\:r$\, par\,
$p\hb^n$\pv on pose\,\hh$k\:r=s\ha_r+t\ha_r\,p\hb^n$\vva avec\,\hh
$s\ha_r\vv t\ha_r\in\N^*$\,\hh et\,\hh$s\ha_r\le\hb p\hb^n-1$\va
Montrons que\,\hh$r_1=r\ha_2\ \sam\ s\ha_{r_1}=\,s\ha_{r\ha_2}$\pv en effet supposons\hh
$s\ha_{r_1}=\,s\ha_{r\ha_2}$\pv alors\vva$p\hb^n\dv k\,(\ha r_1-r\ha_2\ha)$\,\hh donc\,\hh
$p\hb^n\dv r_1-r\ha_2$\hb\pv\ha or\,\hh$|\,r_1-r\ha_2\ha|\le\hb p\hb^n-1$\ha\vg\ha donc\,\hh$r_1=r\ha_2$\va

On peut donc \e crire\hh$\ens{\,s\hb_i\;\|\;1\le i\le p\ha^n-1\ha}
=\,\db{\,1\vv p\ha^n-1\,}$\pv on en d\e duit\,\hh$\all n\ge\ha m$\vvab
$\Big|\;{\Inv{p\ha^n}}\:\sum\pos_{r\,=\,0}^{p\ha^n-1}\,f\ha(\ha k\:r\ha)\,
-\,{\Inv{p\ha^n}}\:\sum\pos_{r\,=\,0}^{p\ha^n-1}\,f\ha(r)\;\Big|\ha
=\hb\Big|\;{\Inv{p\ha^n}}\:\sum\pos_{r\,=\,1}^{p\ha^n-1}\,f\ha(\ha s\ha_r+t\ha_r\,p^n\ha)\,
-\,{\Inv{p\ha^n}}\:\sum\pos_{r\,=\,1}^{p\ha^n-1}\,f\ha(s\ha_r)\;\Big|\ha\le\,\eps$\pv\vvab
en faisant\hh$n\to\pin$\,\hh on obtient\quad$\Big|\Int_{\;\ZP}\hs{-2mm}f\ha(\ha k\,x\ha)\;\del\ha x\,
-\Int_{\;\ZP}\hs{-2mm}f\ha(x)\;\del\ha x\,\Big|\le\hb\eps$\vbc

Supposons maintenant\hh$u\in\ZP$\hh et soit\hh$k\in\N$\,\hh tel\hb\ que\,\hh$u-k\in\ha p\ha^m\:\ZP$\pv
on a\,\hh$\all n\ge\ha m$\vvab
$\Big|\;{\Inv{p\ha^n}}\:\sum\pos_{r\,=\,0}^{p\ha^n-1}\,f\ha(\hw u\:r\hw)\,
-\,{\Inv{p\ha^n}}\:\sum\pos_{r\,=\,0}^{p\ha^n-1}\,f\ha(\ha k\;r\ha)\;\Big|\ha
=\hb\Big|\;{\Inv{p\ha^n}}\:\sum\pos_{r\,=\,0}^{p\ha^n-1}\,f\hb[\,k\:r+(u-k)\,r\,]\,
-\,{\Inv{p\ha^n}}\:\sum\pos_{r\,=\,0}^{p\ha^n-1}\,f\ha(\ha k\:r\ha)\;\Big|\ha\le\,\eps$\pv\vvab
en faisant\hh$n\to\pin$\,\hh on obtient\quad$\Big|\Int_{\;\ZP}\hs{-2mm}f\ha(\ha u\,x\ha)\;\del\ha x\,
-\Int_{\;\ZP}\hs{-2mm}f\ha(x)\;\del\ha x\,\Big|\le\hb\eps$\vcd

\Th{\num}\fbox{\,$\Big[\Int_{\;\ZP}\hs{-2mm}f\ha(x)\;g\ha(x)\;\del\ha x\hb\Big]^{\ha2}
\le\int_{\;\ZP}\hs{-2mm}f\ha^2\ha(x)\;\del\ha x\int_{\;\ZP}\hs{-2mm}g\ha^2\ha(x)\;\del\ha x$\:}\ha\vbc

\Dem$\Big[\,\Inv{p\ha^n}\,\tex\sum\pos_{r\,=\,0}^{p\ha^n-1}\,f\ha(r)\,g\ha(r)\hb\Big]^2
\le\Big[\,\Inv{p\ha^n}\,\tex\sum\pos_{r\,=\,0}^{p\ha^n-1}\,f\ha^2\ha(r)\hb\Big]\
\Big[\,\Inv{p\ha^n}\,\tex\sum\pos_{r\,=\,0}^{p\ha^n-1}\,g\ha^2\ha(r)\hb\Big]$\pv
il suffit ensuite\vvab de faire\hh$n\to\pin$\vbc

\Cor{\num}\fbox{\,$\Big[\Int_{\;\ZP}\hs{-2mm}f\ha(x)\;\del\ha x\hb\Big]^{\ha2}
\le\int_{\;\ZP}\hs{-2mm}f\ha^2\ha(x)\;\del\ha x$\:}\ha\vbc

\Def{\num}$\all u\in\ZP\,\hh\all\el\in\N$\,\hh on pose\,\hh$\Int_{\,\bal{u}}\hs{-8mm}f\ha(x)\;\del\ha x\,
=\int_{\;\ZP}\!\!\un\ha_{\bal{u}}\hb(x)\:f\ha(x)\;\del\ha x$\vc

\Th{\num}\fbox{\,$\Int_{\,\bal{u}}\hs{-8mm}f\ha(x)\;\del\ha x\,
=\ \inv{p\ha^\el}\,\int_{\;\ZP}\hs{-2.5mm}f\ha(\ha u+\ha p\ha^\el\ha x\ha)\;\del\ha x$\:}\ha\vbc

\Dem Supposons d'abord\hh$u=k\in\N^*$\pv on a\,\hh$\Int_{\,\bal{k}}\hs{-8mm}
f\ha(x)\;\del\ha x\,=\Int_{\;\ZP}\!\!\un\ha_{\bal{k}}\hb(x)\:f\ha(x)\;\del\ha x$\vvwa
$=\,\dis\lim_{n\to\pin}\ \inv{p\ha^n}\:\tex\sum\pos_{r\,=\,0}^{p\ha^n-1}\
\un\ha_{\bal{k}}\,(r)\,f\ha(r)\,=\dis\lim_{n\to\pin}\ \inv{p\ha^n}
\tex\sum\pos_{0\:\le\:s\:<\:(\,p\ha^n-k-1\ha)\ha/\ha p\ha^\el}
f\ha(\ha k+s\,p\ha^\ell\ha)$\vva$=\dis\lim_{n\to\pin}\ \inv{p\ha^n}\
\tex\sum\pos_{s\,=\,0}^{p\ha^{n-\ell}}\:f\ha(\ha k+s\,p\ha^\ell\ha)
=\ \Inv{p\ha^\el}\,\int_{\;\ZP}\hs{-2.5mm}f\ha(\ha k+\ha p\ha^\el\ha x\ha)\;\del\ha x$\vc

\newpage
Supposons maintenant\hh$u\in\ZP$\pv soit\hh$\eps>0$\,\hh et soit\hh$\under{-----}{m\ge\ha\el}$\,\hh
tel que\,\hh$\all v\vv w\in\ZP$\vvaab$\big|\,f\ha(\ha v+w\,p\ha^m)-f\ha(v)\,\big|\ha
\le\hb\eps$\pv soit\hh$k\in\N$\,\hh tel que\,\hh$u-k\in\ha p\ha^m\:\ZP$\pv
alors\,\hh$k\in\Bal{u}$\vg\vvab donc\,\hh$\Bal{u}=\ha\Bal{k}$\vg donc\,\hh
$\Int_{\,\bal{u}}\hs{-8mm}f\ha(x)\;\del\ha x\,=\Int_{\,\bal{k}}\hs{-8mm}f\ha(x)\;\del\ha x\,
=\ \Inv{p\ha^\el}\,\int_{\;\ZP}\hs{-2.5mm}f\ha(\ha k+\ha p\ha^\el\ha x\ha)\;\del\ha x$\va
On en d\e duit\vvab$\Big|\Int_{\,\bal{u}}\hs{-8mm}f\ha(x)\;\del\ha x
-\Inv{p\ha^\el}\,\int_{\;\ZP}\hs{-2.5mm}f\ha(\ha u+\ha p\ha^\el\ha x\ha)\;
\del\ha x\,\Big|=\:\Big|\;\Inv{p\ha^\el}\,
\int_{\;\ZP}\hs{-3mm}f\ha(\ha k+\ha p\ha^\el\ha x\ha)\;\del\ha x
-\Inv{p\ha^\el}\,\int_{\;\ZP}\hs{-2.5mm}f\ha(\ha u+\ha p\ha^\el\ha x\ha)\;
\del\ha x\:\Big|$\vvab$\le\:\Inv{p\ha^\el}\,\int_{\;\ZP}\hs{-1.5mm}
\big|\hb f\ha(\ha k+\ha p\ha^\el\ha x\ha)-f\ha(\ha u+\ha p\ha^\el\ha x\ha)\hb\big|\;
\del\ha x\hb\le\,\eps\ha/\ha p\ha^\el$\vcd

\Def{\num}Une partie\, $\AZ$\, de\, $\ZP$\; est\, \underline{\e l\e mentaire}\, ssi\ha\ elle est constitu\e e
d'une\vvw r\e union finie disjointe de boules\, $\B\ha_i$\, de\, $\ZP$\pv on pose alors\hh
$\Int_{\,\AZ}\!f\ha(x)\;\del\ha x=\,\tex\sum\pos_i\hb\Int_{\,\B\ha_i}\hs{-2mm}f\ha(x)\;\del\ha x$\vd

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

\oui La th\e orie des mesures et des fonctionnelles sur\, \ab\, se transpose ais\e ment \ac\, $\ZP$\vg\vvw
avec de plus la circonstance simplificatrice que\hh $\RP=\ha\CP$\vg\cad\ que le compl\e t\e\ de\, $\EP$\vvw
pour la norme\, $\|\ \|$\, est\, $\CP$\pt Reprenons les principales \e tapes de la th\e orie\va
Sur\, $\CP$\, il y a trois\, \underline{normes}\, fondamentales\dd outre la norme\, $\|\ \|$\ha\vg\ha
on pose\,\hh$\all f\in\CP$\vvaab$\none{\ha f\ha}=\Int_{\;\ZP}\hs{-2mm}|\ha f\ha(x)\ha|\;\del\ha x$\,\hh
et\,\hh$\ntwo{\ha f\ha}=\Big[\ha\Int_{\;\ZP}\hs{-2mm}f\ha^2\ha(x)\;\del\ha x\hb\Big]^{1\ha/\ha2}$\pv
on a\,\hh\fbox{\,$\none{\ha f\ha}\le\ntwo{\ha f\ha}\le\|\ha f\ha\|$\,}\ha\vbc

\begin{picture}(,)
\put(8,-2){Le\ha\ N\hw-\hw dual\ha\ de l'espace norm\e\, $\big(\ha\EP\vv\|\ \|\ha\big)$\, se note\, $\MP$\;
et ses \e l\e ments s'appellent }
\put(8,-8){les\, \underline{mesures}\, sur\, $\ZP$\hb.}
\put(5.25,-11){\frm(150.5,14.5)}
\end{picture}\\[7.5mm]

Tout\, $\tmu\in\MP$\, s'\e tend canoniquement \ac\, $\CP$\, en posant\hh$\all f\in\CP$\vvab
\hs{30mm}\fbox{\;$\tmu\:(f)=\,\lim\pos_n\;\tmu\:(f\ha_n)\quad\mrom{avec}\hh
f\ha_n\in\EP\,\hh\mrom{et}\hh f\ha_n\uu f$\;}\ha\vc

\emph{\underline{Notation\ha\ int\e grale}}\dd$\all\tmu\in\MP\,\hh\all f\in\CP$\,\hh
on note\,\hh\fbox{\:$\tmu\:(f)\ha=\Int_{\;\ZP}\hs{-2mm}f\ha(x)\;\tmu\,(x)$\,}\ha\vbc

$\all f\in\CP$\hh la forme lin\e aire\hh\fbox{\,$\ans{f}\,:\ \CP\to\R\::\
g\ha\tto\Int_{\;\ZP}\hs{-2mm}f\ha(x)\;g\ha(x)\;\del\hb x$\:}\hh est la mesure\vvb
\underline{associ\e e}\, \ac\, $f$.\, On note\,\hh$\low{\EP}=\ens{\ans{f}\,\big\|\,f\in\EP}\,
\sub\hb\ \low{\CP}=\ens{\ans{f}\,\big\|\,f\in\CP}\hb\sub\,\MP$\vb

La mesure\hh\fbox{\:$\haar\ha:\ \CP\to\R\::\ g\ha\tto\Int_{\;\ZP}\hs{-2mm}g\ \del\hb x$\;}\hh
est la\; \fbox{\ha mesure de Haar$\ph{\dot{t}}$\lo\hs{-2.25mm}}\; sur\, $\ZP$\vbc

Dans\, $\MP$\, on d\e finit la norme\, $\nst{\ }$\ha\vg\ha duale de la norme\hh$\|\ \|$\ha\vg\ha
en posant\hh$\all\tmu\in\MP$\\
\begin{picture}(,)
\put(2,-2){$\nst{\hb\tmu\hb}\ =\hs{-2mm}\sup\pos_{g\,\in\:\EP\,,\;\|\ha g\ha\|\,=\,1}
\hs{-1mm}|\hb\tmu\hb(g)\ha|$\hh.\, La norme\hh$\nst{\ }$\hh prolonge la norme\hh$\none{\ }$\hh
sur\; $\low{\CP}$\pt}
\put(.25,-8){\frm(49.25,11)}
\end{picture}\\[4mm]

On note\; $\LP^1$\; la fermeture de\: $\low{\EP}$\: dans\: $\MP$\ha\pt
On a\,\hh$\low{\CP}\hb\sub\hb\LP^1$\ha\pt Les \e l\e ments de\, $\LP^1$\vvwa
s'appellent les\, \underline{fonctionnelles sommables}\: sur\, $\ZP$\pt
On note\hh$\all\tdf\in\LP^1$\,\hh\fbox{\,$\bnone{\ha\tdf\hb}\ha=\ha\bnst{\ha\tdf\hb}$}\ha\vb

\emph{\underline{Notation\ha\ int\e grale}}\dd$\all\tdf\in\LP^1$\hh on \e crit\;
$\Int_{\;\ZP}\!\tdf\ha(x)\ \del\ha x$\hh au lieu de\; $\Int_{\;\ZP}\hs{-2mm}\tdf\ha(x)$\pt
on a donc\vva\hs{40mm}\fbox{\ha\,$\Int_{\;\ZP}\hs{-2mm}\tdf\ha(x)\ \del\ha x\ha
=\int_{\;\ZP}\hs{-2mm}\tdf\ha(x)\hb=\hb\tdf\ha(\un)\,$}\quad(\hb avec $\un=\ha\un\hb_{\ZP}\hb)$\vc

Le\ha\ N\hw-\hw dual\ha\ de l'espace norm\e\, $\big(\ha\EP\vv\ntwo{\ }\ha\big)$\, se note\, $\LP^2$\;
et ses \e l\e ments s'appellent les\vvww\underline{fonctionnelles hilbertiennes}\, sur\, $\ZP$\vw
Le\ha\ N\hw-\hw dual\ha\ de l'espace norm\e\, $\big(\ha\EP\vv\none{\ }\ha\big)$\, se note\ha\ $\BP$\:
et ses \e l\e ments s'appellent les\vvww\underline{fonctionnelles born\e es}\, sur\, $\ZP$\va

On a\,\hh$\low{\CP}\hb\sub\,\BP\hb\sub\hb\LP^2\,\sub\hb\LP^1$\,\hh et la norme sur\, $\LP^2$\; se note
encore\; $\ntwo{\ }$\; car elle prolonge\vvw la norme\; $\ntwo{\ }$\; sur\; $\low{\CP}$\pv de plus\:
$\low{\EP}$\: est dense dans\: $\LP^2$\va
On note\hh$\nbb{\ }$\hh la norme sur $\BP$\pv on a\,\hh$\all\tdf\in\ha\BP$\quad
\fbox{\,$\bnone{\ha\tdf\hb}\le\bntwo{\ha\tdf\hb}\le\bnbb{\ha\tdf\hb}$\ha}\ha\vab

$\MP$\hh et\hh$\LP^1$\hh sont des espaces de Riesz\ha-Banach\pv $\LP^2$\: est un espace de Riesz\ha-Hilbert\,;\vvww
$\BP$\; est une alg\ee bre de Riesz\ha-Banach\va

L'ensemble\, $\KKP$\,  des\, \underline{fonctionnelles caract\e ristiques}\, et l'espace\, $\NP$\, des\,
\underline{fonctionnelles}\vvww\underline{totalement singuli\ee res}\, se d\e finissent comme\ha\ sur\, \ab\va

On peut \e noncer le \theo\ha\ de Radon-Nikodym\dd\fbox{\,$\MP\ha=\ha\LP^1\ha\oplus\ha\NP$\,}\ha\vaab

On d\e finit aussi les\ha\ espaces\ha\hh$\BAP$\hh et\: $\WP$\ha\vg respectivement des\,
\underline{fonctions de Baire}\, et\, des\, \underline{fonctions\, universelles}\, sur\, $ \ZP$\vwa

Tous les espaces d\e crits pr\e c\e demment v\e rifient les m\^emes propri\e t\e s que leurs corres-pondants
sur\, \ab\pv quant aux d\e monstrations elles se transposent quasiment sans changement\pt De plus on dispose
toujours des incontournables \theo s de convergence\, \underline{monotone}\, et\, \underline{born\e e}\vwa

Finalement on construit l'alg\ee bre de Riesz\, $\FOP$\: des\, \underline{fonctionnelles}\,
(\ha g\e n\e rales\ha)\ha\  sur\, $\ZP$\vg et on d\e veloppe la m\^eme th\e orie \ac\ leur sujet que
pour\, \ab\vbc

\oui\textbf{\underline{R\e capitulatif}}\vvc

\hs{72.25mm}$\FP$\hs{37mm}$\FOP$\vvww\hs{73.25mm}$\cup$\hs{40.5mm}$\cup$\vvww
\hs{30mm}$\EP\ \sub\ \CP\ \sub\ \BAP\ \sub\ \WP\ \hookrightarrow\,\
\BP\ \sub\ \LP^2\,\ \sub\ \LP^1\ha\ \sub\;\MP$\vvww
\hs{89.5mm}$\cup\hs{38.5mm}\cup$\vvww\hs{89mm}$\KKP\hs{36mm}\NP$\\
\begin{picture}(,)
\put(26,3.5){\frm(113.5,40)}
\end{picture}\vvbc

\oui La th\e orie s'\e tend \ac\hh $\ZP\tim\ZP$\hh et permet de d\e montrer le \theo\ha\ de\,
\underline{Fubini}\vab

\oui Ces r\e sultats se g\e n\e ralisent sans difficult\e\ \ac\ des fonctions ou des mesures complexes\pv
nous utiliserons pour les espaces correspondants les notations\hh$\wh{\CP}$\ha\vg\ha etc \ldots\vvcd

\quad\textbf{\Num\underline{Moyenne d'une mesure sur une boule}}\vvc
\initnum

\Def{\num}$\all\tmu\in\wh{\MP}\,\hh\all v\in\ZP\,\hh\all\el\in\N$\quad on pose\vvab
\hs{35mm}\fbox{\,$\T_\el\,(\tmu)\,(v)\hb=\;p\ha^\el\ \tmu\,(\ha\un\ha_{\bal{v}}\ha)\,
=\ p\ha^\el\!\Int_{\;\ZP}\hs{-2mm}\un\ha_{\bal{v}}\hb(x)\;\tmu\,(x)$\,}\ha\vbc

$\T_\el\,(\tmu)\,(v)$\, est la\, \underline{moyenne}\, de\, $\tmu$\: sur la boule\, $\Bal{v}$\vc

\Th{\num}$\all\tmu\in\wh{\MP}$\quad on a\quad$\T_\el\,(\tmu)\in\ha\wh{\EP}$\quad
et\quad$\big\|\ha\T_\el\,(\tmu)\ha\big\|\ha\le\:p\ha^\el\:\nst{\ha\tmu\ha}$\vb

\Dem Il existe\ha\ exactement\, $p\ha^\el$\, boules\, (\ha disjointes\ha)\, de ``\,rayon\," $p\ha^\el$\hb;\:
$\T_\el\,(\tmu)$\; est constant sur chacune d'entre elles\vg donc\hh$\T_\el\,(\tmu)\in\ha\wh{\EP}$\hb\pt
Par ailleurs on a clairement\,\hh$\all v\in\ZP$\\$\big|\hb\T_\el\,(\tmu)\,(v)\ha\big|
\le\:p\ha^\el\:\nst{\ha\tmu\ha}$\vc

\Th{\num}$\all\tmu\in\wh{\MP}$\quad on a\quad\fbox{\:$\bnone{\hb\T_\el\,(\tmu)\hb}
\le\hb\nst{\ha\tmu\ha}$\hb}\ha\vb

\Dem$\bnone{\hb\T_\el\,(\tmu)\hb}=\Int_{\;\ZP}\hs{-2mm}\big|\hb\T_\el\,(\ha\tmu\ha)\hb(v)\ha\big|\;\del\,(v)\,
\le\ p\ha^\el\!\int_{\;\ZP}\!\Big[\int_{\;\ZP}\hs{-2mm}\un\ha_{\bal{v}}\hb(x)\;|\ha\tmu\ha|\,(x)\hb\Big]\,\del\,(v)$\pv\vvab
or\,\hh$\all v\vv x\in\ZP\,\hh\un\ha_{\bal{v}}\hb(x)=\ha\un\ha_{\bal{x}}\hb(v)$\vg donc\vvb
$\bnone{\hb\T_\el\,(\tmu)\hb}=\:p\ha^\el\!\Int_{\;\ZP}\!\Big[\int_{\;\ZP}\hs{-2mm}
\un\ha_{\bal{x}}\hb(v)\;\del\,(v)\hb\Big]\,|\ha\tmu\ha|\,(x)\,=\int_{\;\ZP}\hs{-2mm}
|\ha\tmu\ha|\,(x)\:=\,\nst{\ha\tmu\ha}$\vc

\Th{\num}$\all f\in\wh{\CP}$\quad on a\quad$\T_\el\,(f)\,\uu f$\quad quand\quad$\el\to\pin$\vb

\Dem Soit\hh$\eps>0$\hh et soit\,\hh$m\in\N$\,\hh tel que\,\hh$\all a\vv b\in\ZP$\,\hh on ait\vva
$\big|\,f\ha(\ha a+b\,p\ha^m)-f\ha(a)\hb\big|\ha\le\hb\eps$\pv on peut alors \e crire\,\hh
$\all\el\ge m\,\hh\all v\in\ZP$\vvab$\big|\hb\T_\el\,(f)\,(v)-f\ha(v)\ha\big|\ha\le\ha\Int_{\;\ZP}\hs{-1.5mm}
\big|\hb f\ha(\ha v+\ha p\ha^\el\ha x\ha)-f\ha(v)\hb\big|\;\del\ha x\ha\le\hb\eps$\ha\vg\vva\cad\quad
$\all\el\ge m\quad\,\big\|\,\T_\el\,(f)\,(v)-f\ha(v)\hb\big\|\ha\le\hb\eps$\vc

\Th{\num}$\all\tdf\in\wh{\LP^1}$\quad on a\quad$\T_\el\,(\tdf)\,\one\,\tdf$\quad quand\quad$\el\to\pin$\vab

\Dem On applique le\, \emph{LFAF}\, aux op\e rateurs lin\e aires\,\hh
$\T_\el\,-\,\I\;:\ \wh{\LP^1}\,\to\,\wh{\LP^1}$\vbc

\Th{\num}$\all\tdf\in\wh{\LP^2}$\quad on a\quad$\T_\el\,(\tdf)\,\two\,\tdf$\quad quand\quad$\el\to\pin$\vab

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

\Def{\num}Soient\ha\hh$\tmu\hb_n\vv\tmu\in\MP$\pv on dit que la suite\: $\tmu\hb_n$\, converge\,
\underline{faiblement}\, vers\, $\tmu$\, ssi\,\hh$\all h\in\CP\,\hh\tmu\hb_n\hb(h)\to\tmu\hb(h)$\pt
On \e crit\,\hh$\tmu\hb_n\,\ww\,\tmu$\vbc

\Th{\num}$\all\tmu\in\wh{\MP}$\quad on a\quad$\T_\el\,(\ha\tmu\ha)\,\ww\,\tmu$\quad quand\quad$\el\to\pin$\vb

\Dem Il faut montrer que\,\hh$\all h\in\CP\hh\Int_{\;\ZP}\hs{-2mm}h\hb(v)\;
\T_\el\,(\ha\tmu\ha)\hb(v)\;\del\,(v)\,\to\int_{\;\ZP}\hs{-2mm}h\hb(x)\;\tmu\,(x)$\vvww
quand\,\hh$\el\to\pin$\hb\pt\vvaab On a\,\hh$\Int_{\;\ZP}\hs{-2mm}h\hb(v)\;
\T_\el\,(\ha\tmu\ha)\hb(v)\;\del\,(v)\hb=\int_{\;\ZP}\hs{-2mm}h\hb(v)\ p\ha^\el\;
\Big[\int_{\;\ZP}\hs{-2mm}\un\ha_{\bal{v}}\hb(x)\;\tmu\,(x)\hb\Big]\;\del\,(v)$\vva
$=\hb\ p\ha^\el\hs{-.5mm}\Int_{\;\ZP}\hs{-2mm}\ \Big[\int_{\;\ZP}\hs{-2mm}
\un\ha_{\bal{x}}\hb(v)\,h\hb(v)\;\del\,(v)\hb\Big]\;\tmu\,(x)\,
=\int_{\;\ZP}\hs{-2mm}\T_\el\,(h)\,(x)\;\tmu\,(x)$\pv\vvaab
comme\,\hh$\T_\el\,(\hw h)\,\uu\,h$\hb\vg on en d\e duit\quad
$\lim\pos_{\el\to\pin}\,\Int_{\;\ZP}\hs{-2mm}\T_\el\,(\hw h)\,(x)\;\tmu\,(x)\,=
\int_{\;\ZP}\hs{-2mm}h\hb(x)\;\tmu\,(x)$\vd

\newpage
\initNum

\centerline{\textbf{CHAPITRE\, XXI}}\vva
\centerline{\textbf{SERIES DE FOURIER SUR\, $\ZP$}}\vvcd

\oui Nous d\e veloppons la th\e orie des s\e ries de Fourier des mesures complexes sur\, $\ZP$\pt
Ces s\e ries sont des combinaisons lin\e aires infinies \ac\ coefficients complexes de la fonction\, $\un$\,
et des fonctions de la forme\, $e^{\:2\:\pi\,i\:m\:x\ha/\hb p\ha^n}$\hh\big(\hb$m\vv n\in\N^*$\vg
$p\ndv m\hb\le\,p\ha^n-1$\hb\big)\vg qui constituent les\ha\ \emph{caract\ee res}\ha\ de\, $\ZP$\vww\oui
Nous proposons ensuite des exemples explicites de calcul de s\e ries de Fourier sur\, $\ZP$\vc

\quad\textbf{\Num\underline{Caract\ee res de\, $\ZP$}}\vvc
\initnum

\underline{\emph{Notations}}\:\ :\ha\hh$\KP$\: =\; compl\e t\e\ p-adique de\; $\Q$\, =\; corps des fractions
de\, $\ZP$\vvwwa
\hs{24mm}$\U=\ens{\hb z\in\:\C\ \|\:|\hw z\hw|=1\hw}$\vbc

\Th{\num}La fonction\hh\fbox{\:$E\,:\hw\ \KP\ha\to\U\,:\,x\,\tto\,e^{\:2\,\pi\,i\:x}$\,}\hh
est bien d\e finie\vb

\Dem Comme\,\hh$\KP\ha/\ha\ZP$\hh est canoniquement isomorphe \ac\,\hh
$\Z\hb[\ha1\hw/\hw p\hb]\hb/\ha\Z\,\sub\,\Q\ha/\ha\Z$\vvww et que la fonction\,\hh
$\Q\ha/\ha\Z\ha\to\U\,:\,x\,\tto\,e^{\:2\,\pi\,i\:x}$\hh est bien d\e finie\vg
il en est de m\^eme de\vvww la fonction\, $E$\vb

\Th{\num*}$E\in\,\widehat{\CP}$\ha\pv de plus\ha\hh$\all x\in\KP\hh\big[\;E\,(x)=\ha0\ \sam\ x\in\ZP\;\big]$.\vvb

\Def{\num}Une fonction\,\hh$F\::\ \ZP\to\U$\,\hh est un\, \underline{caract\ee re}\, de\: $\ZP$\;
ssi\hh$F\in\,\widehat{\CP}$\vvww et ssi\hh$\al\;x\vv y\in\ZP\,\hh F\,(x+y)=F\,(x)\:F\,(y)$\vw
La fonction constante\,\hh$E\ha_0\,:\,\ZP\ha\to\U\,:\,x\,\tto\,1$\,\hh est clairement un caract\ee re de\,
$\ZP$\vbc

\Th{\num*}$\al\ m\vv n\in\N^*$\hh tels que\,\hh\fbox{\:$p\ndv m\hb\le\,p\ha^n-1$\hb}\,\hh la fonction\vvb
\fbox{\,$E\hb_{m\vv n}\,:\,\hb\ZP\to\U\,:\,x\,\tto\,E\,(\ha m\,x\ha /\ha p\ha^n)
=\,e^{\:2\:\pi\,i\:m\:x\ha/\ha p\ha^n}$\,}\hh est un caract\ee re de\, $\ZP$\vc

\Th{\num*}$E\hb_{m\vv n}=\,E\hb_{m\ha',\:n\ha'}\ \sam\ha\ \big[\:m=\ha m\ha'\,\hh
\mrom{et}\,\hh n=\hw n\ha'\,\big]$\vb

\Rem Nous verrons plus loin que\: $E\ha_0$\: et les\: $E\hb_{m\vv n}$\: sont en fait les seuls
caract\ee res\vvww de\, $\ZP$\vg mais nous ne ferons pas usage de ce r\e sultat pour l'instant\vb

\Th{\num*}Les caract\ee res de\, $\ZP$\; forment un groupe multiplicatif\: $\GP$\vb

\Not$\all z\in\:\C$\ha\hh nous notons\: $z\ha^\#$\, le conjugu\e\ha\ de\, $z$\vb

\Th{\num*}$\all F\in\hb\GP\,\hh\all x\in\ZP$\quad on a\quad$F\ha(-\hb x\ha)
=\ha F\ha(x)\ha^\#=\ha1\ha/\ha F\ha(x)$\vb

\Th{\num*}$\all x\in\ZP$\quad$E\hb_{m\vv n}\,(-\hb x\ha)\ha
=\,E\hb_{m\vv n}\hb(x\ha)\ha^\#=\:E\hb_{p\hb^n-\,m\vv n}\,(x)$\vbc

\Lem{\num}Si deux caract\ee res de\, $\ZP$\, sont proportionnels ils sont \e gaux\vab

\Dem Soient\, $F$\, et\, $G$\, deux caract\ee res de\, $\ZP$\; et soit\, $c\in\U$\:
tel que\hh$G=c\:F$\pv\vvw on a\,\hh$c\ha^2\,F\ha(1)\ha^2=\,G\hb(1)\ha^2=\,G\hb(\ha2\ha)
=\ha c\:F\ha(\ha2\ha)=\ha c\;F\hb(1)\ha^2$\vg donc\hh$c\ha^2=\hb c$\vg donc\hh$c\ha=1$\vbc

\Th{\num}\vva Si\, $\F$\, et\, $\G$\, sont deux caract\ee res\, \underline{distincts}\,
de\, $\ZP$\; on a\,\hh$\Int_{\;\ZP}\hs{-2mm}F\ha(x)\:G\ha(x)\ha^\#\;\del\ha x=\hb0$\vb

\Dem On a\,\hh$\all y\in\ZP\,\hh\Int_{\;\ZP}\hs{-2mm}F\ha(x)\:G\ha(x)\ha^\#
\;\del\ha x=\int_{\;\ZP}\hs{-2mm}F\ha(\hw x+y\hw)\:G\hb(\hw x+y\hw)\ha^\#\;\del\ha x$\vvwa
$=\,F\ha(y)\:G\ha(y)\ha^\#\!\Int_{\;\ZP}\hs{-2mm}F\ha(x)\:G\ha(x)\ha^\#\;\del\ha x$\pv
puisque\hh$\F\ne\G$\vg il existe au moins un\hh$y\in\ZP$\vva tel que\,\hh
$\F\ha(y)\:\G\hb(y)\ha^\#=\,\F\ha(y)\hb/\,\G\hb(y)\,\ne\hb1$\pv donc\hh
$\Int_{\;\ZP}\hs{-2mm}F\ha(x)\:G\ha(x)\ha^\#\;\del\ha x=\hb0$\vbc

\Cor{\num}\vvwa$\al\ m\vv n\in\N^*$\hh tels que\,\hh$p\ndv m\hb\le\,p\ha^n-1$\,\hh
on a\quad\fbox{\,$\Int_{\;\ZP}\hs{-2mm}e^{\:\pm\,2\:\pi\,i\:m\:x\ha/\ha p\ha^n}\hb\del\ha x\ha=\hb0$\,}\ha\vd

\quad\textbf{\Num\underline{S\e ries de Fourier sur\, $\ZP$}}\vvc
\initnum

\Def{\num}Soit\hh$\tmu\in\wh{\MP}$\pv on pose\quad
\fbox{\,$c\hb_0\hb=\Int_{\;\ZP}\hs{-2mm}\tmu\,(x)$\,}\ha\ $\in\:\C$\vvab et\,\hh
$\al\ m\vv n\in\N^*$\hh tels que\,\hh\fbox{\,$p\ndv m\hb\le\,p\ha^n-1$\,}\quad
\fbox{\,$c\hb_{m\vv n}\hb=\Int_{\;\ZP}\hs{-2mm}
e^{\:-2\:\pi\,i\:m\:x\ha/\hb p\ha^n}\hb\tmu\,(x)$\,}\ha\ $\in\:\C$\vbc
Ce sont les\, \underline{coefficients de Fourier}\, de\, $\tmu$\vbc

\Th{\num*}$|\ha c\hb_0\ha|\ha\le\ha\nst{\hb\tmu\hb}$\quad et\quad
$|\ha c\hb_{m\vv n}\ha|\ha\le\ha\nst{\hb\tmu\hb}$\vbc

\newpage
\Def{\num}\vva La s\e rie\quad$c\hb_0\ +\ \sum\pos_{n\,=\,1}^{\infty}\,
\Big(\,\sum\pos_{\sbs{m\,=\,1\\[.5mm]p\sndv m}}^{p\ha^n-1}\hb c\hb_{m\vv n}
\ha\ e^{\:2\:\pi\,i\:m\:x\ha/\ha p\ha^n}\,\Big)$\quad est\, \underline{la s\e rie de Fourier}\, de\, $\tmu$\vwa
On pose\,\hh$\all\el\in\N\,\hh\all v\in\ZP\quad\SZ\hb_\el\,(\ha\tmu\ha)\,(v)\,
=\;c\hb_0\ +\ \sum\pos_{n\,=\,1}^{\el}\,\Big(\,\sum\pos_{\sbs{m\,=\,1\\[.5mm]p\sndv m}}
^{p\ha^n-1}\hb c\hb_{m\vv n}\ha\ e^{\:2\:\pi\,i\:m\:v\ha/\hb p\ha^n}\,\Big)$\pv\vvaab
$\SZ\hb_\el\,(\ha\tmu\ha)\,\in\,\wh{\CP}$\,\hh est la\, \underline{somme partielle}\, d'ordre\, $\el$\,
de la s\e rie de Fourier de\, $\tmu$\vcd

\Th{\num}$\all\el\in\N$\,\hh on a
\begin{picture}(,)
\put(5,0){$\SZ\hb_\el\,(\ha\tmu\ha)\hb=\:\T_\el\,(\ha\tmu\ha)$\hh.}
\put(3,-2.5){\frm(33,7.25)}
\end{picture}\vvbc

\Dem On a\,\hh$\all v\in\ha\ZP\hhh\SZ\hb_\el\,(\ha\tmu\ha)\,(v)\,=\;c\hb_0\
+\ \sum\pos_{n\,=\,1}^{\el}\,\Big(\,\sum\pos_{\sbs{m\,=\,1\\[.5mm]p\sndv m}}
^{p\ha^n-1}\hb c\hb_{m\vv n}\ha\ e^{\:2\:\pi\,i\:m\:v\ha/\ha p\ha^n}\,\Big)$\vva
$=\;\Int_{\;\ZP}\hs{-2mm}\tmu\,(x)\ +\,\ \tex\sum\pos_{n\,=\,1}^{\el}\,
\Big(\,\sum\pos_{\sbs{m\,=\,1\\[.5mm]p\sndv m}}^{p\ha^n-1}\,
\Big[\,\Int_{\;\ZP}\hs{-2mm}e^{\:-2\:\pi\,i\:m\:x\ha/\ha p\ha^n}\,\tmu\,(x)\,\Big]\,
e^{\:2\:\pi\,i\:m\:v\ha/\ha p\ha^n}\,\Big)$\\
$=\;\Int_{\;\ZP}\hs{-2mm}\tmu\,(x)\ +\,\Int_{\;\ZP}\Big[\;\tex\sum\pos_{n\,=\,1}^{\el}\,
\Big(\,\sum\pos_{\sbs{m\,=\,1\\[.5mm]p\sndv m}}^{p\ha^n-1}\,
e^{\:2\:\pi\,i\:m\:(\ha v-x\ha)\ha/\ha p\ha^n}\,\Big)\,\Big]\:\tmu\,(x)$\\
$=\Int_{\;\ZP}\Big[\;\tex\sum\pos_{m\,=\,0}
^{p\ha^\el-1}\,e^{\:2\:\pi\,i\:m\:(\ha v-x\ha)\ha/\ha p\ha^\el}\,\Big]\:\tmu\,(x)$\va

Si\hh$x\in\Bal{v}$\vg\cad\, si\hh$(x-v)\ha/\ha p\ha^\el\in\ZP$\ha\vg\ha on a\quad
$\sum\pos_{m\,=\,0}^{p\ha^\el-1}\:e^{\:2\:\pi\,i\:m\:(\ha v-x\ha)\ha/\ha p\ha^\el}\!=\:p\ha^\el$\pv\vvab
si\hh$x\nin\Bal{v}$\vg\cad\, si\hh$(x-v)\ha/\ha p\ha^\el\nin\ZP$\ha\vg\ha on a\vva
$\sum\pos_{m\,=\,0}^{p\ha^\el-1}\:e^{\:2\:\pi\,i\:m\:(\ha v-x\ha)\ha/\ha p\ha^\el}\!
=\:\Frac{e^{\:2\:\pi\,i\:m\:(\ha v-x\ha)}-1}
{e^{\:2\:\pi\,i\:m\:(\ha v-x\ha)\ha/\ha p\ha^\el}-1}\hb=\:0$\pv\vva
donc\quad$\sum\pos_{m\,=\,0}^{p\ha^\el-1}\:e^{\:2\:\pi\,i\:m\:(\ha v-x\ha)\ha/\ha
p\ha^\el}\!=\;p\ha^\el\:\un\ha_{\bal{v}}\hb(x)$\ha\vg donc\vva$\SZ\hb_\el\,(\ha\tmu\ha)\,(v)\,
=\;p\ha^\el\!\Int_{\;\ZP}\hs{-2mm}\un\ha_{\bal{v}}\hb(x)\;\tmu\,(x)\ha=\,\T_\el\,(\ha\tmu\ha)\,(v)$\vc

On en d\e duit le corollaire suivant :\vvb

\Cor{\num*}$\all f\in\wh{\CP}$\quad on a\quad$\SZ\hb_\el\,(\ha f\ha)\,\uu f$\quad quand\quad$\el\to\pin$\vab
\hs{34mm}$\all\tdf\in\wh{\LP^1}$\quad on a\quad$\SZ\hb_\el\,(\ha\tdf\ha)\,
\one\tdf$\quad quand\quad$\el\to\pin$\vab
\hs{34mm}$\all\tdf\in\wh{\LP^2}$\quad on a\quad$\SZ\hb_\el\,(\ha\tdf\ha)\,
\two\tdf$\quad quand\quad$\el\to\pin$\vb

\oui Remarquons que\vg contrairement aux s\e ries de Fourier classiques\ha\ sur\, $\R\ha/\ha2\,\pi\hb\Z$\ha\vg
les r\e sultats obtenus sont les plus favorables possible\vbc

\num\ \underline{Relation de Parseval}\dd $\all\tdf\in\wh{\LP^2}$\,\hh on a\vvc

\begin{picture}(,)
\put(30,0){$|\ha c\hb_0\ha|\ha^2\,+\,\sum\pos\pos_{n\,=\,1}^{\infty}\,\Big(\,\sum
\pos_{\sbs{m\,=\,1\\[.5mm]p\sndv m}}^{p\ha^n-1}\hb|\ha c\hb_{m\vv n}\ha|\ha^2\,\Big)\,
=\hb\Int_{\;\ZP}\hs{-2mm}|\ha\tdf\ha|\hb^2\hb(x)\ \del\ha x$}
\put(27,-9){\frm(91.25,17.5)}
\end{picture}\vvd

\Dem\vvab$\Int_{\;\ZP}\hs{-2mm}\tdf\ha(x)\ha^\#\:\SZ\hb_\el\,(\ha\tmu\ha)\,(x)\;\del\ha x\,
=\ c\hb_0\int_{\;\ZP}\hs{-2mm}f\ha(x)\ha^\#\;\del\ha x\
+\ \tex\sum\pos_{n\,=\,1}^{\el}\,\sum\pos_{\sbs{m\,=\,1\\[.5mm]p\sndv m}}
^{p\ha^n-1}\hb c\hb_{m\vv n}\,\Big[\,\Int_{\;\ZP}\hs{-2mm}f\ha(x)\ha^\#\,
e^{\:2\:\pi\,i\:m\:x\ha/\hb p\ha^n}\,\del\ha x\,\Big]$\vvwa
\hs{43.5mm}$=\ |\ha c\hb_0\ha|\ha^2\,+\,\sum\pos\pos_{n\,=\,1}^{\el}\,
\Big(\,\sum\pos_{\sbs{m\,=\,1\\[.5mm]p\sndv m}}^{p\ha^n-1}\hb|\ha c\hb_{m\vv n}\ha
|\ha^2\,\Big)$\pv\vvwa de plus\,\hh$\SZ\hb_\el\,(\ha\tmu\ha)\,\two\tdf$\vg donc\,\hh
$\lim\pos_{\el\to\pin}\,\Int_{\;\ZP}\hs{-2mm}\tdf\ha(x)\ha^\#\:\SZ\hb_\el\,(\ha\tmu\ha)\,(x)\;\del\ha x\,
=\Int_{\;\ZP}\hs{-2mm}|\ha\tdf\ha|\hb^2\hb(x)\;\del\ha x$\vc

\Cor{\num}Les seuls caract\ee res de\, $\ZP$\, sont la fonction\: $E\ha_0$\: et les fonctions\hh
$E\hb_{m\vv n}$\,\hh avec\,\hh$m\vv n\in\N^*$\hh et\,\hh$p\ndv m\hb\le\,p\ha^n-1$\vb

\Dem Soit\, $\F\in\GP$\, distinct des caract\ee res \e nonc\e s ci-dessus\pv on a\,\hh
$\F\in\,\wh{\CP}\,\sub\,\wh{\LP^2}$\pv\vvw or tous les coefficients de Fourier de\, $\F$\, sont nuls car\,
$\F$\, est orthogonal\, \ac\ tous les autres\vvw caract\ee res de\, $\ZP$\pv on en d\e duit\:
$\Int_{\;\ZP}\hs{-2mm}|\ha\F\ha|\ha^2\hb(x)\ \del\ha x\ha=\,0$\hb\vg donc\hh$\F=\hb0$\vc

\Th{\num}$\all\tmu\in\wh{\MP}\,\quad\all h\in\wh{\CP}$\,\quad on a\vvc

\begin{picture}(,)
\put(10,0){$c\hb_0\Int_{\;\ZP}\hs{-2mm}h\hb(v)\;\del\,(v)\;
+\;\tex\sum\pos_{n\,=\,1}^{\infty}\,\Big[\,\sum\pos_{\sbs{m\,=\,1\\[.5mm]p\sndv m}}
^{p\ha^n-1}\ha c\hb_{m\vv n}\Int_{\;\ZP}\hs{-2mm}h\hb(v)\
e^{\:2\:\pi\,i\:m\:v\ha/\ha p\ha^n}\hb\del\,(v)\,\Big]\,=\ha\int_{\;\ZP}\hs{-2mm}h\hb(x)\;\tmu\,(x)$}
\put(7.5,-9.5){\frm(147.5,18.5)}
\end{picture}\\\vvc

\Dem On a\,\hh$\all\el\in\N^*$\vvab
$c\hb_0\Int_{\;\ZP}\hs{-2mm}h\hb(v)\;\del\,(x)\;+\;\tex\sum\pos_{n\,=\,1}^{\el}\,
\Big[\,\sum\pos_{\sbs{m\,=\,1\\[.5mm]p\sndv m}}^{p\ha^n-1}\ha c\hb_{m\vv n}\dis
\int_{\;\ZP}\hs{-2mm}h\hb(v)\ e^{\:2\:\pi\,i\:m\:v\ha/\ha p\ha^n}\hb\del\,(v)\,\Big]\,
=\,\Int_{\;\ZP}\hs{-2mm}h\hb(v)\;\SZ\hb_\el\,(\ha\tmu\ha)\hb(v)\;\del\,(v)$\vvwa
$=\,\Int_{\;\ZP}\hs{-2mm}h\hb(v)\;\T_\el\,(\ha\tmu\ha)\,(v)\;\del\,(v)\,\to
\int_{\;\ZP}\hs{-2mm}h\hb(x)\;\tmu\,(x)$\quad quand\quad$\el\to\pin$\vc

\Cor{\num}\vvwa Soit\ha\hh$\tmu\in\wh{\MP}$\,\hh tel que tous ses coefficients de Fourier sont nuls\pv
alors\hh\hb$\tmu\ha=\ha0$\vb

\Dem Si tous les coefficients de Fourier de\, $\tmu$\, sont nuls\vg alors\,\hh$\all h\in\EP$\vva
$\Int_{\;\ZP}\hs{-2mm}h\hb(x)\;\tmu\,(x)=\ha0$\pv donc par d\e finition\ha\hh$\tmu\ha=\ha0$\vd

\newpage
\quad\textbf{\Num\underline{Exemples de s\e ries de Fourier sur\, $\ZP$}}\vvc
\initnum

\underline{Exemple\, 1}$^\bul$\hhh$f_s\,:\ \ZP\,\to\,[\hb0\,,1\ha]\ :\ x\,
\tto\,|\ha x\ha|\ha_p^{\ha s}$\hhh avec\,\hh$s\ha\in\ha\R_{\,\star}^{\ha+}$\vc

\underline{\emph{Calcul de\hb\ $c\hb_0$}}\dd\vvab

$c\hb_0\,=\Int_{\;\ZP}\hs{-2mm}|\ha x\ha|\ha_p^{\ha s}\,\ \del\,(x)\,
=\,\tex\sum\pos_{r\,=\,0}^{\infty}\ \Int_{\;p\ha^r\:\ZP\:-\:p\ha^{r+1}\:\ZP}
\hs{-2mm}|\ha x\ha|\ha_p^{\ha s}\,\ \del\,(x)\,=\,\tex\sum\pos_{r\,=\,1}^{\infty}\ \Inv{p\hb^{r\ha s}}
\int_{\;p\ha^r\:\ZP\:-\:p\ha^{r+1}\:\ZP}\hs{-2mm}\del\,(x)$\vvb
$=\,\tex\sum\pos_{r\,=\,0}^{\infty}\ \Inv{p\hb^{r\ha s}}\ \Big(\inv{p\hb^r}
-\inv{p\hb^{r+1}}\Big)\,=\ha\ \frac{p-1}{p}\ \tex\sum\pos_{r\,=\,0}^{\infty}\
\Inv{p\,^{r\,(\ha s+1\hw)}}\ =\ha\ \Frac{(\ha p-1\hw)\;p\,^s}{p\,^{s+1}-1}$\vcd

\underline{\emph{Calcul de\hb\ $c\hb_{m\vv n}$}}\dd\vvab

$c\hb_{m\vv n}\hb=\Int_{\;\ZP}\hs{-2mm}|\ha x\ha|\ha_p^{\ha s}\,\
e^{\:-2\:\pi\,i\:m\:x\ha/\ha p\ha^n}\:\del\,(x)\,=\,\tex\sum\pos_{r\,=\,0}^{\infty}\
\Int_{\;p\ha^r\:\ZP\:-\:p\ha^{r+1}\:\ZP}\hs{-2mm}|\ha x\ha|\ha_p^{\ha s}\,\
e^{\:-2\:\pi\,i\:m\:x\ha/\ha p\ha^n}\:\del\,(x)$\vvb
$=\,\tex\sum\pos_{r\,=\,0}^{\infty}\ \Inv{p\hb^{r\ha s}}\int_{\;p\ha^r\:\ZP\:
-\:p\ha^{r+1}\:\ZP}\hs{-15mm}e^{\:-2\:\pi\,i\:m\:x\ha/\ha p\ha^n}\:\del\,(x)$\vvb
$=\,\tex\sum\pos_{r\,=\,0}^{\infty}\ \Inv{p\hb^{r\ha s}}\ \Big[\ \inv{p\ha^r}
\int_{\ZP}e^{\:-2\:\pi\,i\:m\:x\ha/\ha p\hb^{n-r}}\:\del\,(x)\,-\,\inv{p\ha^{r+1}}
\int_{\ZP}e^{\:-2\:\pi\,i\:m\:x\ha/\ha p\hb^{n-r-1}}\:\del\,(x)\:\Big]$\vvab
$=\,\tex\sum\pos_{r\,=\,n-1}^{\infty}\ \Inv{p\hb^{r\ha s}}\ \Big[\ \inv{p\ha^r}
\int_{\ZP}e^{\:-2\:\pi\,i\:m\:x\ha/\ha p\hb^{n-r}}\:\del\,(x)\,-\,\inv{p\ha^{r+1}}
\int_{\ZP}e^{\:-2\:\pi\,i\:m\:x\ha/\ha p\hb^{n-r-1}}\:\del\,(x)\:\Big]$\vvb

$=-\:\Inv{p\,^{(\ha n-1\hw)\,s}}\ \inv{p\hb^n}\
+\,\tex\sum\pos_{r\,=\,n}^{\infty}\ \Inv{p\hb^{r\hb s}}\ \Big(\ \inv{p\ha^r}
-\,\inv{p\ha^{r+1}}\Big)=-\:\Inv{p\,^{n\,(\ha s+1\hw)\:-\:s}}\,+\
\frac{p-1}{p}\,\tex\sum\pos_{r\,=\,n}^{\infty}\ \Inv{p\hb^{r\,(\ha\s+1\hw)}}$\vvb

$=\,-\:\Inv{p\,^{n\,(\ha s+1\hw)}}\ \Big[\ p\hb^s-\ \frac{p-1}{p}\,\tex\sum
\pos_{r\,=\,0}^{\infty}\ \Inv{p\hb^{r\,(\ha s+1\hw)}}\:\Big]\,
=\,-\:\Inv{p\,^{n\,(\ha s+1\hw)}}\ \Big[\ p\hb^s-\ \frac{p-1}{p}\,\frac{p\,^{\s+1}}
{p\,^{s+1}-1}\:\Big]$\vvb$=\,-\:\Frac{p\,^s}{p\,^{n\,(\ha s+1\hw)}}\
\Big[\:1-\ \frac{p-1}{p\,^{s+1}-1}\:\Big]\,
=\,-\:\Frac{p\,^s}{p\,^{n\,(\ha s+1\hw)}}\hh\frac{p\,^{s+1}-p}{p\,^{s+1}-1}\,
=\,-\:\Inv{p\,^{(\ha n-1\hw)\,(\ha s+1\hw)}}\hh\frac{p\,^s-1}{p\,^{s+1}-1}$\vcd

Comme\hh$f_s\in\CP$\hh on peut \e crire\hh$\all x\in\ZP$\vvb
\hs{10mm}\fbox{\;$|\ha x\ha|\ha_p^{\ha s}\ =\ \Frac{(\ha p-1\hw)\;p\,^s}
{p\,^{s+1}-1}\,-\ \frac{p\,^s-1}{p\,^{\s+1}-1}\,\ \dis\sum\pos_{n\,=\,1}^{\infty}\
\inv{p\,^{(\ha n-1\hw)\,(\ha s+1\hw)}}\ \sum\pos_{\sbs{m\,=\,1\\[.5mm]
p\sndv m}}^{p\ha^n-1}e^{\:2\:\pi\,i\:m\:x\ha/\ha p\ha^n}$\,}\vvbc

Puisque\, $f_s$\, est une fonction r\e elle on en d\e duit\, :\vvb

\quad$(\ha p\,^{s+1}-1\hw)\;|\ha x\ha|\ha_p^{\ha s}\
=\ (\ha p-1\hw)\;p\,^s\hb-\;(\ha p\,^s-1\hw)\,\dis\sum\pos_{n\,=\,1}^{\infty}\
\inv{p\,^{(\ha n-1\hw)\,(\ha\s+1\hw)}}\ \sum\pos_{\sbs{m\,=\,1
\\[.5mm]p\sndv m}}^{p\ha^n-1}\cos\:(\ha2\:\pi\,m\,x\hw/\hw p\ha^n)$\pv\vvab

en particulier pour\, $s=1$\, on obtient\, :\vvb

\hs{35mm}\fbox{\:$(\ha p+1)\;|\ha x\ha|\ha_p\,=\ p\,-\dis\sum\pos_{n\,=\,1}^{\infty}\
\inv{p\,^{2\:(\ha n-1\hw)}}\ \sum\pos_{\sbs{m\,=\,1\\[.5mm]p\sndv m}}
^{p\ha^n-1}\cos\:(\ha2\:\pi\,m\,x\hw/\hw p\ha^n)$\,}\vvb

Par ailleurs on a\,\hh$\all n\ge2$\vvb\hs{10mm}$\sum\pos_{\sbs{m\,=\,1\\[.5mm]
p\sndv m}}^{p\ha^n-1}\cos\:(\ha2\:\pi\,m\,x\hw/\ha p\ha^n)\
=\,\sum\pos_{m\,=\,1}^{p\ha^n-1}\cos\:(\ha2\:\pi\,m\,x\hw/\hw p\ha^n)\,
-\sum\pos_{m\,=\,1}^{p\ha^{n-1}-1}\cos\:(\ha2\:\pi\,m\,x\hw/\hw p\ha^{n-1})$\vbc

ce qui permet d'\e crire\vvb
\hs{35mm}\fbox{\:$|\ha x\ha|\ha_p\,=\;\Frac{p}{p+1}\,-\,(\ha p\ha-1)\,
\sum\pos_{n\,=\,1}^{\infty}\ \inv{p\hb^{2\:n}}\ \sum\pos_{m\,=\,1}^{p\ha^n-1}
\cos\:(\ha2\:\pi\,m\,x\hw/\hw p\ha^n)$\hb}\vvc

Remarquons que ces s\e ries n'ont qu'un nombre fini de termes non nuls\pv les termes\vvw d'indice\:
$n>\ha k+1$\; sont nuls quand\hh$|\ha x\ha|\ha_p\ha=\ha\Inv{p\ha^k}$\vb

\underline{Exemple\, 2}\,$^\bul$\vva

On d\e finit la fonction\,\hh$\sig\ha\ :\,\ \ZP\,\to\,[\hb0\,,1\ha]$\,\hh de la mani\ee re suivante\dd\vvwa
si\quad\underline{$\ldots\,d\ha_k\;d\ha_{k\,-1}\,\ldots\,d\ha_2\ d\hw_1$}\quad est le d\e veloppement p-adique
de\, $x\in\ZP$\vg on pose\vva
$\sig\,(x)\ha=\:0\vv\underline{d\hw_1\:d\ha_2\,\ldots\,d\ha_{k\,-1}\:d\ha_k\,\ldots}$\quad
que l'on consid\ee re comme le d\e veloppement p-adique\vvwa
d'un nombre\, \underline{r\e el}\hb\ $\in\hb[\hb0\,,1\ha]$\pv autrement dit si\,\hh
$x\ha=\sum\pos_{k\,=\,1}^{\infty}d\ha_k\;p\hb^{k\,-1}\in\ha\ZP$\hb\vg\ha avec\,\hh
$\all k\in\N^*$\\$d\ha_k\in\db{\,0\vv p-\hs{-.5mm}1\,}$\ha\vg\ha on pose\,\hh
$\sig\,(x)\ha=\sum\pos_{k\,=\,1}^{\infty}d\ha_k\hb/\hb p\ha^k\ha\in\hb[\hb0\,,1\ha]$\vbc

\underline{\emph{Calcul de\hb\ $c\hb_0$}}\dd\vvab

$c\hb_0\,=\Int_{\;\ZP}\hs{-2mm}\sig\,(x)\ \del\,(x)\,=\lim\pos_{\el\to\pin}\
\Inv{p\hb^\el}\ \tex\sum\pos_{r\,=\,0}^{p\hb^\el-1}\,\sig\,(r)$\pv on a\,\hh
$\all\el\in\N$\vva$\sum\pos_{r\,=\,0}^{p\hb^{\el}-1}\,\sig\,(r)\,
=\ha\sum\pos_{d\hw_1\,=\,0}^{p\,-1}\,\sum\pos_{d\ha_2\,=\,0}^{p\,-1}\,\ldots\,
\sum\pos_{d\ha_\el\,=\,0}^{p\,-1}\,\Big(\,\Frac{d\hw_1}{p}\,+\,\frac{d\ha_2}{p\hb^2}\,
+\,\ldots\,+\,\frac{d\ha_\el}{p\ha^\el}\,\Big)$\vva
$=\ \Inv{p\hb^\el}\ \tex\sum\pos_{d\hw_1\,=\,0}^{p\,-1}\,\sum\pos_{d\ha_2\,
=\,0}^{p\,-1}\,\ldots\,\sum\pos_{c\hb_\el\,=\,0}^{p\,-1}\,\Big(\,d\ha_\el+\,d\ha_{\el-1}\;p
+\,\ldots\,+\,d\ha_2\;p\hb^{\el-2}+\,d\hw_1\;p\hb^{\el-1}\,\Big)$\vva
$=\ \Inv{p\hb^\el}\ \tex\sum\pos_{r\,=\,0}^{p\ha^\el-1}\ha r\:
=\hb\ \Frac{p\hb^\el-1}{2}$\pv\hb donc\quad$c\hb_0\,=\lim\pos_{\el\to\pin}\
\Frac{p\hb^\el-1}{2\;p\hb^\el}\,=\;\inv{2}$\vc

\underline{\emph{Lemme}}\dd$\all m\in\Z$\,\hh tel que\,\hh\fbox{\ha$\,p\ndv m$\,}\,\hh
on a\quad1\ha)\hhh$\dis\sum\pos_{k\,=\,0}^{p\,-1}\ e^{\;2\:\pi\,i\:m\:k\ha/\ha p}\ha=\,0$\vvab
\hs{86.5mm}2\ha)\hhh$\dis\sum\pos_{k\,=\,0}^{p\,-1}\,k\;e^{\;2\:\pi\,i\:m\:k\ha/\ha p}\ha
=\,\frac{p}{e^{\;2\:\pi\,i\:m\ha/\ha p}-1}$\vb

\Dem On a\,\hh$\all u\in\hb\C\ha^*\hh\sum\pos_{k\,=\,0}^{p\,-1}\,e^{\:k\:u}\ha
=\,\Frac{e^{\:p\;u}-1}{e^{\:u}-1}$\ha\pv\ha en d\e rivant par rapport \ac\, $u$\,\vva
on trouve\,\hh$\sum\pos_{k\,=\,0}^{p\,-1}\,k\;e^{\:k\:u}\ha
=\,\Frac{p\;e^{\:p\;u}}{e^{\:u}-1}\,-\;e^{\:u}\:\frac{e^{\:p\;u}-1}{(\ha e^{\:u}-1\hw)\hb^2}$\ha\pv\ha
en posant\,\hh$u=\ha2\:\pi\,i\:m\ha/\ha p$\vvb on obtient les deux r\e sultats\vc

\underline{\emph{Calcul de\hb\ $c\hb_{m\vv n}$}}\dd\vvab

Nous calculons\; $c\hb_{m\vv 3}$\hb\vg le cas g\e n\e ral s'en d\e duisant ais\e ment\vb

$c\hb_{m\vv 3}\:=\Int_{\;\ZP}\hs{-2mm}\sig\,(x)\ e^{\:-\,2\:\pi\,i\:m\:x\ha/
\ha p\ha^3}\ \del\,(x)\ =\lim\pos_{\el\to\pin}\ \Inv{p\hb^\el}\
\tex\sum\pos_{r\,=\,0}^{p\ha^\el-1}\,\sig\,(r)\ e^{\:-\,2\:\pi\,i\:m\;r\ha/\ha
p\ha^3}$\hb;\vvb on a\,\hh$\all\el\hw\ge\hw4\quad
\sum\pos_{r\,=\,0}^{p\ha^\el-1}\,\sig\,(r)\ e^{\:-\,2\:\pi\,i\:m\;r\ha/\ha p\ha^3}$\vvab
$=\,\sum\pos_{d\hw_1\,=\,0}^{p\,-1}\,\sum\pos_{d\ha_2\,=\,0}^{p\,-1}\,\sum\pos_{d\ha_3\,=\,0}^{p\,-1}\,\ldots\,
\sum\pos_{d\ha_\el\,=\,0}^{p\,-1}\,\Big(\,\Frac{d\hw_1}{p}\,+\,\frac{d\ha_2}{p\hb^2}\,
+\,\frac{d\ha_3}{p\hb^3}\,+\,\ldots\,+\,\frac{d\ha_\el}{p\hb^\el}\,\Big)$\vva
\hs{75mm}$\tim\ \exp\,\big\{\!-\ha2\;\pi\,i\;\Frac{m}{p\hb^3}\ \big(\ha d\hw_1
+d\ha_2\;p+\,\ldots\,+d\ha_\el\;p\ha^{\el-1}\ha\big)\ha\big\}$\vvab

$=\,\sum\pos_{d\hw_1\,=\,0}^{p\,-1}\,\sum\pos_{d\ha_2\,=\,0}^{p\,-1}\,\sum\pos_{d\ha_3\,=\,0}^{p\,-1}\,\ldots\,
\sum\pos_{d\ha_\el\,=\,0}^{p\,-1}\,\Big(\,\Frac{d\hw_1}{p}\,+\,\frac{d\ha_2}
{p\ha^2}\,+\,\frac{d\ha_3}{p\ha^3}\,+\,\ldots\,+\,\frac{c\hb_\el}{p\hb^\el}\,\Big)\
\exp\ \{-\,2\;\pi\,i\;m\ \frac{d\hw_1}{p\hb^3}\,\}$\vva
\hs{25mm}$\tim\ \exp\ \{-\,2\;\pi\,i\;m\ \Frac{d\ha_2}{p\hb^2}\,\}\
\exp\ \{-\,2\;\pi\,i\;m\ \frac{d\ha_3}{p}\,\}\,\ldots\ha\
\exp\ \{-\,2\;\pi\,i\;m\ d\ha_n\ p\,^{\el\,-\ha4}\,\}$\vvab

$=\,\sum\pos_{d\hw_1\,=\,0}^{p\,-1}\,\sum\pos_{d\ha_2\,=\,0}^{p\,-1}\,\sum\pos_{d\ha_3\,=\,0}^{p\,-1}\,\ldots\,
\sum\pos_{d\ha_\el\,=\,0}^{p\,-1}\,\Big(\,\Frac{d\hw_1}{p}\,+\,\frac{d\ha_2}
{p\ha^2}\,+\,\frac{d\ha_3}{p\ha^3}\,+\,\ldots\,+\,\frac{d\ha_\el}{p\hb^\el}\,\Big)$\vvab
\hs{35mm}$\tim\ \exp\ \{-\,2\;\pi\,i\;m\ \Frac{d\hw_1}{p\hb^3}\,\}\
\exp\ \{-\,2\;\pi\,i\;m\ \frac{d\ha_2}{p\hb^2}\,\}\ \exp\ \{-\,2\;\pi\,i\;m\ \frac{d\ha_3}{p}\,\}$\pv\vvab

or\,\hh$\sum\pos_{d\hw_3\,=\,0}^{p\,-1}\ \exp\ \{-\,2\;\pi\,i\;m\ \Frac{d\ha_3}{p}\,\}\,=\ 0$\ha\vg\ha
donc l'expression ci-dessus peut s'\e crire\vvab
$\sum\pos_{d\hw_1\,=\,0}^{p\,-1}\,\sum\pos_{d\ha_2\,=\,1}^{p\,-1}\,\sum\pos_{d\ha_3\,=\,0}^{p\,-1}\,\ldots\,
\sum\pos_{d\ha_\el\,=\,0}^{p\,-1}\ \Frac{d\ha_3}{p\ha^3}\hb\
\exp\;\{-\,2\;\pi\,i\;m\ \frac{d\hw_1}{p\hb^3}\,\}\ \exp\;\{-\,2\;\pi\,i\;m\
\frac{d\ha_2}{p\hb^2}\,\}\ \exp\;\{-\,2\;\pi\,i\;m\ \Frac{d\ha_3}{p}\,\}$\vvab

$=\ p\hb^{\el\,-\,6}\;\sum\pos_{d\hw_1\,=\,0}^{p\,-1}\,\sum\pos_{d\ha_2\,=\,0}^{p\,-1}\,\sum\pos_{d\ha_3\,=\,0}^{p\,-1}\:d\ha_3\ha\
\exp\;\{-\,2\;\pi\,i\;m\ \Frac{d\hw_1}{p\hb^3}\,\}\ \exp\;\{-\,2\;\pi\,i\;m\
\frac{d\ha_2}{p\hb^2}\,\}\ \exp\;\{-\,2\;\pi\,i\;m\ \frac{d\ha_3}{p}\,\}$\pv\vvab

donc\,\hh\mbox{$c\hb_{m\vv 3}\hb=\hb\Inv{p\ha^6}\,\ \frac{e^{\:-\,2\;\pi\,i\;m\ha/\hb p\hb^2}-1}{e^{\:-\,2\;\pi\,i\;m\ha/\hb p\ha^3}-1}\,\
\frac{e^{\:-\,2\;\pi\,i\;m\ha/\hb p}-1}{e^{\:-\,2\;\pi\,i\;m\ha/\hb p\hb^2}-1}\,\ \frac{p}{e^{\:-\,2\;\pi\,i\;m\ha/\hb p}-1}\hb
=\hb\Inv{p\ha^5}\,\ \Inv{e^{\:-\,2\;\pi\,i\;m\ha/\ha p\hb^3}-1}$}\vvc

En g\e n\e ralisant on trouve donc
\begin{picture}(,)
\put(4,0){$\,c\hb_{m\vv n}\,=\:\Inv{p\,^{2\:n\,-1}}\,\ \Inv{e^{\:-\,2\;\pi\,i\;m\ha/\ha p\hb^n}-1}$\,\ \hh;}
\put(2.5,-5.75){\frm(63.5,13.25)}
\end{picture}\vvcd

or\,\hh$\all\alf\in\R\quad\Inv{e^{\:-\,i\:\alf}-1}\ =\,-\:\inv{2}\;+\;\frac{i}{2}\;\cot\Big(\frac{\alf}{2}\Big)$\vg\vvb
donc\quad$c\hb_{m\vv n}\:=\ \Inv{2\ p\,^{2\:n\,-1}}\,\Big[\hs{-.5mm}-\hs{-.2mm}1+\,i\,\cot\Big(\frac{\pi\:m}{p\hb^n}\Big)\ha\Big]$\vc

Comme\hh$\sig\in\CP$\hh on peut \e crire\hh$\all x\in\ZP$\vvb

\hs{10mm}\fbox{\;$2\;\sig\,(x)\,=\ 1\:+\:\dis\sum\pos_{n\,=\,1}^{\infty}\ \inv{p\,^{2\;n\,-1}}\ \sum\pos_{\sbs{m\,=\,1\\[.5mm]p\sndv m}}^{p\ha^n-1}
\Big[\hs{-.5mm}-\hs{-.2mm}1+\,i\,\cot\Big(\frac{\ha m\:\pi}{p\hb^n}\hb\Big)\ha\Big]\ e^{\:2\:\pi\,i\:m\:x\ha/\ha p\ha^n}$\:}\vvbc

Puisque\, $\sig$\, est une fonction r\e elle on en d\e duit\, :\vvb

\quad$2\;\sig\,(x)\,=\ 1-\,\dis\sum\pos_{n\,=\,1}^{\infty}\ \inv{p\,^{2\;n\,-1}}\,\sum\pos_{\sbs{m\,=\,1\\[.5mm]
p\sndv m}}^{p\ha^n-1}\,\Big[\cos\:(\ha2\:\pi\,m\,x\hw/\hw p\ha^n\ha)\ha+\,\cot\Big(\frac{\ha m\:\pi}{p\hb^n}\hb\Big)\ha
\sin\:(\ha2\:\pi\,m\,x\hw/\hw p\ha^n\ha)\ha\Big]$\vab

Par ailleurs on sait que\vvab
\hs{20mm}$1-\,\dis\sum\pos_{n\,=\,1}^{\infty}\ \inv{p\,^{2\;n\,-1}}\,\sum\pos_{\sbs{m\,=\,1\\[.5mm]
p\sndv m}}^{p\ha^n-1}\,\cos\:(\ha2\:\pi\,m\,x\hw/\hw p\ha^n\ha)\,=\hb\ \Frac{p+1}{p}\ |\ha x\ha|\ha_p$\pv\vvab

on obtient donc\vvbc

\hs{10mm}\fbox{\;$2\;\sig\,(x)\,=\ha\ \Frac{p+1}{p}\ |\ha x\ha|\ha_p\
-\:\sum\pos_{n\,=\,1}^{\infty}\ \inv{p\,^{2\;n\,-1}}\,\sum\pos_{\sbs{m\,=\,1\\[.5mm]
p\sndv m}}^{p\ha^n-1}\hb\cot\Big(\frac{\ha m\:\pi}{p\hb^n}\hb\Big)\hb
\sin\:(\ha2\:\pi\,m\,x\hw/\hw p\ha^n\ha)$\:}\ha\vg\vvbc

ce qui montre d'ailleurs que\,\hh$\Frac{p+1}{2\;p}\ |\ha x\ha|\ha_p$\,\hh
est la partie\, \underline{paire}\, de\, $\sig\,(x)$\vbc

Par ailleurs on a\,\hh$\all n\ge2\hhh\dis\sum\pos_{\sbs{m\,=\,1\\[.5mm]p\sndv m}}
^{p\ha^n-1}\cot\Big(\frac{\ha m\:\pi}{p\hb^n}\hb\Big)\sin\:(\ha2\:\pi\,m\,x\hw/\ha p\ha^n)$\\
$=\ \dis\sum\pos_{m\,=\,1}^{p\ha^n-1}\cot\Big(\frac{\ha m\:\pi}{p\hb^n}\hb\Big)\sin\:
(\ha2\:\pi\,m\,x\hw/\hw p\ha^n)\,-\!\sum\pos_{m\,=\,1}^{p\ha^{n-1}-1}\cot
\Big(\frac{\ha m\:\pi}{p\hb^{n\hb-1}}\hb\Big)\sin\:(\ha2\:\pi\,m\,x\hw/\hw p\ha^{n-1})$\vb

ce qui permet d'\e crire\vvb

\hs{10mm}\fbox{\;$2\ \Frac{p}{p+1}\ \sig\,(x)\,=\:|\ha x\ha|\ha_p\,
-\:(\ha p\ha-1)\:\sum\pos_{n\,=\,1}^{\infty}\ \inv{p\hb^{2\:n}}\,\sum\pos_{m\,=\,1}
^{p\ha^n-1}\hb\cot\Big(\frac{\ha m\:\pi}{p\hb^n}\hb\Big)\ha\sin\:(\ha2\:\pi\,m\,x\hw/\hw p\ha^n\ha)$\:}\ha\vd

\newpage
\underline{Exemple\, 3}\,$^\bul$\dd\underline{g\e n\e ralisation de l'exemple\, 2}\,$^\bul$\vva

On d\e finit la fonction\,\hh$\tau\ha\ :\,\ \ZP\,\to\:\C$\,\hh de la mani\ee re suivante\dd\vva
si\,\hh$x\ha=\sum\pos_{k\,=\,1}^{\infty}d\ha_k\:p\hb^{k\hb-1}$\hh avec\,\hh
$\all k\in\N\,\hh c\hb_k\in\db{\,0\vv p-\hs{-.5mm}1\,}$\ha\vg\ha on pose\,\hh
$\tau\,(x)\ha=\sum\pos_{k\,=\,1}^{\infty}d\ha_k\:\lam^{k\hb-1}$\vva avec\hh$\lam\in\,\C$\,\hh
et\,\hh$|\ha\lam\ha|<1$\vbc

\underline{\emph{Calcul de\hb\ $c\hb_0$}}\dd\vvab

$c\hb_0\,=\Int_{\;\ZP}\hs{-2mm}\tau\,(x)\ \del\,(x)\,=\lim\pos_{\el\to\pin}\
\Inv{p\hb^\el}\ \tex\sum\pos_{r\,=\,0}^{p\ha^\el-1}\,\tau\,(r)$\pv on a\,\hh
$\all\el\in\N$\vva$\sum\pos_{r\,=\,0}^{p\ha^\el-1}\,\tau\,(r)\,
=\ha\sum\pos_{d\hw_1\,=\,0}^{p\,-1}\,\sum\pos_{d\ha_2\,=\,0}^{p\,-1}\,\ldots\,
\sum\pos_{d\ha_\el\,=\,0}^{p\,-1}\,\Big(\ha d\hw_1\,+\,d\ha_2\,\lam
+\,\ldots\,+\,d\ha_\el\,\lam^{\el\ha-1}\Big)$\vva
$=\,\Inv{2}\ p\hb^\el\,(\ha p-1)\ \frac{1-\lam^\el}{1-\lam}$\pv
donc\quad$c\hb_0\,=\lim\pos_{n\to\pin}\ \Inv{2}\ (\ha p-1)\
\frac{1-\lam^\el}{1-\lam}\ha=\;\inv{2}\,\ \frac{p-1}{1-\lam}$\vc

\underline{\emph{Calcul de\hb\ $c\hb_{m\vv n}$}}\dd\vvab

Nous calculons\; $c\hb_{m\vv 3}$\hb\vg le cas g\e n\e ral s'en d\e duisant ais\e ment\vb

$c\hb_{m\vv 3}\:=\Int_{\;\ZP}\hs{-2mm}\tau\,(x)\ e^{\:-\,2\:\pi\,i\:m\:x\ha/
\ha p\ha^3}\ \del\,(x)\ =\lim\pos_{\el\to\pin}\ \Inv{p\hb^\el}\
\tex\sum\pos_{r\,=\,0}^{p\ha^\el-1}\,\tau\,(r)\ e^{\:-\,2\:\pi\,i\:m\;r\ha/\ha
p\ha^3}$\hb;\vvb on a\,\hh$\all\el\hw\ge\hw4\quad
\sum\pos_{r\,=\,0}^{p\ha^\el-1}\,\tau\,(r)\ e^{\:-\,2\:\pi\,i\:m\;r\ha/\ha p\ha^3}$\vvab
$=\,\sum\pos_{d\hw_1\,=\,0}^{p\,-1}\,\sum\pos_{d\ha_2\,=\,0}^{p\,-1}\,
\sum\pos_{d\ha_3\,=\,0}^{p\,-1}\,\ldots\,\sum\pos_{d\ha_\el\,=\,0}^{p\,-1}\,
\Big(\,d\hw_1\,+\,d\ha_2\,\lam\,+\,{d\ha_3}\,\lam^2\,+\,\ldots\,+\,d\hb_\el\,\lam^{\el\ha-1}\,\Big)$\vva
\hs{75mm}$\tim\ \exp\,\big\{\!-\ha2\;\pi\,i\;\Frac{m}{p\hb^3}\ \big(\ha d\hw_1
+d\ha_2\;p+\,\ldots\,+d\ha_\el\;p\hb^{\el\ha-1}\ha\big)\ha\big\}$\vvab

$=\,\sum\pos_{d\hw_1\,=\,0}^{p\,-1}\,\sum\pos_{d\ha_2\,=\,0}^{p\,-1}\,\sum\pos_{d\ha_3\,=\,0}^{p\,-1}\,\ldots\,
\sum\pos_{d\ha_\el\,=\,0}^{p\,-1}\,\Big(\,d\hw_1\,+\,d\ha_2\,\lam\,
+\,{d\ha_3}\,\lam^2\,+\,\ldots\,+\,d\ha_\el\,\lam^{\el\hb-1}\,\Big)\
\exp\ \{-\,2\;\pi\,i\;m\ \Frac{d\hw_1}{p\hb^3}\,\}$\vva
\hs{25mm}$\tim\ \exp\ \{-\,2\;\pi\,i\;m\ \Frac{d\ha_2}{p\hb^2}\,\}\
\exp\ \{-\,2\;\pi\,i\;m\ \frac{d\ha_3}{p}\,\}\,\ldots\ha\
\exp\ \{-\,2\;\pi\,i\;m\ d\ha_\el\ p\,^{\el\ha-\ha4}\,\}$\vvab

$=\,\sum\pos_{d\hw_1\,=\,0}^{p\,-1}\,\sum\pos_{d\ha_2\,=\,0}^{p\,-1}\,\sum\pos_{d\ha_3\,=\,0}^{p\,-1}\,\ldots\,
\sum\pos_{d\ha_\el\,=\,0}^{p\,-1}\,\Big(\,d\hw_1\,+\,d\ha_2\,\lam\,
+\,{d\ha_3}\,\lam^2\,+\,\ldots\,+\,d\ha_\el\,\lam^{\el\hb-1}\,\Big)$\vvab
\hs{35mm}$\tim\ \exp\ \{-\,2\;\pi\,i\;m\ \Frac{d\hw_1}{p\hb^3}\,\}\
\exp\ \{-\,2\;\pi\,i\;m\ \Frac{d\ha_2}{p\hb^2}\,\}\ \exp\ \{-\,2\;\pi\,i\;m\ \frac{d\ha_3}{p}\,\}$\pv\vvab

or\,\hh$\sum\pos_{c\ha_2\,=\,0}^{p\,-1}\ \exp\ \{-\,2\;\pi\,i\;m\ \Frac{c\hb_2}{p}\,\}\,=\ 0$\ha\vg\ha
donc l'expression ci-dessus peut s'\e crire\vvab
$\sum\pos_{d\hw_1\,=\,0}^{p\,-1}\,\sum\pos_{d\ha_2\,=\,0}^{p\,-1}\,\sum\pos_{d\ha_3\,=\,0}^{p\,-1}\,\ldots\,
\sum\pos_{d\ha_\el\,=\,0}^{p\,-1}\ d\ha_3\,\lam^2\ \exp\ \{-\,2\;\pi\,i\;m\ \Frac{d\hw_1}{p\hb^3}\,\}\
\exp\ \{-\,2\;\pi\,i\;m\ \Frac{d\ha_2}{p\hb^2}\,\}\ \exp\ \{-\,2\;\pi\,i\;m\ \frac{d\ha_3}{p}\,\}$\vvab

$=\ p\hb^{\el\,-\,3}\ \lam^2\;\sum\pos_{d\hw_1\,=\,0}^{p\,-1}\,
\sum\pos_{d\ha_2\,=\,0}^{p\,-1}\,\sum\pos_{d\ha_3\,=\,0}^{p\,-1}\:d\ha_3\
\exp\ \{-\,2\;\pi\,i\;m\ \Frac{d\hw_1}{p\hb^3}\,\}\ \exp\ \{-\,2\;\pi\,i\;m\ \Frac{d\ha_2}{p\hb^2}\,\}\
\exp\ \{-\,2\;\pi\,i\;m\ \frac{d\ha_3}{p}\,\}$\vvb

donc finalement\vvb$c\hb_{m\vv 3}\:=\:\Frac{\lam^2}{p\hb^3}\,\
\frac{e^{\:-\,2\;\pi\,i\;m\ha/\hb p\hb^2}-1}{e^{\:-\,2\;\pi\,i\;m\ha/\hb p\hb^3}-1}\,\
\frac{e^{\:-\,2\;\pi\,i\;m\ha/\hb p}-1}{e^{\:-\,2\;\pi\,i\;m\ha/\hb p\hb^2}-1}\,\
\frac{p}{e^{\:-\,2\;\pi\,i\;m\ha/\hb p}-1}\:
=\;\Frac{\lam^2}{p\hb^2}\,\ \Inv{e^{\:-\,2\;\pi\,i\;m\ha/\ha p\hb^3}-1}$\vc

En g\e n\e ralisant on trouve donc
\begin{picture}(,)
\put(4,0){$\,c\hb_{m\vv n}\,
=\:\Big(\Frac{\lam}{p}\Big)^{\hs{-.35mm}n\hb-1}\inv{e^{\:-\,2\;\pi\,i\;m\ha/\ha p\hb^n}-1}\,\hh,$}
\put(2.5,-5.75){\frm(63,13.25)}
\end{picture}\vvcd

\cad\quad$c\hb_{m\vv n}\:=\ \Inv{2}\ \Big(\frac{\lam}{p}\Big)^{n\,-1}\,
\Big[\hs{-.5mm}-\hs{-.2mm}1+\,i\,\cot\Big(\frac{\ha m\:\pi}{p\hb^n}\Big)\ha\Big]$\vc

Comme\hh$\tau\in\wh{\CP}$\hh on peut \e crire\hh$\all x\in\ZP$\vvb

\hs{10mm}\fbox{\;$2\ \tau\hb(x)\,=\ \Frac{p-1}{1-\lam}\:+\:\sum\pos_{n\,=\,1}
^{\infty}\ \Big(\Frac{\lam}{p}\Big)^{n\,-1}\ \sum\pos_{\sbs{m\,=\,1\\[.5mm]p\sndv m}}
^{p\ha^n-1}\Big[\hs{-.5mm}-\hs{-.2mm}1+\,i\,\cot\Big(\frac{\ha m\:\pi}{p\hb^n}\hb\Big)
\ha\Big]\ e^{\:2\:\pi\,i\:m\:x\ha/\ha p\ha^n}$\:}\vvcd

On pose\hh$\nn^*$\, \fbox{\,$d\hb_n\hb(x)\ha=\hb n\,\super{i\ee me}$ chiffre dans le d\e veloppement
p-adique de\, $x\in\ZP$\,}\ha\vbc

En identifiant les coefficients des puissances de\, $\lam$\, on obtient\hh$\nn^*$\vvb

\hs{25mm}\fbox{\ $2\ d\hb_n\hb(x)\hb=\:p\ha-1\:+\;\Inv{p\hb^{n\,-1}}\;
\sum\pos_{\sbs{m\:=\,1\\[.5mm]p\sndv m}}^{p\ha^n-1}\Big[\hs{-.5mm}-\hs{-.2mm}1+\,i\,
\cot\Big(\frac{\ha m\:\pi}{p\hb^n}\hb\Big)\ha\Big]\ e^{\:2\:\pi\,i\:m\:x\ha/\ha p\ha^n}$\:}\ha\vg\vvbc

ou encore\vvb

\hs{10mm}\fbox{\ $2\ d\hb_n\hb(x)\hb=\:p\ha-1\,-\:\Inv{p\hb^{n\,-1}}\;
\sum\pos_{\sbs{m\:=\,1\\[.5mm]p\sndv m}}^{p\ha^n-1}\,\Big[\cos\:(\ha2\:\pi\,m\,x
\hw/\hw p\ha^n\ha)\ha+\:\cot\Big(\frac{\ha m\:\pi}{p\hb^n}\hb\Big)\ha
\sin\:(\ha2\:\pi\,m\,x\hw/\hw p\ha^n\ha)\Big]$\,}\vvcd

\underline{\emph{Relation de Parseval}}\dd\vvb

Calculons\hh$\Int_{\;\ZP}\hs{-2mm}d\ha_n\hb(x)\ha^2\ \del\,(x)\,
=\lim\pos_{\el\to\pin}\ \Inv{p\hb^\el}\ \tex\sum\pos_{r\,=\,0}^{p\ha^\el-1}\,d\ha_n\hb(r)\ha^2$\pv
on a\,\hh$\all\el\ge n$\vva$\sum\pos_{r\,=\,0}^{p\ha^\el-1}\,d\ha_n\hb(r)\ha^2\,
=\ha\sum\pos_{d\hw_1\,=\,0}^{p\,-1}\,\sum\pos_{d\ha_2\,=\,0}^{p\,-1}\,\ldots\,
\sum\pos_{d\ha_\el\,=\,0}^{p\,-1}\,d\ha_n^{\ha2}\,=\;\Inv{6}\hb\ p\ha^\el\:(\ha p-1)\:(\ha2\;p-1)$\pv\vva
donc\,\hh$\Int_{\;\ZP}\hs{-2mm}d\ha_n\hb(x)\ha^2\ \del\,(x)\,=\;\Inv{6}\ha\ (\ha p-1)\:(\ha2\;p-1)$\vb

\newpage
En appliquant la formule de Parseval \ac\ la s\e rie de Fourier de\, $d\ha_n\hb(x)$\,
on trouve donc\vvaab
$\nn^*\quad\Frac{2}{3}\ha\ (\hb p-1\hw)\:(\ha2\;p-1\hw)\,=\,(\hb p-1\hw)\ha^2\,
+\,\inv{p\hb^{2\,(\ha n\,-1)}}\;\sum\pos_{\sbs{m\:=\,1\\[.5mm]p\sndv m}}^{p\ha^n-1}
\inv{\sin^{\ha2}\Big(\Frac{\ha m\:\pi}{p\hb^n}\Big)}$\,\vg\\\vs{-2mm}
\cad\quad$\dis\sum\pos_{\sbs{m\:=\,1\\[.5mm]p\sndv m}}^{p\ha^n-1}\inv{\sin^{\ha2}
\Big(\Frac{\ha m\:\pi}{p\hb^n}\Big)}\:=\;\inv{3}\ (\hb p\hb^2-1\hw)\ p\hb^{2\,(\ha n\,-1)}$\ha\vg donc\vva
$\dis\sum\pos_{m\:=\,1}^{p\ha^n-1}\ \inv{\sin^{\ha2}\Big(\Frac{\ha m\:\pi}{p\hb^n}\Big)}\:
=\sum\pos_{m\:=\,1}^{p\hb-1}\ \inv{\sin^2\Big(\Frac{\ha m\:\pi}{p}\Big)}\,
+\,\sum\pos_{r\,=\,2}^n\,\Big[\sum\pos_{m\:=\,1}^{p\ha^r-\ha1}\ \inv{\sin^{\ha2}
\Big(\Frac{\ha m\:\pi}{p\hb^r}\Big)}-\sum\pos_{m\:=\,1}^{p\hb^{r-1}\ha-\ha1}\
\inv{\sin^{\ha2}\Big(\Frac{\ha m\:\pi}{p\hb^{r-1}}\Big)}\Big]$\vva

$=\dis\sum\pos_{r\,=\hb1}^n\ \sum\pos_{\sbs{m\:=\,1\\[.5mm]p\sndv m}}^{p\ha^n-1}
\inv{\sin^{\ha2}\Big(\Frac{\ha m\:\pi}{p\hb^n}\Big)}\hb
=\:\Inv{3}\ (\hb p\hb^2-1\hw)\hb\sum\pos_{r\,=\hb1}^n\,p\hb^{2\,(\ha r\,-1)}\ha
=\: \inv{3}\ (\hb p\hb^2-1\hw)\ \frac{p\hb^{2\:n}-1}{p\hb^2-1}\ha=\,\inv{3}\ (\hb p\hb^{2\:n}-1\hw)$\vg\vvb

\cad\hs{40mm}\fbox{ $\dis\sum\pos_{m\:=\,1}^{p\hb^n-1}\ \inv{\sin^{\ha2}\Big(\dis
\frac{\ha m\:\pi}{p\hb^n}\Big)}\:=\ \inv{3}\ \big(\hb p\,^{2\:n}-1\hw\big)$\:} ,\vvbc

qui est un cas particulier de la formule\, (\hb$n\ge\ha2$\hb)

$$\boxed{\ \sum\pos_{m\:=\,1}^{n\ha-\ha1}\ \inv{\sin^{\ha2}\Big(\Frac{\ha m\:\pi}
{n}\Big)}\:=\ \inv{3}\ \big(\ha n\ha^2-1\ha\big)\:}\ .$$\vvc


\underline{Exemple\, 4}\,$^\bul$\dd\vva

On d\e finit la fonction\,\hh$\vphi\ha\ :\,\ \ZP\,\to\,\C$\,\hh de la mani\ee re suivante\dd\vva
si\,\hh$x\ha=\sum\pos_{k\,=\hb1}^{\infty}d\hb_k\;p\hb^{k\hb-1}$\hh
avec\,\hh$\all k\in\N\,\hh c\hb_k\in\db{\,0\vv p-\hs{-.5mm}1\,}$\ha\vg\ha on pose\vvw
$\vphi\,(x)\ha=\sum\pos_{k\,=\,0}^{\infty}\hb\lam^{k\hb-1}\hb
\exp\;\{\hb2\,\pi\:i\:w\;\Frac{d\ha_k}{p}\ha\}$\,\hh avec\hh$\lam\in\hb\C\hb\vg
|\ha\lam\ha|<1$\,\hh et\,\hh$w\ha\in\db{\hb1\hb,\hb p-\hs{-.3mm}1\hb}$\vbc

\underline{\emph{Calcul de\hb\ $c\hb_0$}}\dd\vvaab

$c\hb_0\,=\Int_{\;\ZP}\hs{-2mm}\vphi\,(x)\ \del\,(x)\,=\lim\pos_{\el\to\pin}\
\Inv{p\hb^\el}\ \tex\sum\pos_{r\,=\,0}^{p\ha^\el-1}\,\vphi\,(r)$\pv on a\,\hh$\all\el\in\N$\vva
$\sum\pos_{r\,=\,0}^{p\ha^\el-1}\,\vphi\,(r)\,
=\ha\sum\pos_{d\hw_1\,=\,0}^{p\,-1}\,\sum\pos_{d\ha_2\,=\,0}^{p\,-1}\,\ldots\,
\sum\pos_{d\ha_\el\,=\,0}^{p\,-1}\,\Big[\hb\exp\;\{\hb2\,\pi\:i\:w\;\Frac{d\hw_1}{p}\ha\}\hb+\:\lam\:
\exp\;\{\hb2\,\pi\:i\:w\;\Frac{d\ha_2}{p}\ha\}$\vva
\hs{50mm}$+\,\ldots\,+\,\lam^{\el\ha-1}\,\exp\;\{\hb2\,\pi\:i\:w\;
\Frac{d\ha_\el}{p}\ha\}\hb\Big]=\:0$\ha\pv\ha donc\,\hh$c\hb_0=\ha0$\vb

\underline{\emph{Calcul de\hb\ $c\hb_{m\vv n}$}}\dd\vvab

Nous calculons\; $c\hb_{m\vv 3}$\hb\vg le cas g\e n\e ral s'en d\e duisant ais\e ment\vb

$c\hb_{m\vv 3}\:=\Int_{\;\ZP}\hs{-2mm}\vphi\,(x)\ e^{\:-\,2\:\pi\,i\:m\:x\ha/
\ha p\ha^3}\ \del\,(x)\ =\lim\pos_{\el\to\pin}\ \Inv{p\hb^\el}\
\tex\sum\pos_{r\,=\,0}^{p\ha^\el-1}\,\vphi\,(r)\ e^{\:-\,2\:\pi\,i\:m\;r\ha/\ha
p\ha^3}$\hb;\vvb en raisonnant comme \ac\ l'exemple\, 3\,$^\bul$\, on trouve\,\hh
$\all\el\hw\ge\hw4$\vvb$\sum\pos_{r\,=\,0}^{p\ha^\el-1}\,\vphi\,(r)\
e^{\:-\,2\:\pi\,i\:m\;r\ha/\ha p\ha^3}=\ p\,^{\el\,-\,3}\ \lam^2\;
\sum\pos_{d\hw_1\,=\,0}^{p\,-1}\,\sum\pos_{d\ha_2\,=\,0}^{p\,-1}\,
\sum\pos_{d\ha_3\,=\,0}^{p\,-1}\:\exp\;\{\hb2\,\pi\:i\:w\;\Frac{d\ha_3}{p}\ha\}$\vvab
\hs{35mm}$\tim\ \exp\;\{-\,2\;\pi\,i\;m\ \Frac{d\hw_1}{p\hb^3}\,\}\ \exp\;\{-\,2\;\pi\,i\;m\
\frac{d\ha_2}{p\hb^2}\,\}\ \exp\;\{-\,2\;\pi\,i\;m\ \frac{d\ha_3}{p}\,\}$\vvb
$=\ p\,^{\el\,-\,3}\ \lam^2\;\sum\pos_{d\hw_1\,=\,0}^{p\,-1}\,\sum\pos_{d\ha_2\,=\,0}^{p\,-1}\,
\sum\pos_{d\ha_3\,=\,0}^{p\,-1}\:\exp\;\{-\,2\;\pi\,i\;m\
\Frac{d\hw_1}{p\hb^3}\,\}\ \exp\;\{-\,2\;\pi\,i\;m\ \frac{d\ha_2}{p\hb^2}\,\}$\vk0mm
\hs{102.5mm}$\times\,\exp\,\big\{\hb2\;\pi\,i\;(\hw w\hw\hw-\hw\hw m\hw)\;\Frac{d\ha_3}{p}\hb\big\}$\vvb

$=\ p\,^{\el\,-\,3}\ \lam^2\,\ \Frac{e^{\:-\,2\;\pi\,i\;m\ha/\hb p\hb^2}-1}
{e^{\:-\,2\;\pi\,i\;m\ha/\hb p\hb^3}-1}\,\ \frac{e^{\:-\,2\;\pi\,i\;m\ha/\hb p}-1}
{e^{\:-\,2\;\pi\,i\;m\ha/\hb p\hb^2}-1}\ \tex\sum\pos_{d\ha_3\,=\,0}^{p\,-1}\,
\exp\,\big\{\hb2\;\pi\,i\;(\hw w\hw-\hw m\hw)\;\Frac{d\ha_3}{p}\hb\big\}$\vvb

$=\ p\hb^{\el\,-\,3}\ \lam^2\,\ \Frac{e^{\:-\,2\;\pi\,i\;m\ha/\hb p}-1}
{e^{\:-\,2\;\pi\,i\;m\ha/\hb p\hb^3}-1}\ \tex\sum\pos_{d\ha_3\,=\,0}^{p\,-1}\,
\exp\,\big\{\hb2\;\pi\,i\;(\hw w\hw-\hw m\hw)\;\Frac{d\ha_3}{p}\hb\big\}$\vvb

donc\,\hh$c\hb_{m\vv 3}\:=\ \Frac{\lam^2}{p\hb^3}\ \frac{e^{\:-\,2\;\pi\,i\;m\ha/\hb p}-1}
{e^{\:-\,2\;\pi\,i\;m\ha/\hb p\hb^3}-1}\ \tex\sum\pos_{d\ha_3\,=\,0}^{p\,-1}\,
\exp\,\big\{\hb2\;\pi\,i\;(\hw w\hw-\hw m\hw)\;\Frac{d\ha_3}{p}\hb\big\}$\vvb
\hs{23mm}$=\ \begin{cases}\ \Frac{\lam^2}{p\hb^2}\
\frac{e^{\:-\,2\;\pi\,i\;m\ha/\hb p}-1}{e^{\:-\,2\;\pi\,i\;m\ha/\hb p\hb^3}-1}$\quad
ssi\quad$m\ha\cgr\ha w\hh(\ha\mrom{mod}\ p\hb)\\\hh0$\quad ssi\quad$m\ha
\not\cgr\ha w\hh(\ha\mrom{mod}\ p\hb)\pt\end{cases}$\vvb

En g\e n\e ralisant on trouve\quad$c\hb_{m\vv n}\ =\ \begin{cases}\
\Big(\Frac{\lam}{p}\hw\Big)^{\hs{-.35mm}n\hb-1}\:\frac{e^{\:-\,2\;\pi\,i\;m\ha/
\hb p}-1}{e^{\:-\,2\;\pi\,i\;m\ha/\hb p\hb^n}-1}$\quad ssi\quad$m\ha\cgr\ha w\hh
(\ha\mrom{mod}\ p\hb)\\\quad0$\quad ssi\quad$m\ha\not\cgr\ha w\hh(\ha\mrom{mod}\ p\hb)\vg\end{cases}$\vvb

\cad\quad$c\hb_{m\vv n}\ =\ \begin{cases}\
\Big(\Frac{\lam}{p}\hw\Big)^{\hs{-.35mm}n\hb-1}\:\frac{e^{\:-\,2\;\pi\,i\;w\ha/
\hb p}-1}{e^{\:-\,2\;\pi\,i\;m\ha/\hb p\hb^n}-1}$\quad ssi\quad$m\ha\cgr\ha w\hh
(\ha\mrom{mod}\ p\hb)\\\quad0$\quad ssi\quad$m\ha\not\cgr\ha w\hh(\ha\mrom{mod}\ p\hb)\pt\end{cases}$\vvc

Comme\hh$\vphi\in\wh{\CP}$\hh on peut \e crire\hh$\all x\in\ZP$\vvc

\quad\fbox{\;$\Big[\hs{-.5mm}-\hs{-.2mm}1+\,i\,\cot\Big(\Frac{\ha w\:\pi}
{p}\hb\Big)\ha\Big]\,\vphi\,(x)\,=\ \dis\sum\pos_{n\,=\,1}^{\infty}\
\Big(\frac{\lam}{p}\hw\Big)^{\hs{-.35mm}n\hb-1}\hs{-5mm}\sum
\pos_{\sbs{m\,=\,1\\[.5mm]m\hb\cgr\hb w\ (\ha\mrom{mod}\ p\hb)}}^{p\ha^n-1}\hs{-5mm}
\Big[\hs{-.5mm}-\hs{-.2mm}1+\,i\,\cot\Big(\frac{\ha m\:\pi}{p\hb^n}\hb\Big)\ha\Big]\
e^{\:2\:\pi\,i\:m\:x\ha/\ha p\ha^n}$\:}\vvbc

\fbox{\;$\Big[\hs{-.5mm}-\hs{-.2mm}1+\,i\,\cot\Big(\Frac{\ha w\:\pi}{p}\hb\Big)\ha\Big]\,\vphi\,(x)\,
=\dis\sum\pos_{n\,=\,1}^{\infty}\ \Big(\frac{\lam}{p}\hw\Big)^{\hs{-.35mm}n\hb-1}
\sum\pos_{r\:=\:0}^{p\ha^{n\ha-1}\ha-\ha1}\Big[\hs{-.5mm}-\hs{-.2mm}1+\,i\,\cot\Big(\ha\Frac{w+r\:p}
{p\hb^n}\;\pi\Big)\ha\Big]\ e^{\:2\:\pi\,i\:(\hw w\ha+\ha r\,p\ha)\,x\ha/\ha p\ha^n}$\,}\vc

En identifiant les coefficients des puissances de\, $\lam$\, on obtient\hh$\nn^*$\vvbc

\begin{picture}(,)
\put(13,-3){$\Big[\hs{-.5mm}-\hs{-.2mm}1+\,i\,\cot\Big(\Frac{\ha w\:\pi}{p}\hb\Big)\ha\Big]\,
\exp\hb\Big\{\hb2\,\pi\:i\:w\;\Frac{d\hb_n\hb(x)}{p}\ha\Big\}$}
\put(20,-16.5){$=\ \Inv{p\hb^{n\ha-1}}\ \exp\hb\Big\{\hb2\,\pi\:i\:w\;\Frac{x}{p\hb^n}\ha\Big\}
\sum\pos_{r\:=\:0}^{p\ha^{n\ha-1}\ha-\ha1}\Big[\hs{-.5mm}-\hs{-.2mm}1+\,i\,
\cot\Big(\ha\Frac{w+r\:p}{p\hb^n}\;\pi\Big)\ha\Big]\ e^{\:2\:\pi\,i\:r\,x\ha/\ha p\ha^{n\ha-1}}$}
\put(10.5,-23.5){\frm(146,28.5)}
\end{picture}\vk35mm

\underline{\emph{Relation de Parseval}}\dd\vvbc
On a\quad$\Int_{\;\ZP}\Big|\exp\,\Big\{\hb2\,\pi\:i\:w\;\Frac{d\hb_n\hb(x)}{p}
\ha\Big\}\,\Big|^{\hb2}\ \del\,(x)\,=\Int_{\;\ZP}\hs{-2mm}\del\,(x)\,=\hb1$\ha\vg donc\,\hh$\nn$\vvb

\hs{40mm}\fbox{\ $\dis\sum\pos_{r\:=\:0}^{p\ha^n\ha-\ha1}\ \inv{\sin^{\ha2}
\Big(\ha\Frac{w+r\:p}{p\hb^{n\hb+\hw1}}\;\pi\Big)}\:=\:\frac{p\hb^{2\:n}}
{\sin^{\ha2}\Big(\Frac{\ha w\:\pi}{p}\Big)}$\:}\ \vvd

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

\centerline{\textbf{CHAPITRE\, XXII}}\vva
\centerline{\textbf{CONVOLUTION SUR\, $\ZP$}}\vvcd

\oui Nous \e tudions la\ha\ \emph{convolution}\ha\ de deux mesures r\e elles ou complexes sur\, $\ZP$\,
avec les m\^emes m\e thodes que sur\, $\R$\pt Nous pr\e sentons deux exemples explicites de calcul de
convolution sur\, $\ZP$\vc

\Lem{\numm*}\vvwa
Soit\, $h\in\EP$\pv posons\,\hh$H\,:\ \ZP^{\hb2}\,\to\R\;:\;(\ha r,\ha s\hw)\tto h\,(r+s)$\,;\:
alors\ha\ on a\,\hh$H\in\CP\ha(\ha\ZP^{\hb2}\ha)$\vbc

\Def{\numm}La\, \underline{convol\e e}\, de\hh$\tmu\vv\tnu\ha\in\MP$\hh est d\e finie par
$$\boxed{\;\tmu*\tnu\ :\ \EP\to\R\ :\ h\,\tto\int\hs{-2mm}\int_{\,\ZP^{\ha2}}\!h\,(r+s)\;
\tmu\hb(r)\;\tnu\ha(s)\,}\pt$$\vvab

\Th{\numm*}$\all\tmu\vv\tnu\ha\in\MP$\quad on a\,\quad$\tmu*\tnu\in\MP$\,\quad
et\,\quad$\nst{\hb\tmu*\tnu\hb}\le\nst{\ha\tmu\ha}\,\nst{\hw\tnu\hw}$\vb

\Th{\numm}$\big(\MP\:,\hb*\hb\big)$\; est une alg\ee bre de Banach associative\vg commutative et \mbox{unitaire\ha.}\vvb

\Dem Classique\vbc

\Th{\numm}$\all\tmu\in\MP$\quad on a\:\quad\fbox{\:$\tmu\hb*\un\ha
=\hb\tmu$\ho$(\un)$\,}\ha\pv\ha en particulier\,\hh$\un*\un=\un$\vb

\Dem$\all h\in\EP\quad(\hb\tmu\hb*\un\hw)\,(h)\,
=\Int\hs{-2mm}\int_{\,\ZP^{\ha2}}\!h\,(r+s)\;\tmu\hb(r)\;\del\ha(s)
=\,\Int_{\;\ZP}\hs{-.5mm}\Big[\int_{\;\ZP}\hs{-1mm}h\,(r+s)\;\del\ha(s)\hb\Big]\:\tmu\,(r)$\vva
$=\,\Int_{\;\ZP}\hs{-.5mm}\Big[\int_{\;\ZP}\hs{-1mm}h\,(s)\;\del\ha(s)\hb\Big]
\:\tmu\,(r)\,=\;\tmu\:(\un)\Int_{\;\ZP}\hs{-2mm}h\,(s)\;\del\ha(s)$\vbc

\Th{\numm}\vva$\all u\vv v\in\ZP\quad\all k\vv\el\in\N$\,\hh avec\quad
\fbox{\,$k\le\hb\el$\:}\quad on a\quad \fbox{\,$\un\ha_{\bak{u}}\ha*\ha
\un\ha_{\bal{v}}\,=\,\Inv{p\ha^\el}\ \un\ha_{\bak{u\ha+\hb v}}$\,}\ha\vb

\Dem$\all h\in\EP\quad\Int\hs{-2mm}\int_{\,\ZP^{\ha2}}
h\,(r+s)\;\un\ha_{\bak{u}}\hb(r)\ \un\ha_{\bal{v}}\hb(s)\ \del\ha(r)\ \del\ha(s)$\vvab
\qquad$=\,\Int_{\;\ZP}\hs{-.5mm}\Big[\int_{\;\ZP}\hs{-1mm}h\,(r+s)\:\un\ha_{\bak{u}}
\hb(r)\ \del\ha(r)\hb\Big]\:\un\ha_{\bal{v}}\hb(s)\ \del\ha(s)$\vvab
\qquad$=\:\Inv{p\ha^k}\int_{\;\ZP}\hs{-.5mm}\Big[\int_{\;\ZP}\hs{-1mm}h\,
(u+\ha p\ha^k\hb r+s)\ \del\ha(r)\hb\Big]\:\un\ha_{\bal{v}}\hb(s)\ \del\ha(s)$\vvab
\qquad$=\:\Inv{p\ha^k}\int_{\;\ZP}\hs{-.5mm}\Big[\int_{\;\ZP}\hs{-1mm}h\,
(u+\ha p\ha^k\hb r+s)\:\un\ha_{\bal{v}}\hb(s)\ \del\ha(s)\hb\Big]\ \del\ha(r)$\vvab
\qquad$=\:\Inv{p\hb^{k+\el}}\int_{\;\ZP}\hs{-.5mm}\Big[\int_{\;\ZP}\hs{-1mm}
h\,(u+\ha p\ha^k\hb r+v+\ha p\ha^\el\ha s\ha)\ \del\ha(s)\hb\Big]\ \del\ha(r)$\vvab
\qquad$=\:\Inv{p\hb^{k+\el}}\int_{\;\ZP}\hs{-.5mm}\Big[\int_{\;\ZP}\hs{-1mm}h\ha
\big[\hb u+v+\ha p\ha^k\,(\ha r+p\hb^{\el-k}\hb s\ha)\,\big]\ \del\ha(r)\hb\Big]\ \del\ha(s)$\vvab
\qquad$=\:\Inv{p\hb^{k+\el}}\int_{\;\ZP}\hs{-2mm}h\,(u+v+\ha p\ha^k\hb r)\
\del\ha(r)\,=\:\inv{p\hb^\el}\int_{\;\ZP}\hs{-1mm}h\hb(r)\:\un\ha_{\bak{u\ha+\hb v}}\hb(r)\ \del\ha(r)$\vcd

\Cor{\numm*}$\EP\ha*\hb\EP\,\sub\,\EP$\vbc

\Th{\numm}$\all f,\,g\in\CP\,\hh\all u\in\ZP$\quad on a\quad$(f*\hw g\ha)\,(u)\,
=\Int_{\;\ZP}\hs{-2mm}f\ha(x)\:g\hb(u-x)\;\del\ha x$\ha\pv\vvw on en d\e duit\hh$f*g\in\CP$\vab

\Dem C'est une cons\e quence du \theo\ha\ de Fubini\vbc

\Lem{\numm}$\all f,\,g\in\CP$\quad on a\quad$(f*\hw g\ha)\ha^2\,\le\,f\ha^2*\ha g\ha^2$\vb

\Dem On a\,\hh$\all u\in\ZP$\vva$\quad(f*\hw g)\ha^2\,(u)\,=\,\Big[\Int_{\;\ZP}
\hs{-2mm}f\ha(x)\:g\hb(u-x)\;\del\ha x\Big]^{\ha2}\le\Int_{\;\ZP}\hs{-2mm}
f\ha^2\ha(x)\:g\ha^2\ha(u-x)\;\del\ha x\,=\,(f\ha^2*\ha g\ha^2)\,(u)$\vbc

\Th{\numm}$\all f,\,g\in\CP$\quad on a\quad1)\hh$\|\ha f*\hw g\hb\|\,\le\,\|\ha f\ha\|\:\|\ha g\ha\|$\vva
\hs{67.5mm}2\ha)\hh$\none{\ha f*\hw g\hb}\,\le\,\none{\ha f\ha}\ \none{\ha g\ha}$\vva
\hs{67.5mm}3\ha)\hh$\ntwo{\ha f*\hw g\hb}\,\le\,\ntwo{\ha f\ha}\ \ntwo{\ha g\ha}$\vvab

\Dem 1)\hh Trivial\hhh2\ha)\hh Cons\e quence du\, \Theo\, 3\vvab
3\ha)\hh$\ntwo{\ha f*\hw g\hb}^{\hw2}\,=\,\none{\ha(f*\hw g)\ha^2\ha}\,
\le\,\none{\ha f\ha^2*\ha g\ha^2\ha}\,\le\,\none{\ha f\ha^2\ha}\ \none{\ha g\ha^2\ha}\,
=\,\ntwo{\ha f\ha}\ \ntwo{\ha g\ha}$\vc

\Cor{\numm*}$\CP\ha*\hb\CP\,\sub\,\CP$\,\hh;\,\hh$\LP^1\,*\,\LP^1\,\sub\,\LP^1$\,\hh;\,\hh
$\LP^2\,*\,\LP^2\,\sub\,\LP^2$\vbc

\Th{\numm}$\BP\;*\hb\BP\,\sub\:\BP$\quad et\quad$\al\;\tdf\,\,\tdg\ha\in\,\BP$\quad
$\nbb{\ha f*\hw g\hb}\,\le\,\nbb{\ha f\ha}\ \nbb{\ha g\ha}$\vb

\Dem Soient\,\hh$\tdf,\,\tdg\ha\in\,\BP$\,\hh et soit\hh$h\in\hb\EP$\pv on a\vvab
$\big|\ha(\ha\tdf*\tdg\ha)\,(h)\ha\big|\,\le\Int\hs{-2mm}\int_{\,\ZP^{\ha2}}
|\hb h\hb(r+s)\ha|\:\vtdf\ha(r)\:\vtdg\hb(s)\ \del\ha(r)\ \del\ha(s)$\vvab
\hs{23mm}$=\Int_{\;\ZP}\Big[\int_{\;\ZP}\!\big|\hb h\hb(r+s)\ha\big|
\:\vtdf\ha(r)\ \del\ha(r)\hb\Big]\:\vtdg\hb(s)\ \del\ha(s)$\pv\vva
or la fonction\quad$\ZP\to\R\ :\ s\,\tto\Int_{\;\ZP}\!|\hb h\hb(r+s)\ha|
\:\vtdf\ha(r)\ \del\ha(r)$\,\hh est continue\vg\vva donc\quad
$\big|\ha(\ha\tdf*\tdg\ha)\,(h)\ha\big|\:\le\;\nbb{\ha\tdg\ha}\Int_{\;\ZP}
\Big[\int_{\;\ZP}\!|\hb h\hb(r+s)\ha|\:\vtdf\ha(r)\ \del\ha(r)\hb\Big]\ \del\ha(s)$\vvab\hs{36mm}
$=\;\nbb{\ha\tdg\ha}\Int_{\;\ZP}\Big[\int_{\;\ZP}\!|\hb h\hb(r+s)\ha|\
\del\ha(s)\hb\Big]\:\vtdf\ha(r)\ \del\ha(r)$\pv\vvab de m\^eme la fonction\quad
$\ZP\to\R\ :\ r\,\tto\Int_{\;\ZP}\!|\hb h\hb(r+s)\ha|\;\del\ha(s)$\,\hh est continue\vg\vvab
donc\quad$\big|\ha(\ha\tdf*\tdg\ha)\,(h)\ha\big|\:\le\;
\nbb{\ha\tdf\ha}\ \nbb{\ha\tdg\ha}\Int_{\;\ZP}\Big[\int_{\;\ZP}\!
|\hb h\hb(r+s)\ha|\ \del\ha(s)\hb\Big]\ \del\ha(r)$\vvab\hs{36mm}
$=\;\nbb{\ha\tdf\ha}\ \nbb{\ha\tdg\ha}\Int_{\;\ZP}\!|\ha h\hb(s)|\
\del\ha(s)\:=\:\nbb{\tdf}\ \nbb{\ha\tdg\ha}\ \none{h}$\pv\vvb donc\quad
$\tdf*\tdg\hb\in\,\BP$\quad et\quad$\nbb{\ha f*\hw g\hb}\,\le\,\nbb{\ha f\ha}\ \nbb{\ha g\ha}$\vc

\Not$\all\tmu\in\wh{\MP}$\hh on note\,\hh$c\hb_0\hb(\ha\tmu\ha)$\hh et\hh$c\hb_{m\vv n}\hb(\ha\tmu\ha)$\hh
les coefficients de Fourier de\, $\tmu$\vbc

\Th{\numm}\vva$\all\tmu\vv\tnu\in\wh{\MP}$\quad on a\quad$c\hb_0\hb(\hb\tmu*\tnu\ha)
\hb=\,c\hb_0\hb(\ha\tmu\ha)\ c\hb_0\hb(\hw\tnu\hw)$\:\hh et\:\hh$c\hb_{m\vv n}\hb(\hb\tmu*\tnu\ha)
\hb=\,c\hb_{m\vv n}\hb(\ha\tmu\ha)\ c\hb_{m\vv n}\hb(\hw\tnu\hw)$\vbc

\Dem$c\hb_{m\vv n}\hb(\hb\tmu*\tnu\ha)\,=\Int\hs{-2mm}\int_{\,\ZP^{\ha2}}
e^{\:-2\:\pi\,i\:m\:(\ha r+s\hw)\ha/\hb p\ha^n}\,\tmu\hb(r)\:\tnu\ha(s)$\vvab
$=\Int_{\;\ZP}e^{\:-2\:\pi\,i\:m\: r\ha/\hb p\ha^n}\,\tmu\hb(r)\,
\int_{\;\ZP}e^{\:-2\:\pi\,i\:m\:s\ha/\hb p\ha^n}\,\tnu\ha(s)\:
=\ c\hb_{m\vv n}\hb(\ha\tmu\ha)\,c\hb_{m\vv n}\hb(\hw\tnu\hw)$\vd

\numm\,\ \underline{Calcul\, de\hh$|\ha x\ha|\ha_p^{\ha r}\ha
*\ha|\ha x\ha|\ha_p^{\ha s}$}\,\quad(\hb$r\hb,\hb s\ha\in\ha\R^+$\hw)\vb

A partir des s\e ries de Fourier de\: $|\ha x\ha|\ha_p^{\ha r}$\hh et\: $|\ha x\ha|\ha_p^{\ha s}$\,\vg
et en vertu du \theo\ pr\e c\e dent\vg on peut \e crire\hh$\all x\in\ZP$\vvc

$|\ha x\ha|\ha_p^{\ha r}\ha*\ha|\ha x\ha|\ha_p^{\ha s}\
=\ \Frac{(\ha p\ha-1\hw)\ha^2\ p\,^{r\ha+\ha s}}{(\ha p\,^{r\ha+\ha1}-1\hw)\:(\ha p\,^{s\ha+\ha1}-1\hw)}
+\ \frac{(\ha p\,^r-1\hw)\;(\ha p\,^s-1\hw)}{(\ha p\,^{r\ha+\ha1}-1\hw)\:(\ha p\,^{s\ha+\ha1}-1\hw)}$\vvab
\hs{70mm}$\tim\ \dis\sum\pos_{n\,=\,1}^{\infty}\ \inv{p\,^{(\ha n\ha-\ha1\hw)\,
(\ha r\ha+\ha s\ha+\ha2\ha)}}\,\ \Big(\,\sum\pos_{\sbs{m\,=\,1\\[.5mm]p\sndv m}}
^{p\ha^n-1}e^{\:2\:\pi\,i\:m\:x\ha/\ha p\ha^n}\,\Big)$\pv\vva

or on a\vvb
$|\ha x\ha|\ha_p^{\ha r\ha+\ha s\ha+\ha1}\ =\ \Frac{(\ha p\ha-1\hw)\;
p\,^{r\ha+\ha s\ha+\ha1}}{p\,^{r\ha+\ha s\ha+\ha2}-1}\,
-\ \frac{p\,^{r\ha+\ha s\ha+\ha1}-1}{p\,^{r\ha+\ha s\ha+\ha2}-1}\,\
\dis\sum\pos_{n\,=\,1}^{\infty}\ \inv{p\,^{(\ha n\ha-\ha1\hw)\,
(\ha r\ha+\ha s\ha+\ha2\ha)}}\,\ \Big(\,\sum\pos_{\sbs{m\,=\,1\\[.5mm]
p\sndv m}}^{p\ha^n-1}e^{\:2\:\pi\,i\:m\:x\ha/\ha p\ha^n}\,\Big)$\pv\vva

on en d\e duit\vvb

$|\ha x\ha|\ha_p^{\ha r}\ha*\ha|\ha x\ha|\ha_p^{\ha s}\
=\ \Frac{(\ha p\ha-1\hw)\ha^2\;p\,^{r\ha+\ha s}}{(\ha p\,^{r\ha+\ha1}-1\hw)\:
(\ha p\,^{s\ha+\ha1}-1\hw)}$\vvab
\hs{30mm}$+\,\ \Frac{(\ha p\,^r-1\hw)\;(\ha p\,^s-1\hw)}{(\ha p\,^{r\ha+\ha1}-1\hw)\:
(\ha p\,^{s\ha+\ha1}-1\hw)}\,\ \frac{(\ha p-1\hw)\;p\,^{r\ha+\ha s\ha+\ha1}
-(p\,^{r\ha+\ha s\ha+\ha2}-1)\ |\ha x\ha|\ha_p^{\ha r\ha+\ha s\ha+\ha1}}{p\,^{r\ha+\ha s\ha+\ha1}-1}$\vvab

\hs{12.5mm}$=\ \Inv{p\,^{r\ha+\ha s\ha+\ha1}-1}\ \Big[\,(\ha p\ha-1)\;p\,^{r\ha+\ha s}
-\ \frac{(\ha p\,^r-1\hw)\;(\ha p\,^s-1\hw)}{(\ha p\,^{r\ha+\ha1}-1\hw)\:
(\ha p\,^{s\ha+\ha1}-1\hw)}\ (p\,^{r\ha+\ha s\ha+\ha2}-1)\
|\ha x\ha|\ha_p^{\ha r\ha+\ha s\ha+\ha1}\,\Big]$\vvcd

En particulier pour\ha\hh$r=\ha s=\ha1$\hh on obtient\vvbc
\hs{30mm}\fbox{\:$|\ha x\ha|\ha_p\ha*\ha|\ha x\ha|\ha_p\:=\ \Inv{p\hb^2+p+1}\
\Big(\hb p\hb^2-\ \frac{p\hb^2+1}{p+1}\ |\ha x\ha|\ha_p^3\hb\Big)$\,}\ha\vd


\end{document}
