TRANSFORM EE de FOURIER

TRANSFORM EE ^de FOURIER

1) ^{Petite bearish}

^'que

Rien de vnaimenl

rigour

^{eux dans ce}

qui

^{sail !}

Si

f

^est

^peiiodique

^de

^peiiode

^{T he's}

^grande

^{. Ou bien}

^f

^{que eco -que eh o -}

faicg-fsmfth.tk ^]

+ T-

pehiodicile

^{'. Zac se parse t'il}

^quand

^{c- → + p ?}

f-

^{Cx) =} _, ^{Cn Cf}

⁾

^e

'

'

nx

avec Cn

Cfl

^.-

if Ifa

^{, e-i}

^2¥

^{' - die}

f

^{Cx )=}

Fez ^fate

- i " "

di

e'

'

ah "

own , oui

wn=2

^{n , own =}

A la ^le mile ( ^{somme de Riemann} )

fix

⁼

_{I !} ^It

^{fate - is " DL e''9"}

^dy

Si on pose

I

^Cx)=

far

^{) e- " " du} , aeons

flock It ^Icy

^{) e'is "} dy ^.

2)

^Dans

^{L' C}

^IR)

Si

f

^E

^L'

^{CIR), o-}

^definite

^{, t 3 CR}

(3) is I ^{±fr these}

^{- "}

^dsc

^.

( ^Iesh la

transformed

^{de Fourier de}

^f

^.

Proposition

^I

Si

f

^{E L'(IR) alon}

f

^{esh continue sun IR , elle head vers O a- e' y . On a}

IIF Ho

^{E HfIt},

.

Dem

fade

^{- "}

i3

^{est continue .}

^}

^{done par cu dom}

^I

^{est continue .}

•

Ifcxse^{- "}

^I

^{e IfCall}

i3

^{E C'}

•

If (g) ^I

^e

fr

^{IfCall da = Hfll, done}

^Iif

^{H - E Kfk , .}

• Soil Cfn

)

une suite dans

CL ?

IR) Lg Hf ^- fall _, ^{→ o .} Alon It

f

^-

In

^{K- =}

ltffnll

^{- e} Hf^- fall, → O

Done

In

^→

f ^unit

^.

FIB

Main ^{'s next}

is

^{e- " da =}

^{[ fu ]}

⁺

^{÷ !}

^.

^ftxl

^{e- ''"}

Sdx

3 ³

=

it

Ents

).

I

Incest

Is

^{, "f}

^{'d 's} ¥

,

I

⁼

fin

,

EH

^•

^{Zuelques f=}

^}

^If

^,,

^{examples ,j4}

^E⇒

^qui

^{nous servin = ooh}

^ein

^}

^In

^{-- o. DE}

FBI

^•

^Ffg xscoscxesldx fog ^{sins since}

^•.•

^{flock fc.ch}

^{Dans- I}

^{f {}

^On

%)

^1-

⁽

^{> ox)0Ie-}

e? ^la

^{,=sincepose=t'e-}

⁽

^{"'a "suite.C-Iaeessixx2) )}

^I

^,

^aeons

^{aeons:§=pour S (⇒}

^its

^Hy

_zaf

⁾

is ^f

⁽

^]

^x

^?

^a

^{Fcs I}

(^{ol) )}

^f

⁼x⁼

⁽³⁾

^{e- 2x}) ^E)t-='

⁾

^"x-=

e ^{L' ↳}

⁼f^{e-=C IT"C IR}

^{Lie J ¥}

^ix,3

^hit

^{.d -}

⁼

^"

^!

^)a

⁽

^{3If)daI,-IRe--"txtdaon}

^{tf 3%1 1¥}

^da

⁾

^=e-

^{! )}

^pose

^fo

⁼

'

^+i3,,

^'s

^e-"idemondx

^I

^3"eeda

⁽³⁾

^3=TD

^fence

^da2.==sSo=

^{'ll If}

^=+en

^÷

^{-- T}

³⁾

^{g23. D=.t 3}

^fjffx

^-⇒

^÷

^es

^I

²

⁽³⁾

^{)=e "sincet'"=3da}since,. On}

⁽

^a

^{FEGER )}

^. i

Ea^{Cas = e}

i * x

Proposition

²

Soienl

f

^, g E L (

"

^IR)

• tf I

, m E a _,

Iftar

^{= I}

ftp.g

~ r v

•

fo

⁵ ⇐

fo

S =

f

T

•

I

⁼

ITS

⁼

^f

. Fat IR* _,

Tota

⁼

^Fa

^,

FOH

^{- n ya}

• taek

,

Taf

⁼ e- a

f

• Fa EIR _,

eat

^{= Tea}

^I

•

fig

⁼

fig

^.

Towles ces

demonstrations

soot

facies

^{et Caisse 'g e- exercise .}

Proposition

³

Soit

f

^{E L'CIR) .}

2)

^si

f

^{eh c' eh si}

f

^{'c- L'}

^aeons f

^{O eh}

F' (3)

⁼ is

IG ₎

2) ^Si

Lf

^{Cx) =}

^ixfcxl

^{E L'CIR) , alon}

I

^{est de Livable eh}

I

^{' = -}

LI

^.

Dem

1) f ⁽

^{x ) =}

^f

^Cost

Iff

^{'C Hdl . Done}

^f

^{admel une finite en to . Comme}

^f

^{E L' CK) cette}

Ci

^{mile ne}

peut

^entree que O.

fr ^fix

^{, e- '' " ' dx =}

^[

⁺

^¥ Inf ^'m

^{e- is " da =}

fits ₎

2) I (5)

⁼

fr ^flat

^{e- its " dx .}

Thon ^de'ni ration som e^' in ^e'

grace

^{: 5 i→}

^flat

^{e- '}

'5" est

derivable

, de define ^'e ^-

ixfcxs

^{e- is " .} I^-

ixfcxs

^e-its"

I

^{= ILf Call EL ! Done} pan ^{th -} delineation :

I

^"

⁽³⁾

⁼

fr

^{- in}

^flute

^{- is" do =}

^{-1mL fees}

^{e- it " dsc = -}

^{tf (}

^{51. •}

2)

^La

clause YCIR )

On

dit qu

'

une

f

^{one lion}

f

^{E C ( IR}

⁾

^{est ai de' doin once}

^rapid

^{e si f k, n E IN,}

I Mean

^O

^he Idk

^I"

> ^{k If}

^{' "}

^call

^s

^Me

^{,n .}

C ^'est un s. e. r. de C ^- ( IR)

, note ^' TCM .

On pose

Me

_, n

(f)

⁼

s.yy.fr talk _If

^{' "}

cool

,

puis

^S

(f)

⁼

^{I 2-}

^{k t" min}

(

^I,

^Me

,. Cfl

)

k,n E IN

eh en

fin ^DC fig )

^{= S}

(

f ^-

g)

^.

Proposition

⁴

1) d ^{est une distance sun Y}(^IR) ^.

2) One suite

( f )

^{c TCR)} converge ^{pour d vers}

f

^{E YARI si eh scale meal si}

Liz

^,

^Me

^{, p}

⁽

^for-

^f)

^{= O}

Hh, p E IN.

3)

Y

^est

complete

pour d .

Dem

s) On a la syme Erie e' vide moment . On a S

(

^{h th '}

)

s Sch ) *

Sch

^'

)

d^'oui

line 'g

^.

^tniang

^.

On a 8

(

^h

)

^{= o ⇒}

Mo

,^.

(h)^-- o ⇒

11h40

^{-- o ⇒ h-- o .}

2)

^Si

fn d- of

^{, aeons}

Me

, p

( fn

^-

f )

^e

2kt

^r

_{d ( fn}

,

f)

^done

Me

_{, p}

(

for^-

f )

^{o .}

Inverse ment, si cha _que Mee

,p

( fn

^-

f

⁾

_I

^>

₀

, aeons For _>o poisons

An

_,n

=

¥=

^,

^2-

^{" +" min}

⁽

^{I, Me, , ( fn- ft}

⁾

^E

^I

^{Crtc ) .}

De

plus

,

lion ^Arm

^{= O par}

^hypo

^{these .}

La seki e

§ ^An

^{,n Comr . non male meat par}

^rapport

^{a' n , done}

lion

,

dlfnif )

⁼

Lion

,

⇐ ^Ann

⁼

^{I lion}

^{An,. = 0 .}

Ainsifndsf

^.

3)

^Si

( fn )

^{c Y CK}

⁾

^{est de}

Cauchy

pour d

, aeons

( fn )

^{est de}

Cauchy

^{pour It-}

^Ily

, done

fn

^"

f

eh da coup

f

^E

Coo

ar ).

Les suites

( ^f )

^{son anni de}

^Cauchy

^{pour It. He (}

^regard

^a

^No

^{,p (fn -f )}

)

, done

elle

CVO

vers an

gr

^{E C:} Cir ). Aimee _, par les thin us nets

f

^{est C Pel}

for

^{) =}

GP

^.

Aim i

f

^{esh c- .}

Roshe ^{a- moorhen} que

f ^{EYCK )}

^et que

fn f

^.

HE

> o_,

ttkcp

^{, F}

Ne

_{, qp}

to

^{him 7}

Ne

_,e_,p

⇒

dlfn.fm )

^{s 2- teeth} e .

=3 ^min

(

^I, Me

, p Cfn^-

fm ) )

^{s 2 " k d}

^{( fun}

^,

^fu )

^{e E at}

⇒ Mean

(

fu,fm) ^{= min}

(

l ,

new (

^fu-

f

m)

)

^{E E}

On a aim i

lxlk _If !

^" cus ^-

flmmcxsl

^{s E Fx . En} parran Loi la limit e sun m :

ixtle I f 'T

^{' (x) -}

fc2.es/EEtx

, ^a

'-e .

Me

, pCf ^-

fn

) ^{E E}.

Da _coup Me_{, p}Cf ) ^E Et

Me

_{, p}

( ^f

_we

,ee_,,) ^est

find

^{. On a montu '}

f

^{E YCIR).}

On ^a de

plus Me

, p

( fn

^-

f ) =

^{sO . Do -a pan 21 :}

dffnif

) ^{→ O .}

⑤

Proposition

⁴

CF

^{(IR) esh dense dam YC IR).}

Dem

'she ^une

faction

^X,E

^Cf

^{t elle que}

"

⇐ ⁽

^{Caisse ' en exercise}

⁾

Poisons

fix

^{) = X, ( Mn )}

^f

^{Cx), aeons}

fnf CF

^{. Dont - on} que

fn Is f

^.

= Xn

f

^{Cx )}

Not ons D e^'op^{. de} de're ration . On a

Dr Cfn

^-

f)

⁼

^{Dr ( f}

^{(xn- 1)}

)

^⇐

¥0 ⁽

^Pe

^{) De (}

^{xn - I}

⁾

^DP

-

tf )

⁼

Dff ^{) (}

^{Xn -}

¹⁾

^t

⇐ ⁽

^er

^{) DYK}

^-t)

^{DM f}

Rais

De

_Xn Cx ) ⁼

# ^{④ ex}

^,

^{) ( In )}

^{, done}

^No

^{,e lXn) =}

^ht

^{no,e ( Xi}

^I

^E

^{In no}

^{, eCx,I.}

Finale

^{meal :}

Reap ^Cfn

^-

f)

^a-

Me

_,.

( Drift I

Xn ^-I

) ) ^{(f) Me}

^{, ,}

tf

^.

^{eff )}

^-

^No

^,e

⁽

^Xn

⁾

she ,o( ^Dolf

⁾

⁽

^{Xn - D}

)

^t

I (E) _Me

, ,^.

elf ^{) no}

,e

I

Xi

I

⇐ sup I ^{'ll '}

IDF call

^t

I ( I ) nee

, p^.

elf ⁾ no

_,e (Xi

)

1×17, n

⇐

I ( Me

"

.ph

^t

⇐ ^{( i ) neem}

^.

^{elf ) no}

^,

^eh

^.l

⁾

Done

Me

_{, p} ⁽

^fn

^-

f ⁾

^{- so .} Coro Elaine 5

D KIRI

^C

Csar

) ⁿ

? ²⁴ⁱⁿ

^{) .}

2)

YCIR

) esh dense dam LP (^{IR) f I} Epc ^{- .} 3) YCIR ) esh dense dam

⁽

^CR

⇐ ⁾

^{, H-}

^Ho )

^.

4)

^{L '}

injection ^Y

^→

^{LPCIR )}

^{, Is} pete ^{, est continue .}

Dem

-

1) Si

f

^EY

⁽

^IR

⁾

^,

^{along lfcxsf}

^{s no,oCf )}

_µ

^{t Ma,. Cft E}

Cg

^ar ) n

Lock )

^{t is} p ^{c- .}

2)

^et 3) ^{vieanenl du} coup ^avec

CL

^{C YAR}

)

^{ec la deasihe ' de CF .}

4) ftp.lffxs/rdxJ%eCno.oCfIt-MqolfHf1r#,pdx)*=Cp(no.offltMz

^,

^off

^I

⁾

Hfllx

^E

11¥

^.

^{He (}

^no,

^off

^{) t}

^Nz

,

off

⁾

)

The

^' one '

me 6

Une

^{a. e. T: TCIR) →} TCIR ) ^esh continue ^{mi t} (k_, ^{) E IN}

p

^{2, F Cee},p 70 et Nee

,p E IN

to

tf

^{E TCR}

⁾

,

Me .pl

^Tf

⁾

^{E Cee}_{, p}

^I

^Mr,.Cf

⁾

^.

nts ENee

, p

Deme

⇐

^: Si

for Isf

, ^{on a}

Me

_{, ,}

(

Tcfn ^-

ft )

^{E Cee}_{, ,}

I

Mr_,

Cfn

^-

f )

→n^O , done

Tfn I Tf

^{. ✓}

rts € Nee_, p

- (ktp t I)

⇒ si Test continue I Lee_{, p} ^>

old ( fro )

^s Lee_{, ,} ^⇒ d

( Tf :

, ^o

)

^{E 2} .

• d

( Tf

,o

)

E 2-^{lht " "})

, ga ^- eat dine

Me

, p

(

Tf

)

^E

Yz

. F Nee

, a

4 ¥

, n

,,

!

^{" " E}

^th

^{, done si}

^¥

^. ,

.fr

^..

^(f)

^E

^FI

^,

^{aeons dffio )}

^e

^dem

(

^{car E Mr}.

.CH

^s

HI )

rts >Neap

On

^{a montee '} _que ^:

^I

^Mr,,

( ft

^E

teh

^⇒

^Me

^{, pCTf )E}

^I

^.

nts ENee_,P

eh les

arguments

On en here

facile

^meal

^Me

, p (

Tf )

^{I Cee}, p

Ifs

,we

!

^'s

⁽

^f

⁾

^{I avec}

^Cher

⁼

÷

asada

fan

^{etc ....}

•

Proposition

⁶

Les ope^{'sat ears}

Df

⁼

f

^{' ele}

Lf

^{(x) =}

^ixf

^{Cx) soul continues Y → Y. De}

^plus

^{on a}

pour ^tout

f

^{E Y}_,

DI

⁼

LI

^{el D}

I

^{= -}

#

^.

Demme

, p ( Df

)

⁼

Me

, pH ( f ) Me_{, p}

( Lf )

^E

_Mean

, p ( f) ^t p

Me

_{, p}^.,

(f)

^car

(

Lf

)

^{cxs = x}

DP

^D

Pf

^{he ) t} p D^"

- '

f ^.

Le neste ^on l ^'a

deja

^{ru .} De

Coro

Elaine

7

Si

f

^{E Y CIR}

⁾

^,

^aeons I

^C-

^{YCCR )}

^{. De}

^{plus f}

^no

^I

^{est continue Y → Y.}

Demy

Lk DPI

⁼

C- Dr

^Lee

( Ff )

± C-

Dr DEFF

⇒

Me

_,p

(E)

^{= H Lee}

Drf He

^{= H}

Deify

,

e

HD

^{- L}

Pf

It_, ^{E IT}

(

no_,. C

Deaf )

^t

Nz

_,o

( Deaf

)

)

en vertu de la

maj

^{oration oblenue}

an

Corolla

're 5, I ) .

< ^{I en} vertu de la continue ite ^'

de ^{Del L :} Y → T .

Ce la ^{no - he} que

I

^{E Y. EL mime} que

Me

_{, p}

( In

^-

_{I ) -20}

^si

_fu Is f

^.

Proposition ⁸

Soil

f

^E

^Y

^CIR

⁾

^.

^Aeons

pour ^Lout

h

> o _,

la

se

' nie

Izz ^f

^Cuh

^{) h converge}

^eh

Limo Fez ^{f (}

^nh

^{) h}

⁼

fr ^for

^{) DL .}

Dems

^{a --}

!

a, a

hfcnh

⁾ , _a ⁼

a a

hfcnh ⁾

Par

la

thionic usuelle dg

^sommer

^{de Riemann}

^:

leino

⁺

^SE

^{,a =}

I !fHdl

^.

Si h E

Co

,Go

]

, aeons

e.E.ae#nhYs2ngdtInEn.ntzs2nE*nE..lE-IIs2a7oijfY-

e

2n

A ^-ho La se 've Ste

,a converge done

,un

if

^{sun Co,h.}

^]

^.

La se 've

I

,

h

f

^Cnh

)

⁼

Sj

_,a

t Set

,A

Cv done .

Quand

^{A Lead vers to} ,

Sf

,a ^{con v un}

if

^vers

A ,

SI

_,a

=

fin

,

Sfa

^{t Ste},a

=

¥

,

efcne

^).

On

peal

^{done e'}

changer

^{Ces Ei miles :}

Eino If

,

hfhht =L off ^?

^{, ,} a a

hftnh ⁾

⁼

fin

^,

^fi ;oE

a

hffnh

^{) --}

fi ^Iaaf , ^Hdt

=

fjfctsdh

^.

Coro

Elaine

9

Si

f

^E

^{TCR )}

^et

^SE

^{IR ,}

^aeons

Ffs ⁾

⁼

Limo ¥

,

hfcnhl

^{e- ink .}

Theorie

^{me 10}

Si

f

^E

^YURI

,

aeons

fcxs

^--

¥ ^{Its )}

^{i' "}

^{as f- It} I

⁼

_It I

Demy

Commer sons par

f

^E

^{CT (}

^IR

⁾

^.

Suppose

^as

_que support ( f )

^c

C-

^R,r]. Soil

g Ra fo

^nation

2K ^-

peke

^-

odique

, e'

gale

^at

f

^{sun C- R,R}

^]

^{. Aeons}

goes

⁼

If

_, ^{Cn e}

^""

^{, avec an =}

In ! ! ^f

^{Ch e- in t ok =}

^{In I ( net )}

Done pour KIER ^{on a}

nhx

f-

^{Get =}

Ip ^Ee

,

at ^FC )

^{e "}

"

=

¥ Fez

^en

Fcnh )

^{e" area en=}

^The

^.

2 wa ^-d R → to _, aeons h ^- so ec on oblieal _, via la

Proposition

⁸,

f-

^{Cx) =}

# ! ! ^Its

^{) e " " da .}

Si f

^E

^T

^CIR

⁾

^{maiden aah}, ^on l ^'_{ap pro}

che par

^{une suite}

( ^{fn )}

^c

^Can

^{, pour Ea distance d de Y .}

On

^a

fncxs

⁼

¥ ^{files ) eis}

^{" dsc.}

La

transf

^{. de Fourier est continue sun Y} , done

f-

n

-s

I

^{et o- ow ien}

f

^{Cx) =}

If f( ^{5) eisa da}

^.

•

Corollaries

La

transformer

^{de Fourier I table .L une}

bijection

^de

^Y

^CIR

^{) dam}

^{lui - meme .}

The^' anime

12

Pour ^Lout

f

^{E T CIR}

⁾

^{, on a}

Hf Hear

_, ⁼

It ^{" F}

^"

^Kari

Dems

i

f

^E

^CT

^{C IR) , on re}

fail

^{comme co - denies a-ee} g^. On a done Parse val :

j III ^fail

^{' all =}

^E.

^,

¹⁴¹²

^{, avec ca =}

^{Is I ( net )}

^.

If ^{all ' di =}

Lin

,

f ! ^{If HI}

^{' all ±}

^{It Eez E tf (}

ⁿ

^{E) I}

^{' =}

^{It Fish des}

^{' .}

The^' on 'e me 13

Si

f

^, g

ETC

^IR

) ^aeons

{

,

f

^{he )}

ojcxsdx

⁼

fr ^{ICH gcxsdsc}

^.

Dem

Polarisation

^:

÷

^{④ ft g}

^Hi

^{- Itf -91122t c}

' Itf ^- is HE - i

llftigll ! )

^{= Lf , g)}

done

Sir ^I

^{he ) goes de =}

¥ fr Ecsisgcxsdx

^.

Pre non

f

^--

E

^⇒

fr ^E

^{g =}

^{It f IF 5}

E-

=

E

⁼

^E

^{= ate}

Feis

^.

3) ^Retour

^a-

^L

^'

^C

^IR

⁾

The

toneme 14

Si

f

, g ^{E L}

' (IR

) aeons

↳ ^fcxigcxsdx

⁼

fr ^fcxsgcxsdx

^.

Dem

Tied In )

,

Cgm

) ^C

KIRI ^teller

que

fn Is f

, gn

g

,

aeons

Hfoi

^-

FSH

, e

Hf 5

^-

fall

_,^t

It foin

^-

fn 5h11

_, ^t

Hfn5-f ^It

^{, t}

^HEA

^-

^{Ign It}

^{, +}

^HES

^.-

^Fall

^,

E Hfll

, 115^-

5. He

^{t Hf -}

fall

_,

11916

^t

HE

^-

Elle

^{119mV, t}

^HIM

^-

^{I Ign}

^-g11,

E Hfll_,

Hg

-911

, t

Hf

^- fall _,

llgnll

_, ^t Hfn ^-

fll

, 119mV

, t Hfll

, 119N^-911

,

=3

^{O , car It}

^fall

^,

^ecllgnll

,

so -L bonnets .

Proposition

^is

n

Si f

,g E L

'

CIR )

_, ^{on a}

f ^of

^E

^Lt

^CIR

⁾

^{, on a}

I

^*_g ^{E C: Cir}

⁾

^eh

f

^{gu =}

I

^* _g

Demy

.

If

^Cx)

5C

^{- x})

I

^{E 11911}_,

If

^Cx)

I

^{E L'CK) .}

•

f

^{E L'CIR) done}

If

^{C : art}, done

I

^* g ^{f C}:C^IR).

~

.

fr ^Fcc

^{) gCx -t) DL =}

^fr

^fu

^g

^{all =}

fr tg

^{'" '}

^flute

^{(t) all =}

^{f. of}

^. DE

Proposition

¹⁶

Soil

GE

^L'(IR

)

_,

aeons

g ⁼

fig ÷ fill

^{- '}

^{It 5cm}

^{eu da}

an seas de la

convergence

^{en non me dans L}

'CIR).

Dems

oil och ⁼

(

I ^- IH

)t

, aeons

SC

5)^{= sine '}

( % )

^{. On a}

LTE

L^'CIR ) ⁿ

Coo

CIR

)

_, ^{done les}

fonclions faces

⁼

OKI Cu

^,

oars ,

=L 1¥ ^8¥

^{Hill ,}

re

' aldsenl ^une

approximation

^{de l ' unite '}

quand

^{a → O.}

( of

^{TD .} ). Done

f

_, ^* g g ^.

L^-so

Mae

's on a :

A -

^{V v-}

DTA

^.) * g ⁼ OK .I g ⁼ OC^{- a.)}

5

⁼ ok ^.) of A Imi

← yr

94

⁼

him ^1¥

,

frock ! ^{) 5cal}

^eats_~ ^du.

Le nesle a- calmer 11811_,^. On a sine 2cal = IT 0( uh

) (

of ^T.D.

)

^{. Done}

4811

_, ⁼

£ ₍

o

)

Ecu

el ^{- ) >o == 2 IT2 IT.OG-O}

⁾

( ^{I (3)}

^{= nine '( 3G ) ,}

^{② (3)}

^{= ni -c'(3G) = 2 lC 23) = LID}

⁽

⁹

⁾

^{en O i IT .}

⁾

DE

Coro Elaine17

La

transformer

^{de Fourier est}

injective

^{sun Ll (IR}

⁾

Dem

IgE

^L'(K)

^{vehifie of}

^{=o , aeons pan la}

Proposition

¹⁶ g ^{I O} dans L' CIR).

•

On

^{note A (IR) =}

Lf

^{E L'CIR)};

FEL

^'CR)

}

^{. On man it A(IR}

⁾

^{de Ea name}

^Hfll

^{, t}

HEH

_, ^.

The ^' or e- me 18

Si

f

^E

^{ACIR )}

^,

^aeons quel

^{que soil la}

fond

^ion

represent

^ant

^{he Classe}

^de

f

^{, on a}

f- ⁽

^{x) =}

¥ fr ^{Ffs ) e'}

^{'5 " des presque}

^parlour

^.

On

CC peal

^supposes

^{ACIR )} :

(IR

)

^et, dans ce cos

,

tf

^{E ACK )},

fact

⁼

fr ^Ffs

^{) e'' 5'dg the .}

Dem

I

Soi _fi

IR ^-s Cl an re pre

' sealant d^'un e'le^' meal de ACIR ). On pose h=

2¥ ^f

^{E Coo (IR), mont - on}

que

fcxs

^{=hCx) pour presque tout x . D}

'

a pre^'s la

Proposition

^{16, il exist e une suite}

(

Rn) ^{→ to}

tell

_que

f

^{Cx) =}

lion ¥ I !! ⁽

^I-

^{II )} Iculeucxidu

pour p^.p^{. x .}

Mais I

^{C-L', done}

Her ,% !

,

(

^I-

Inn )

I

Ecutleucxll

^s

Iffy

^EL', on

peace

^done

appliques

^{cone .}

dominie eh

f

^{Cx) =}

Its ^flu

^{, each du .}

Le nesle da tem est

facile

^. ⑤

Proposition

¹⁹

i) A (IR) est an Banach

2) ^Soil

f-

^{EL ', aeons}

f

^{E ACIR}

⁾

^ssi

I

^{E ACIR}

⁾

3)

^si

f

, g E A CIR

)

aeons

fg

^{E A (IR}

⁾

^el

FT

⁼

If ^F

^*

^of

^.

_×

4) ^Si If,g) ^E ICIR)x ACIR

)

^aeons

f

^* g E ACIRI et

f

^* g ⁼

fit ⇐5)

Dem Tsi _{( fn )}

^est

^de Cauchy

^ds

^{ACIR ) alon} ( ^{fr )}

^et

⁽ In )

^{sont de}

^Cauchy

^{dam L', done}

{ ^{Fg II I!}

^.

^na

^'s

^In

⁼

^{¥ E.}

^done

^HE

^-

^{5K£ ÷ HEI He 's}

^.

^HE

^{- ask. - so}

Done

for ¥5

^{E Coo , Ce}

qui

^{real dine en}

particulier

que

f= 5

^.

A imig

^{e L' et done}

is

^{EL ', i-e} g ^E A CIR

)

. Ce qui ^donne

I

⁼

2¥ §

^{= g .}

Hfn

^-

f Haar

, ⁼ HE ^-

f Il

_, ^t k

fi

^-

I

¹¹_, ⁼ HE^-

f It

, +

HE

^-911

, → o . ✓

2)

^Si

f

^{E ACK ) alon}

FEL

^{' et}

E

^{ours i}

⇐

^an

I

⁼

^25ft

^.)

⁾

^{. Done}

^I

^{E ACIR ).}

In ^{verse me -}L si

FE

^ACIR)

@ ^f

^{E L' pan}

^{hyv )}

^sales

^felt

^{C- C' =,}

^f

^{E A CIR ).}

3)

^Saint

f

^, g ^E ACK )_, on pose h=

¥ ^of

^{EL ' . On sail que}

^E

^{= g CdsL'}

⁾

^{, done}

^{f E}

^--

^fog

^{E C'}

et

fig

⁼

fei

⁼

I

^{* h E L' par les homie .Le's de * . par Pros inion 15 .}

On ^{a montu '}

fg

^{E ACIR}

⁾

^.

4)

^{Soi ent ,}

Cf ^g)

^{E L'(IR) x A CIRI}, alon

f

^{* g E L'CK)}

⁽

^can

^GEL

^{'Ccr )}

⁾

^{. De}

^plus f*I

⁼

fig

^avec

If

Coo

of

Ccr) et ^{E L'CIR). Done}

fig

^{E L'CK)} , Ce qui ^{mood re}

f

^* g ^C- A(IR

)

^.