T D Transf Fourier

T D Transf ^Fourier

Exercise I

Pour

f ⁽

^{x) = e-}

⁷² calculus I

^.

Exercise

²

L ^'

object if

^est

^{de caecaler} I

pour

f

^{Cx)= sine 'Cx}

⁾

^.

1) Montez

que

¥1 ! ^{¥ (}

^{' e- " " ex --}

^G

^-

^¥15

^.

2) Montney

^que

ein

^.

^final ¥ ⁾

^'

^e-

^{'' ' " da .}

^÷ ¥ ^¥ joie

^{. a.}

^{1=0 unite .ee}

^.

3) Monterey ^que I (5)

⁼

Till

^-

^{1¥ )}

^t.

Exercise

3

1) ^Soit

f

^{: IR - s Q ,}

^f-

^{EL 'CIR}

⁾

^et

fr ^{f (}

^{Hdt to . Si on pose Fn E IN}

^fn

^{Cx) =}

_↳ ^nf _find

talons

( fn )

^{est une}

approximation

^de

l

^' unite ^'.

2)

Pour ^{Lout R 71} , ie exist e une

fo

^{- clio -}

^Xp

^E

^Cf

^(IR)

tg

al ft EIR_, ^of

Xp

^{(t) E I}

b) FLEER ,R] _,

Xp

^{(t) = I}

c)

^tf

^IH

^{3 Rtl ,}

^Xr

^{( 4=0}

3)

^Soil

^f

^{: IR -s Q , il exist e une suite}

( fn ) ECL

^{CIR) tee} que ai si

f

^E

^{Lack )}

,

Hfnllq

^E

^Hf Hq

^et

^Hfn

^-

f Hq TO

I E g ^{c -}

b) ^si

f

^{E C}

^:

^{Ck ) , aeons}

^llfn

^-

^fell

^{, O-}

Exercise

⁴

2 welle fonchaon ^FEL

^'CIR

^{) vehifie} fr ^{f (}

^{x -}

^{g) fly ) dy}

⁼

^×¥

^tx

^?

Exercise 5

caecaeez f ?

^-

^e.

^{'t "}

^dsc

Exercise 6

2

wel

a E

L'

C IR

) vdnifie

^{, FKECR}

next ⁼

e-

^{' "} t

p ! ^e-

^{" " "}

^{ucssds (}

^{p > o}

⁾

Exercise

^7-

-

S oil gn=

HE

n ,ng

eh h =

HC

- I_, IT ^'

a)

Calcalez explicit

^{em eat}

^gn*h

^.

b) Montney que gn* h

=

In

^out

fix

^{) = I sincnx ) sense .}

Tx ²

c) Monkey

que

Hfnll

,

→ to

d) Monkey

que F^:

L'

→ Coo ^{n 'est} pas

surjective

^. e)

Montney

que F

(

^L'

I

^{est dense dam Coo .}

Exercise 8

Calculus ^Sot

-

sin÷dx

^eh

[

^{" da .}

Exercise

9

Mq

^si

^f

^, g EL ^'

aeons f

^* g ^E

Coo

. , It

f

^*

⁹¹¹

- E

Kfk

, 119112

Nq

^F

^(f)

^{* Fog ) = 2T}

fat

I sin ax

fancy

⁼

a-

^.

Caecaeeg ^fa

^*

^fb

^{, a,b so .}

Comb

ien g ^{- a t}

'

il de solution

a-

f*f= f

^dam

^{L2 ?}

Exercise to

D Soil f

^{E L?}

^C

^IR

^{) else}

^C-

^IR

^.

^Montney que FCC

.

f) (3)

^{= e- '}

'33

Tff )

^Ces

) 2) Montney

que ^si I

(f)

⁼ o sun un em ^. A de mesne > o

,

aeons

F g to ^{E Ved}

Leaf

, KEIR

}

^-

En die 'd wine que real Leaf , a EIR

}

^{dense ⇒ Fff}

)

_presque

jamais ^noelle

^.

3) Recipe que

^{meal ,}

monkey

^{que si}

^Fff ) _p.p.IO ^{alois Leaf}

^{,- KEIR}

^}

^{est dense .}

Exercise ¹¹

Soit f

^E

^{L2 CIR )}

^{, on pose}

Pfcxl

⁼

¥ fr ^fix

^-as)

^{sings dy}

1) Monkey

que

Pf

^{esh been}

define

^{eh continue .}

2) Monkey

que

Pf

^{EL ?}

3) Monkey

que

KPFK

_, e

Hflk

^et que

off

^P.

Exercise

12

q

^{ex) =}

I

- e

- "

It

^{too .}

- ,

t

I

4ft

Monkey que 9

_,.

*

9

_, ⁼

9++5

Carriage

^'

^I

(

Caine

ment

f

^{E L'CR}

⁾

^{. On a}

I (3)

^⇐

IF

^{e- " e- '' 3k da .}

÷ ⁽

^{e- ' '' e- is"}

⁾

^{= - in}

^e-

^"

'

e- ^'

'

3"

,

Izzie

^{- ' ' 'e- ik}

⁾ I

^{e 1×1 e-"2 E L'}

^C

^K

⁾

^{. Done} par th m de desecration ^sours

f

^{, on a}

I

^'

(3)

⁼

I !?

^{- ix e- ' ' ?e- is " dsc =}

[ ^I

^{e- ' '' e- ''3"}

] !?

^-

^{I ! !e-}

^{"'c- is ) e-is" da}

= ^- Zz

Its )

^.

Done

I

^est

^{solution de}

y

'

= ^-

Ig

^{, ie.}

y

^{= yCo ) e}

- "

%

y

^{Co) =}

Ico )

^{-- e- " '}dx =

hit

.

Its ⁾

⁼

^ith

^{e -}

^5%

^,

Corri

get

1) ^For

^--

Een ^C

^{' -}

It lei I

^,

^Freak # ( si;!I ^I

^'

For ^* ee ⁼

(

^{I -}

^{IFT )tea}

^.

÷ I ^C

^'

^Ie

^.

^{incase Ci}

^-

^{II. It}

(

^{N = 2 Ntl}

)

2) ^La fondo

^on

g

^{( x) =}

µ,

^{, - z est continue sun}

^C-

^{Fit ] I to}

^}

En

O

, ^on

fail

^{an D.C .}

"

^{" '=}

c÷÷÷* ^,

^-

^÷

^{, =}

÷fu÷÷75p

^-

^{I ]}

=

÷(i-±

12 ^{, - '}

^] YI

⁼

^÷

^,

⁽

^{It to Cah - I}

⁾

⁼

^'s

^to

⁽

^x

⁾

^.

Done

g

^se

prolonge

^{par cont invite ' en O.}

⇐ I f

^-

f I

^E

÷

,

11911_, converge ^{vers O,}

unif

^{. en k.}

3)

^Soil

kn

^{= Lentil u}

⁾

,

celom

en

^.

^÷

^.

Ye

^-

^im

^{" ex --}

_en l

^{I -}

:# i ^t.ci

^{- '}

^¥5

,

I :# ⁽

^'

^je

^{- '' "}

^=L

^.

^{÷ .sc}

^'

^e-

^it

^:#

^a

= 2

! ! !!!! ^@

^{inch ' e- it}

^1¥

^d,

→ 2

flu

⁾

N

D

'

oui

Fcu

^{)= it}

(

^{i -}

If It

.

Carriage

'3

1)

.

ftp.fn

^{= I -}

•

einmsuopkfnll

^{, c- :}

^HEH

^{, =}

¥

,

frnlfcnxsldx

^{= -}

• f Soo_,

Can

^'m

fr

.gs

.gg/fnC4ldL=o:fgTfCnxldsc=fn?fly

⁾

^{dy I}

^car

^rested

^'une

^{integrate convergent}

2)

^.

f ^ECHR

⁾ ,

f

⁼

{

^{> °o "x soto . par}

^{example f}

^{e-O" "} ,^, x^a I^{> o} O .

•

Fgc

^{- no inane}

4=0

^{' ' fo : h case}

^ffxjfci

^{- x}

)

^{en c- ,}

{

^{> °} " " ' '

= I se 7, I ₌

O Sinon .

Has =

to

^{" hands esh}

^CT {

^→ate^to 70^{x soa} 7 ^{, . 0-}

^{mend goes}

^{-. .}

• g

(

^Rt I^- x

)

⁼

f

^{OI xx 3E RRtl}

g ( ^Rt c t x ) =

f

⁰I ^Icx s^c, ^-- ⁽R^{Rtc) o - Mad}

^gcxt

^{Rti ) g ( Rtl - x) .}

3) _4=1×1,1

, ,

oci X

, vie -C da 2) pour ^{R=l .}

\

Ying thnx

^{= n ) est a-}

^support

^dans

ft ^it ]

^ec

⁽

^Yn

^In

^{approx unite}

^!

Soil In

⁼

Into

a

[ ^{gas fcx}

^{- 4 de}

fr

^{: Yn *}

( Inf )

^.

Aeons free of

^{, car Ynec}

^Inf

^{so -L a' SUPP.}

^Cpd

^{-A =}

:(

^{A > R}

⁾

Hfnllq

^e

^114nA

,

Hftnllqellftnllq

^e

^{Hf Hq}

Hf

^-

fake

^E

^Hf

^{- 4h *}

^f

^ItYn

Hat

^*

^{( f}

^-

^f-

^{In ) Hq E K}

^f

^-

Ynxfllg

^tell

^Tall

^{, Hf -}

^f

^In

^Hq

E

Hf

^-

^4h

^*

fllqt ^Yf

^-

^f InHg

→ o ^{→ o}

car approx unite^.

Cv dom .

Si

f ^Ecg

^:

^Hf

^-

^fall

^{- E}

^Hf

^-_→

^4h

_o^*

^{f Hot Hf}

^-- so

^flat

^,

approx ^{unite' care sup}

If

^{cast → 0 car}

f

^O.

txt? ^n-c

Comige

'

4

On

reconnoitre f

^*

floes

⁼

÷

^{, - Comme}

^f

^{C- C'}

^{salon f*f}

^{aemi . On a}

^done

- i3 x

fff (5)

⁼

fr

^It

^e-

^{x2 da .}

11

Fess

^'

On a vu que

la transform

^{' de Fourier de e- I "I}

^e'

^tail

,÷z

^{. Mais e- " 'I E A}

⁽

^IR

⁾

^done

↳

^{dsc -- T}

¥

^.

)

^C.

³⁾

^{= te - B'.}

D ^'oi

I (5)

⁼

^of

^{e -}

^131/2

^.

^Par

^{Founder inverse}

^{f (}

^x

⁾

⁼

÷!÷

^,

±Z

⁼

^¥

^,

Corri get

⁵

- t

e est

integrable

^{en t - car o}

( Is )

• -

A

er ^{~ en} O do - a

integrable

^{e n O.}

• it

eit

^" = in e-

teitx

I in e- ^t e

" "

I

⁼

^The

^{- t}

integrable

^{sun Co,t - C.}

.

f 'cxs= fo-ihe-teitxdr-f.ir ^et

^{" " - " dr =}

^{( ] ? !} ir

^-

^{I? d}

^'

f

^':

f- ^f

¥

⁼

^z¥

^t

^26¥

^,

^{en If I}

^{= -}

^I

^,

^en

^{I setti}

⁾

^t

^iz

^{an clan x t K}

f-

Ix)⁼ C I^x 't'

5¥

exp

( Farc

^{Lan x}

)

C =

flo ⁾

⁼

of

^{do =}

21%

^{- " du =}

^hit

^.

Corri age

'

6

f

^{hes -- e- I " ,}

I (5)

⁼

^I

It

32

u =

f

^t

p f

^{* a ⇒}

^I

⁼

It

p

f ^a

^⇒

^I

⁼

ftp.p-u.z#+gz

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2ps

^so

⁽

^{i.e. B > Yz}

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^{ion n '}

^{et pas continue}

^car

^din

^{on in}

^ahem

^{s 'an na}

^le

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si o .

B' I

a

=

÷

.

=

÷pH%⇒ ⁾

=

u ^--

I

,

E

=

It FT%f ⁾ EH

⁼

^{It ÷}

^,

^{Viers f )} fruits ^{=v÷q e- Visit}

^'

³¹

Carriage

^'

⁷

a)

_gn ^{* e}

Cx

) =

francs

⁾

^ha

^-

^{9) dy}

⁼

! ! ^He

^{, is'" - m}

^dy

⁼

^{fr Henny}

^'s'

^Ha

^.,

^{Iii 's dog}

=

fr ^HC

^{- n, is nCoc-, ,x+ ,] ' S)}

^dy

• Si C- ⁿ,n] ⁿ

(

x ^-I, set IT =

¢

^{i i-e , x s - n -I ou x > ntl . → O}

. Si Cx ^-s, sets ] C C- him ]

, i.e x E

f-

^ntl ,n -I

]

^{i → 2}

. Si x E

C-

^{n -I},^- htt

]

^{: x tht n}

Si x C- (n-i ,ⁿ) i n - x t I

-

b) FC He

a _,a,Cal

)

⁼ x

^Ii

I ( green ⁾

^{= g}

? E

⁼

÷

^, sincnxssinx

hncx

)

.

-

Rais

F(gn*h )

^{E L'CIR) ⇒}

gn*h= en

^{C- x) =}

¥

^sinnera2 ^f,, (x)

IC

^{- x) --}

fix

^{) si} f haie ^:

↳

^{fig , e i "}

^9dg

⁼

fjfc

^{-a) e- ''"9}

The

e) Henk

^'

tr ^E

^{" " da x}

^of

^I

lose

⁼

fi

^"

^Ilose

^.

^It

^{- "}

^da

^--

^II. ¥!

^""

O

"

t.at

,

! !

^""

^'t

^{" 'd'} ."

s into .

2¥

^{nm Co .tk}

^] )

in

^.

Carl -Yet

→

- rial -

d)

Feel

inj

^{. Si F -}

mj

^{⇒ F:}

^! @

^{11114 →}

^Cc

^{:," lls}

⁾

^{his, core . ⇒}

^I

^{' cone .}

⇒ 7C

/ HEH

, e c

Kelly

^{. th .}

HE ftp.kgnxhll

^{- = 2 , Kkk, → - i -}

^namable

^.

e)

Fly ) ⁼

Y

^.

Come .ge

' 8

sin '

( Kh )

L a

fonclion

,

--

f

a pour

ICH

⁼

z

IE

^{L' ⇒}

J

^{sin 'C sea,}

µ

else ⁼

§ (

^o

⁾

^{= 2T}

^flo

^{) = 2T}

4

215

^{on =}

ja

^,

I 1-

^'

^da

^-

E

^.

f- ^Ed

^,

^Plancherel

^:

HE 'll !

^{-_ at}

^{Kfk !}

2/5

⁼

^# flu

^{- seeds.}

24/5 ^nigh

^{dy = htt}

^{f P ] !}

1- ⁱ

^{" --}

^ta

^.

Corri

age

^'

9

•

I fr ^{fly ) g Cx}

^{- g}

^{) dy /}

^e

^{Cuss If} #

^{I '}

^{dy ) "} ( In ^Igcse

^-

^{g) I 2dg} ) ^"

⁼

Hfllzllgllz

^.

Hf

^{* as}

^He Ellfllzllgllz

^.

•

( fr )

_,

(

ga

)

^C

CE ^fn -5 _f

9

.

g

fan

^* oh est

aeons CE

^{. ✓}

Hfn

^* gn ^-

f

^*

^SH

, E

Il fn

^*

Gn

^- 9)

It

_, ^t

H Cfn

^-

f)

^*

94

_, ^E

HEH

_,

Agn

^-

GHz

^t

HE

^-

flk ¹¹⁹¹¹

, → O^.

⇒

fig

^{E Coo . -}

•

It FIFI

^*

Ffs )

^-

He

^e

HFC ft

^*

Ffs

^-

tfg grill

, +

HFH

^-

^fat

^*

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^{t at}

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^-

fig ^He

-

⇐

⁺ it

ltfna

^-

Fa

.

He

*

In

^{= 2 IT}

fngn )

f HICE

Ilk

^{11719- 9ns 112 t It Fff -}

^fill

^,

^Haga

^{)Hz t at}

^llfncg

^{- grill} ,

t 2 IT

Hffn

^-

f) 94

,

E Iff ^Cl, kg^- grill_, ^t Itf ^-

fnllz

^{C t 25}

llfnll

,119-9mHz ^{t 25 Kk-}fllz KgIf_, ^{etc ...}

•

ga

⁼

It ^Haa

^{,as ⇒}

^Fa

⁼

^¥ since

⁼

^fa

r

fa

^*

fb

⁼

^{Flsa )}

^*

Ffs b)

⁼

259%

^{b =}

9am

⁼

fans

•

fat f

a

=

fa

^{Ka .}

(

_orrige ^{' 10}

1) La for ^male

^{est vnaie}

^{pour f}

^E

^{L' no}

^.

^{Pour fed}

^o-

^{mend (E)}

^{c L'}

^{nd -9 f}

^.

Ex

^{esh convince}

L2

→

L2

:

It Caf

Hz ⁼

llfllzj

^{Feel anni continue L2 → Cl. ←}

2) Soil g_, ^{# o},

nuke

hors de A

,

et

g

_, ^E

Ll

^.

Aeons

_,

th

=

?

^2;

Zog

^{j .}

^f

^{E Ved}

^{LEX fix}

^{EIR }, on a}

< g. ,

Fch

)

)

^{= ;}

_fr ⁵

^{, Cal e-i " is}

Ed ^Iffley

⁾

^dy

^{= O .}

Mais g_, ^E

Ll

⇒

g.

⁼

Ffg )

→ CF

Cg

) , Ich ) 7=0 ⇒ Los ,h 7=0 . Aim .^- g ^E Ved ^- et

red ^t #

Lots

. No - dense .

3)

^{Si g E Ved - ⇒}

fr ^FIG

^Cgi

^Fff

^{)Cy) e- i "9}

^dy

^{= O tx E IR}

-

i^.e ^.

ITS

) Fcf ) = o ⇒

FIT

⁾

Tff

^{1=0 ⇒}

^Fcg

^{) = O.}

Carriage ^'ll

1) Convolution 22*22

^.

2)

^{b ee}

f

^{E L ' ⇒}

¥ ^sings

⁼

^FCN

^et

^f

^--

^{FG )}

Pf

⁼

¥ tiny

^*

^f

⁼

^FC

^{as *}

^{7cg )}

^{= 2T}

^ET

Mais

h (g)

⁼

If ^He

^{, , ,}, ^, bonnet

, done

hg

^{E L'n}

^L2

Done

itchy Pf

^--

⁾

^{E L}

^?

³¹²

3)

¹¹

^Pf

^{112 = 2T}

^{It Flghlllz}

⁼

⁽

^ut

^{) Hgh}

^{112 E}

^{At 119112}

^⇐

Hfllz

pop ( f )

⁼

PC

^2T

Flash ) )

^{= 4T ' I}

⁽

^g

^{h2 )}

^{= act}

^Hgh

⁾

^{=P I fl}

^.

Si

k

=

Ice )

E L

} ^Ck

^I

^{Pf >}

^{= 2 it s}

^Fel

^,

^{Flag ) )}

^{= 2 TL e,}

^{Eng )}

^{= 2 it}

^LEE

^{, g}

^>

= 2T the ,

g)

^{= CLIF Che}), Icy) > ^{= L} Pk

If

^{) .}

PIP

^.

Corri ge^' 12

9TH ⁾

⁼

^2A

^{e- " "'}

⁴⁴

= e- sit

I. focus

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

fuscus

^.

9¥

^{, =}

gigs

⁼

9€

^⇒

^4*9

^{, = 9ns -}