2075 TC WB 06 16 corr .pdf


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MINISTÈRE DE L’ÉDUCATION NATIONALE
MINISTÈRE DE L’ENSEIGNEMENT SUPÉRIEUR ET DE LA RECHERCHE

T

Agrégation externe de mathématique
Epreuve d’admissibilité
Analyse

Corrigé
du devoir n°2

Jean-Etienne Rombaldi

1-2075-CT-WB-02-13
407



 


   
 
  wk 
  
     wk = ρk eiθk  ρk =
|wk | > 0  θk ∈ [−π, π[ .
  

⎧ 
2
n
n
n








2



w
=
ρ
+
2
ρ
ρ
cos


θ
)
=
ρ2k + 2
 (wj w )
k
j


j
⎨ 
k

k=1
k=1
1≤j<≤n
k=1
1≤j<k≤n
 n
2
n






|w
|
=
ρ2k + 2
ρj ρk
k

k=1

1≤j<k≤n

k=1



n
n

 

     wk  =
|wk |      
k=1

k=1




ρj ρ (1 − cos (θ − θj )) = 0

1≤j<≤n

 
     
  

   ρj ρ > 0,
 
   cos (θ − θj ) = 1  θ − θj ∈ ]−2π, 2π[ 
 1 ≤ j <  ≤ n 
 
−π ≤ θj < π  −π ≤ θ < π 
 −π < −θj ≤ π  −2π < θ − θj < 2π  
 
 θj − θ = 0  

 (wj w ) = ρj ρ ei(θ −θj ) = ρj ρ = |wj | |w |
  
 θ    
  θj ,
  wj = ρj eiθ = |wj | eiθ 
 

 j 
    1  n.
     (ej )1≤j≤n    
  Cn ,   (aij )1≤i≤n = Aej  
 

   
    
 aij > 0 
 
 i, j 
   
1  n.
! "   A |z| = |Az|  A   
   


n
n







aij |zj | = 
aij zj 


j=1

k=1


 
 i 
    1  n. #  i = 1,    
  
 $
n
n



(a1j zj )1≤j≤m       a1j zj  =
|a1j zj | ,  $  
 θ ∈ [0, 2π[  
j=1

k=1

a1j zj = e |a1j zj | 
 
 j 
    1  n,     zj = eiθ |zj |
  a1j > 0 
 
 j.
% 
   

    & 
  

 


 
n
n
n
n
 





 

 


  iθ
aij zj  = e
aij |zj | = 
aij |zj | =
aij |zj |

 


 


k=1

k=1

k=1

k=1

  aij > 0 
 
 i, j,     A |z| = |Az| .

______________________________________________________________________________________________
1-2075-CT-WB-02-13
2

   

n



Ax | e =

(Ax)k =

n
n




akj xj ≥ 0

k=1 j=1

k=1


 x ∈ C    A   
 n


akj 
       α = max
1≤j≤n

Ax | e =

n
n




k=1

akj xj =

k=1 j=1

n





j=1

xj

n




akj

≤α

(Ae)k =



akj ≥ (βe)k = β = min

1≤k≤n

j=1

xj = α x | e

j=1

k=1

   
 x ≥ 0 
x ∈ C.
  
 k 

 1  n 
n



n



n




akj

k=1

 Ae − βe ≥ 0  β > 0  e ∈ C \ {0} .  
 t ∈ [0, β] 

Ae − te ≥ Ae − βe ≥ 0
 E   

 n  {0}
      [0, β] .


t > α = max
akj   x ∈ C \ {0} ,   
1≤j≤n

k=1

Ax − tx | e = Ax | e − t x | e
≤ α x | e − t x | e = (α − t)

n



xk < 0

k=1

 Ax − tx ∈
/ C 
     
   t ∈
/ E.
  E   
 
α,  E ⊂ [0, α] .



    E  
 

  
   C  
 !C =
n

πk−1 (R+ ) , "  πk   
      Rn
R,  
 
k=1

     #      S 1  $
    Rn       


E = t ≥ 0     %  x ∈ C ∩ S 1 
& Ax − tx ∈ C

1
x,   y ∈ C ∩ S 1  Ay − y ≥ 0#

x

' (tk )k∈N        E  
( 

 t ≥ 0  (xk )k∈N 
      
 C ∩ S 1    Axk − tk xk ≥
 0 
 k ∈ N, 
1
  C ∩ S ,   %

   )  xϕ(k) k∈N  
( 

 
x ∈ C ∩ S 1    


Ax − tx = lim Axϕ(k) − tϕ(k) xϕ(k) ≥ 0
! Ax − tx  x ∈ C \ {0} ,    y =

k→+∞

______________________________________________________________________________________________
1-2075-CT-WB-02-13
3

 t ∈ E   
 E    
     E    

     E = [0, ρ]  ρ = sup (E) .
 E       ρ ∈ E  E ⊂ [0, ρ] .
 x ∈ C ∩ S 1  
 Ax − ρx ≥ 0,   t ∈ [0, ρ] ,   
Ax − tx ≥ Ax − ρx ≥ 0

 t ∈ E.    [0, ρ] ⊂ E 
 
 
  E = [0, ρ]    {0} ,   ρ > 0.
!"  x ∈ C \ {0} 
 y = Ax − ρx ≥ 0.   y = 0.   
 
y ∈ C \ {0}   A  #    
   Ax
 Ay           
   
ε=

min (Ay)k

1≤k≤n

max (Ax)k

∈ ]0, 1]

1≤k≤n

   

(Ay)k − ε (Ax)k ≥ min (Ay)k − ε max (Ax)k = 0
1≤k≤n

1≤k≤n

  k     1  n,  Ay − εAx ∈ C.
$ 
  
A (Ax) − (ρ + ε) Ax = A (y + ρx) − (ρ + ε) Ax = Ay − εAx ∈ C

 Ax ∈ C \ {0} ,  ρ + ε ∈ E,      ρ = max (E) .
   Ax = ρx.
! " % Ax = ρx  x ∈ C \ {0} ,    ρ  
     A  
x            &   
ρxk = (Ax)k > 0

  k     1  n,      x   
      
'
!"  z ∈ Cn  
 Az = ρz,   
  
|Az| = |ρz| = ρ |z|

 



n
n






|Az|k = |(Az)k | = 
akj zj  ≤
akj |zj | = (A |z|)k


j=1

j=1

  k     1  n,  A |z| − |Az| ≥ 0  
A |z| − ρ |z| = (A |z| − |Az|) + (|Az| − ρ |z|) = A |z| − |Az| ≥ 0

______________________________________________________________________________________________
1-2075-CT-WB-02-13
4

   A |z| − ρ |z| ∈ C  |z| ∈ C, 
  A |z| = ρ |z|  

     
|z| = 0,         

 
   θ ∈ [0, 2π[ 
 z = eiθ |z| .

  
z | v = eiθ |z| | v

 

  z | v = 0,     
|z| | v =

n



|zj | vj = 0

j=1

 |zj | vj = 0 

j     1  n, 

 
    
   
  vj         |zj | = 0 

j
    1  n,  z = 0.
    ! v ∈ ker (A − ρId)  
 
  
 
 ker (A − ρId)
     1. " #        
 

 $

   z = 0   !  
   ρ
  %& ! v, 

   
' λ ∈ C 
 
    A 
 |λ| = ρ,   & 
z ∈ Cn \ {0}
  
         
|Az| = |λz| = |λ| |z| = ρ |z|

 A |z| − |Az| ≥ 0  
A |z| − ρ |z| = (A |z| − |Az|) + (|Az| − ρ |z|) = A |z| − |Az| ≥ 0

 A |z| − ρ |z| ∈ C  |z| ∈ C, 
  A |z| = ρ |z|     
θ ∈ [0, 2π[ 
 z = eiθ |z| .
 |z| ∈ ker (A − ρId) = Vect {v} ,   

 z ∈ ker (A − ρId) 
λ = ρ.

   |λ| < ρ 

 
   λ  A  #    ρ.
" (       

  ' A 
       
 Mn (R)   ρ  

 
    A  
  

ρ = ρ (A)   )    A        
   
   &  
  
    
 A  t A  * )+    
    t A   

       

 ρ  
  )    tA 

           1 &  
  
   Φ !
      
 v  Φ  !         
  !
v | Φ = 1.

, '  x = 0
  
    A !      λ  
  
      


λ x | Φ = Ax | Φ = x | t AΦ = ρ x | Φ

______________________________________________________________________________________________
1-2075-CT-WB-02-13
5

x | Φ  
       Φ 
    x    
     
ker (A − ρId) = Vect {v} 
   v.
 y      
d
y = Ay, y (0) = y0 ∈ Rn
dt
−tA
y (t)      
  
  z : t → e
 




 
 
  

λ = ρ



x ∈



z  (t) = e−tA y  (t) − Ae−tA y (t) = e−tA (y  (t) − Ay (t)) = 0

 

e−tA

z (t) = z (0) = y0  y (t) = etA y0 .

  
  
         (1) .
 y0  
        y (t)  t ≥ 0   A    
!    t ∈ R,   
  

 
y (t) | Φ = etA y0 | Φ = y0 | t etA Φ
  


A




    





t

 tA 
e
= et

tA

+∞ k


t

=

k=0
 

t



tA

e



Φ=

+∞ k


t
k=0


      

 ! 

t ≥ 0,

t

k!

A Φ=
k

t

k!

+∞ k


t
k=0

k!

Ak

ρk Φ = eρt Φ

y (t) | Φ e−ρt = y0 | Φ

  

0 ≤ e−ρt yj (t) Φj ≤ y (t) | Φ e−ρt = y0 | Φ
 


0 ≤ e−ρt yj (t) ≤
 

yj (t)

     

"  
 



t → e−ρt yj (t)

Φj

y0 | Φ
Φj


    

  
  

#   $  

πA (X) =

p


[0, +∞[ .

(X − λk )αk

k=1

σ (A) = {λ1 , · · · , λp } .
αk
(      k
   1  p, Ek = ker (A − λk Id)
    



   
       λk ,  ) *   
   
p
n
 %+      C = ⊕ Ek    
  
 %&   



A,

'

k=1

y0 =

p



xk

k=1

______________________________________________________________________________________________
1-2075-CT-WB-02-13
6

 xk ∈ Ek   k ∈ {1, · · · , p} .
   k ∈ {1, · · · , p} ,   yk
        
 
y (t) = Ay (t) (t ∈ R)
y (0) = xk

z =

p


yk            y. 

k=1

p


           y =

yk .

k=1

 (A − λk Id)j xk = 0   j > αk , 

yk (t) = e xk = e
tA

λk t t(A−λk Id)

e

xk = e

λk t


α

k −1 j
j=0

    

Pk (t) =

t
(A − λk Id)j xk
j!

α

k −1 j
j=0



  



y (t) =

t
(A − λk In )j xk
j!

p



eλk t Pk (t)

k=1

   k ∈ {1, · · · , p} , Pk               
   αk − 1     Cn .
          


yj (t) =
Pλ,j (t) eλt (1 ≤ j ≤ n)
λ∈σ(A)

Pλ,j    !        
(X − λ)α(λ) .

α (λ) − 1,   πA (X) =

λ∈σ(A)

"#$  ρ = ρ (A)       A,  α (ρ) = 1 %  !
      !     &  Pρ,j    !   $
"'$ 



e−ρt y (t) = C +




e(λ−ρ)t Pk (t)

λ∈σ(A)\{ρ}

 C = (Pρ,j )1≤j≤n ∈ C .
 |λ| < ρ   λ ∈ σ (A) \ {ρ} , 
n

e(λ−ρ)t = e((λ)−ρ)t  

 (λ) − ρ ≤ | (λ)| − ρ ≤ |λ| − ρ < 0
  lim e(λ−ρ)t Pk (t) = 0  lim e−ρt y (t) = C.
t→+∞

t→+∞


      C      y0 ,  

    ρ,

______________________________________________________________________________________________
1-2075-CT-WB-02-13
7

  
    
       Cn =



ker (A − λId)α(λ) ,

λ∈σ(A)

     C ∈ ker (A − ρId) = Vect {v} .
   C = γv 
y (t) | Φ e−ρt = y0 | Φ

    
y0 | Φ = lim


e−ρt y (t) | Φ = C | Φ = γ v | Φ



t→+∞

       
y0 | Φ
v = y0 | Φ v
v | Φ

lim e−ρt y (t) = C =

t→+∞


      v | Φ = 1



 
     

    
       T ∈ L (E)    

       
        Rλ (T ) = (λId − T )−1 ∈ L (E)
 λ ∈ Res (T ) .






 ! T k L(E) ≤
T
kL(E)
  k ∈ N,  
T k    
 " 

T
L(E) < 1,   "    
  E.
#       
   
  
    
+∞




Tk


(Id − T ) =

lim

k



k→+∞

k=0

= lim



k→+∞


Tj


(Id − T ) = lim

j=0

Id − T

  " 
(Id − T )

k+1



k→+∞

k




Tk


(Id − T )

j=0

= Id

+∞




Tk

= Id

k=0

   "  (Id − T )    $   

+∞



T k   1 ∈ Res (T )

k=0

!        

+∞




(Id − T )−1 


T
kL(E) =
L(E)
k=0

1
1 −
T
L(E)

______________________________________________________________________________________________
1-2075-CT-WB-02-13
8

 


1 
1

  |λ| >
T
L(E) ,   
< 1, 
Id − T     
λT 
λ
L(E)


1
 (λId − T ) = λ Id − T 

λ
Rλ (T ) = (λId − T )

−1

1
=
λ


−1
+∞
1
1
1 k
Id − T
=
T ∈ L (E)
λ
λ k=0 λk

  
λ ∈ Res (T ) 


Rλ (T )
L(E) ≤

1
1
1
=
T

L(E)
|λ| 1 −
|λ| −
T
L(E)
|λ|



|λ|→+∞

0

   λ0 ∈ Res (T ) T0 = (λ0 Id − T ) .
  h ∈ C,   



(λ0 + h) Id − T = h · Id + (λ0 Id − T ) = (λ0 Id − T ) Id + h (λ0 Id − T )−1


= T0 Id + h · T0−1 = T0 (Id + h · Rλ0 (T ))


       h   


h · Rλ0 (T )
L(E) = |h|
Rλ0 (T )
L(E) < 1

      h ∈ C   |h| <

Res (T ) .



1
    D λ0 ,

Rλ0 (T )
L(E)
    
1
,   
  |h| <

Rλ0 (T )
L(E)

1
,   λ0 + h ∈

Rλ0 (T )
L(E)




   Res (T )


−1 −1

Rλ0 +h (T ) = ((λ0 + h) Id − T )−1 = Id + h · T0−1
T0
= (Id + h · Rλ0 (T ))−1 Rλ0 (T )
+∞


=
(−1)k hk (Rλ0 (T ))k+1
k=0


  ! 

Φ (λ0 + h) =  (Rλ0 +h (T ) · x) =

+∞



αk hk

k=0



k
k+1
" αk = (−1)  (Rλ0 (T ))
·x .
 #
 Φ 
!   !  $    λ0 .

______________________________________________________________________________________________
1-2075-CT-WB-02-13
9

  Res (T )  
    σ (T ) = C \ Res (T )     C.
 |λ| >
T
L(E) ,   λ ∈ Res (T ) ,  λ ∈
/ σ (T ) ,  σ (T )   


     D 0,
T
L(E)     
 
 σ (T )        C,  
           

        Res (T ) = C,     (x, ) ∈ E × E  ,   

Φ : λ →  (Rλ (T ) · x)
     C.
 

lim
Rλ (T )
L(E) = 0  | (Rλ (T ) · x)| ≤

E 
Rλ (T )
L(E)
x


|λ|→+∞

 

lim Φ (λ) = 0.   Φ   !  !     C, 

|λ|→+∞

       
"    (Rλ (T ) · x) = 0     (x, , λ) ∈ E × E  × C      


∀ (x, ) ∈ E × E  ,  (R0 (T ) · x) = − T −1 · x = 0
         = 0 #  $     y ∈ E     (y) = 0 
x = T (y)%
"   σ (T ) = ∅.
&
#%  x ∈ E \ M #  M = E %   δx = d (x, M) .    
         n > 0,  ' yn ∈ M    

δx ≤
x − yn
< δx +

1
n

"   lim
x − yn
= δx . ( δx = 0,      x =
n→+∞

lim yn ∈ M,

n→+∞

   M         x ∈ E − M. "   δx > 0.
1
 '   n > 0 ) *     δx +
≤ 2δx   
n
zn = x − yn ,   

d (zn , M) = inf
x − (yn + y)
= inf
x − z
= d (x, M)
y∈M

z∈M


 0 <
zn
≤ 2δx .
1
  u =
zn ,  
u
= 1  

zn



 1

1
1

zn − z 
inf
zn −
zn
z
=
inf
zn − y

d (u, M) = inf 
=

z∈M
zn


zn
z∈M

zn
y∈M
δx
1
d (zn , M)
=

=

zn


zn

2

______________________________________________________________________________________________
1-2075-CT-WB-02-13
10

  
  

(xn )n∈N





 

x0 = u,     
 x0 , · · · , xn , 
n ≥ 0, 
M  
 
   
    
M          
    xn+1  
xn+1
= 1
1
d (xn+1 , M) ≥ ,      
2
1

xk − xn+1
≥ d (xn+1 , M) ≥
2

 k 
 
0 n.

 
 

  


M

    

0 ∈ Res (T ) , T  
  "     T −1 ∈ L (E) .

xn
= 1 
 n ∈ N,  
T −1 xn

T −1
L(E) 
 n ∈ N
 T      

   (T −1 xn )n∈N   
 −1

 



T xϕ(n) n∈N     T T −1 xϕ(n) n∈N = xϕ(n) n∈N  
  

 1

 n = m.
    
     xϕ(n) − xϕ(m)  ≥
2
#    0 ∈
/ Res (T ) ,  0 ∈ σ (T ) .
! λ = 0  


 T. #   Bλ (0, 1) = B (0, 1) ∩ ker (λId − T ) 
    
  ker (λId − T ) ,   

 !

T (Bλ (0, 1)) = λBλ (0, 1)


T (Bλ (0, 1)) = T (Bλ (0, 1))

  

T (Bλ (0, 1))

 ( )    
 

   $     % 

    

ker (λId − T )

1
x → x
λ

Bλ (0, 1)

 &
'

       

*
 !      
 

α > 0   
 x ∈ E,
(λId − T ) x
≥ α
x
,
n ≥ 1,   
xn ∈ E  

  

 
 
 


1

(λId − T ) xn
<
xn
. +   
     %
    
n
1
1
yn =
xn ,     (yn )n∈N∗  
yn
= 1
(λId − T ) yn
<

xn

n

 n ≥ 1.
,
T     
 

   
 (yn )n∈N∗ , 




    T yϕ(n)
 
  
 
   yϕ(n)

n∈N
n∈N∗
 
z ∈ E.
-  


 
 
 

λyϕ(n) − z  = (λId − T ) yϕ(n) + T yϕ(n) − z  ≤ (λId − T ) yϕ(n)  + T yϕ(n) − z 


1
<
+ T yϕ(n) − z 
ϕ (n)


     λyϕ(n)

 
 z      
n∈N∗


1
yϕ(n) n∈N∗ 
 
 y = z.
λ
  
 

(λId − T ) y = lim (λId − T ) yϕ(n) = 0
n→+∞

______________________________________________________________________________________________
1-2075-CT-WB-02-13
11




y
= lim yϕ(n)  = 1,   
 
 

  λId − T.
n→+∞

 
(yn )n∈N = ((λId − T ) xn )n∈N   
  
  (λId − T ) (F ) 
      
 y ∈ E.
   
 

α
xn − xm

(λId − T ) (xn − xm )
=
yn − ym

  
 
 (xn )n∈N 
       F .     
     
 x ∈ F 


y = lim (λId − T ) xn = (λId − T ) x ∈ (λId − T ) (F )
n→+∞

   (λId − T ) (F ) 
   !


 " 
  # 

n ∈ N,  Im (λId − T )n+1 ⊂ Im ((λId − T )n ) .
 Im (λId − T ) = E,  $
 x ∈ E \ Im (λId − T ) 
 

y = (λId − T )n (x) ∈ Im ((λId − T )n )


 y ∈
/ Im (λId − T )n+1 .
 %
#     
 #  $
 x ∈ E
  y = (λId − T )n+1 (x ) 

n
 (λId − T ) 
 
   
      
 λId − T

 
    x = (λId − T ) (x ) ∈ Im (λId − T ) ,   
 !
&   Im (λId − T )n+1  Im ((λId − T )n ) .


  '

n ∈ N, Im (λId − T )n+1 = (λId − T ) (Im ((λId − T )n )) 

  



   Im ((λId − T )n ) ,    $
 xn ∈ Im ((λId − T )n )


 1

 
xn
= 1 
d xn , Im (λId − T )n+1 ≥ .
2
    (λId − T ) 
   
  
(    Im (λId − T ) = E

 
   
    
 (xn )n∈N
  # 

n ∈ N, xn 



 1
 
   Im ((λId − T )n ) 
d xn , Im (λId − T )n+1 ≥ .
2
 

(yn )n∈N = (T xn )n∈N ,  

n ∈ N 






d yn , Im (λId − T )n+1 = d T xn , Im (λId − T )n+1



= d λxn − (λId − T ) xn , Im (λId − T )n+1


 (λId − T ) xn ∈ Im (λId − T )n+1 ,   






d yn , Im (λId − T )n+1 = d λxn , Im (λId − T )n+1



= |λ| d xn , Im (λId − T )n+1
 λ = 0 
    

 |λ|


d yn , Im (λId − T )n+1 ≥
2

______________________________________________________________________________________________
1-2075-CT-WB-02-13
12





 m > n,   xm ∈ Im ((λId − T )m ) ⊂ Im (λId − T)n+1  


T
n
n+1 


 (λId − T ) ,     ym = T xm ∈ Im (λId − T )
 

yn − ym


|λ|
2

 

 T  
   (xn )n∈N      (yn)n∈N = (T xn )n∈N
             
       
  (λId − T )    
! " # λ ∈ σ (T ) \ {0} . # ker (λId − T ) = {0} , $ 
%
 (λId − T ) 
          $      $   $ 
   &       !% 
  
%
 
' %"  λ ∈ Res (T ) ,   $ 
(   ker (λId − T ) = {0}  λ      T.


 
   
  
 

)
!" &     $     T        
* 
 x
 1
|T (f ) (x)| ≤

0

|f (t)| dt ≤

0

|f (t)| dt ≤
f


  f ∈ F   x ∈ [0, 1] ,   
T (f )
∞ ≤
f
∞  
f ∈ F  T     
T
≤ 1.

  T  
   + 
   B (0, 1)  
  
  C 0 [0, 1] ,  T (B (0, 1))  
     
 % 
 $*,* 
  f ∈ B (0, 1) ,  
T (f )
∞ ≤
f
∞ ≤ 1,  T (B (0, 1)) 
 

  
  f ∈ B (0, 1)   x, y  [0, 1] ,   


|T (f ) (y) − T (f ) (x)| = 

y
x



f (t) dt ≤ |x − y|
f
∞ ≤ |x − y|

 T (B (0, 1))   
!" #  T 
    λ ∈ C. -     
f ∈ C 0 [0, 1] \ {0}   


∀x ∈ [0, 1] , T (f ) (x) =

0

x

f (t) dt = λf (x) = λT (f ) (x)

# λ = 0,    T (f ) = 0  f = T (f ) = 0,   $ 
# λ = 0,   T (f ) (x) = αe x  α = T (f ) (0) = 0,  T (f ) = 0 
f = T (f ) = 0,   $ 
&       
   T $
    
1
λ

______________________________________________________________________________________________
1-2075-CT-WB-02-13
13

   f ∈ Cp0 ,  x ∈ R   n ∈ Z, 



 


f (n)
 e2iπnx


f

(n) =

f
 a2 + 4π 2 n2
 a2 + 4π 2 n2 ≤ a2 + 4π 2 n2




e2iπnx 
f (n)  
      R
a2 + 4π 2 n2
 
          1 


 T      Cp0
 Cp0 .

  

  f → f(n) 
  
        T.
! 





1

f


T (f )
∞ ≤
a2 + 4π 2 n2
n∈Z
 
    

    T    
 
"
  T ∈ L Cp0 .
#   f ∈ Cp0   x ∈ R, 


 1
2
e2iπnx
T (f ) (x) =
e−2iπny f (y) dy
2
2
2
1
a + 4π n − 2
n∈Z

 1
2iπn(x−y)
2
e
f (y) dy
=
a2 + 4π 2 n2
− 12
n∈Z





1 1

       − ,
 
       
2 2
! 
    x
k (x) =



n∈Z

e2iπnx
a2 + 4π 2 n2

        Cp0  



T (f ) (x) =

1
2

− 12

k (x − y) f (y) dy

$  %  

 k        &  
 


1 1
1  −a|x|
e
ϕ    − ,

 ϕ (x) =
+ J ch (ax)     R 

2 2
2a
1   '    
k = ϕ( ) J  %  *   

______________________________________________________________________________________________
1-2075-CT-WB-02-13
14

  
  
 ϕ 
  
ϕ
 (n) =
=
=
=

 1
2


1
e−2iπnx e−a|x| + J ch (ax) dx
2a − 12
 1


1 2
cos (2πnx) e−ax + J ch (ax) dx
a 0


 1
e−ax + eax
1 2
−ax
dx
cos (2πnx) e
+J
a 0
2

 1
 1
2
2
1
J
J
−ax
1+
cos (2πnx) e dx +
cos (2πnx) eax dx
a
2
2a
0
0

  
          α 


Jn (α) =

  


Jn (α) = 

1
2

1
2

0

cos (2πnx) eαx dx


e2iπnx eαx dx

0


=

1
2


e(α+2iπn)x dx

0




α
(−1)n e 2 − 1
e(α+2iπn)x
=
=
α + 2iπn 0
α + 2iπn



α
(−1)n e 2 − 1 (α − 2iπn)
=
α2 + 4π 2 n2


α
α (−1)n e 2 − 1
=
α2 + 4π 2 n2


1
2

    
 
ϕ
 (n) =
=
=
=


 

 J 

1
J 
1
n −a
n a
(−1) e 2 − 1 + a (−1) e 2 − 1
−a 1 +
a a2 + 4π 2 n2
2
2

 





J
1
J
n − a2
n a2
(−1) e − 1 +
(−1) e − 1
− 1+
a2 + 4π 2 n2
2
2



 
J
1
n  a2
n − a2
− a2
(−1) e − e
− (−1) e − 1
a2 + 4π 2 n2 2

a 

1
n
n −a
2
− (−1) e − 1
J (−1) sh
a2 + 4π 2 n2
2

    
  
 J    
J (−1)n sh

 

a
2



a
− (−1)n e− 2 − 1 = 1
a

e− 2
J = a
sh 2

______________________________________________________________________________________________
1-2075-CT-WB-02-13
15

     ϕ (n) = a2 + 14π2n2 

n ∈ Z        

  
 ϕ = k.       
1
ϕ (x) =
2a



a

e− 2
e−a|x| +  a  ch (ax)
sh 2



=



n∈Z

e2iπnx
a2 + 4π 2 n2

  
  
          C 1   

  k = ϕ
 

 x,   
1
k (x) =
2a



a

e− 2
e−a|x| +  a  ch (ax)
sh 2



>0

 
f ∈ Cp0 \ {0} !  
     

T (f ) (x) =

1
2

− 12

k (x − y) f (y) dy > 0

         
     

" 
  
 T      
#    
  B (0, 1)   $


     Cp0,    T (B (0, 1))      
  
 
 
%& ' (& 




f ∈ B (0, 1) ,  
T (f )
∞ ≤ a2 + 14π2n2 ,  T (B (0, 1)) 
n∈Z

  $ 
)   k ∈ Cp0  
   
 
R, 

 ε > 0,   

  η > 0  
 |k (v) − k (u)| < ε 

   u, v   
 |v − u| < η. *
 
 
 

f ∈ B (0, 1)  
 x, y  R   
 |y − x| < η   
 1

 2



|T (f ) (y) − T (f ) (x)| = 
|k (y − u) − k (x − u)| f (u) du ≤ ε
f
∞ ≤ ε
 −1

2

+

 T (B (0, 1))  
 

 , f ∈ Cp0 \ {0}    
 T (f ) = λf,      
∀n ∈ Z, T
(f ) (n) = λf(n)

   
    
T
(f ) (n) =
=



1
2

− 12



k∈Z



e−2iπnx



k∈Z


e2iπkx 
f (k) dx
a2 + 4π 2 k 2

 1
2
f(k)
f(n)
−2iπ(k−n)x
e
dx
=
=
k (n) f(n)
a2 + 4π 2 k 2 − 12
a2 + 4π 2 n2

______________________________________________________________________________________________
1-2075-CT-WB-02-13
16

 

∀n ∈ Z,





λ − k (n) f(n) = 0

 f = 0,
     n 
 f(n) = 0  
   
k (n) .


   
     λ = 
  f = 1,   f(n) =



1
2

− 12

e−2iπnx dx = 0   n = 0  f(0) = 1, 

1
1
1
= 2 f     2  
   T.
2
a
a
a
1
 f  
      
 
  2 ,   
 
a


1

∀n ∈ Z,
− k (n) f(n) = 0
a2

   T (f ) =

 

∀n ∈ Z∗ , 
k (n) =

1
1
=

a2 + 4π 2 n2
a2

 f(n) = 0   n = 0  f 
− f(0) (n) = 0   n ∈ Z,  
  f = f(0)     
   

1

      
 
  2  
   ! 
a
!     
1
k (n) =
 λ ∈ C   
   T "   2 ,   
 λ = 
a

1
 n = 0,  
2
a + 4π 2 n2

0<λ<

1
a2

#    
 
   T  

     
1
 2 
  
   T  

a

1

  2  
   T,
 !     
  1   
a
        
1
= |T (1)| ≤
T
L(Cp0 )
1
∞ =
T
L(Cp0 )
a2

 T      !    !  f ∈ Cp0  
 

 
  
− |f | ≤ f ≤ |f | ⇒ −T (|f |) ≤ T (f ) ≤ T (|f |)

 |T (f )| ≤ T (|f |) .
 f ∈ Cp0  
 
         
x 
T (f ) (x) = eiθ(x) |T (f ) (x)|

______________________________________________________________________________________________
1-2075-CT-WB-02-13
17





|T (f ) (x)| = e−iθ(x) T (f ) (x) = T e−iθ(x) f (x)
 
 
 

=  T e−iθ(x) f (x) = T  e−iθ(x) f (x)



 
≤ T  e−iθ(x) f  (x) ≤ T e−iθ(x) f  (x) = T (|f |) (x)
  |T (f )| ≤ T (|f |) 
        
 
    

|T (f )| ≤ T (|f |) ≤ T (
f
∞ ) =
f
∞ T (1) =

T
L(C 0 ) ≤
p


f

a2

1
1
,     
 
T
L(Cp0 ) = 2 .
2
a
a

   V = ker (ϕ) ,  ϕ      


ϕ : g →

1
2

− 21

g (t) dt = 
g (0)

  V          Cp0 .
             g ∈ Cp0 



1
2

− 12

T (g) (t) dt =



n∈Z

g (n)
a2 + 4π 2 n2



1
2

− 21

e2iπnt dt =

g (0)
=0
a2

  V    T.
!   f ∈ Cp0  n ∈ Z,   

T
(f ) (n) =



1
2

− 12


e2iπ(k−n)x
f(n)
(k) dx =
f
a2 + 4π 2 k 2
a2 + 4π 2 n2
k∈Z

 "           
2

T (f ) (x) =



n∈Z



e2iπnx 
e2iπnx
T (f ) (n) =
f(n)
2 + 4π 2 n2 )2
a2 + 4π 2 n2
(a
n∈Z

     

T n (f ) (x) =



k∈Z

    n.
#  

a T (f ) (x) = f(0) +
2n

n

e2iπkx
f(k)
(a2 + 4π 2 k 2 )n



k∈Z∗

a2
a2 + 4π 2 k 2

n

e2iπkx f(k)

______________________________________________________________________________________________
1-2075-CT-WB-02-13
18

 


n
n



2


a2
a



f

e2iπkx f(k) ≤


 ∗ a2 + 4π 2 k 2
a2 + 4π 2 k 2
k∈Z
k∈Z∗
+∞ 
n


a2

f

≤2
2 + 4π 2 k 2
a
k=1

n−1


+∞
a2
a2

f

≤2
2 + 4π 2 k 2
a2 + 4π 2
a
k=1
 n ≥ 2.

  lim

n÷+∞



a2
a2 + 4π 2

n−1
= 0,      
lim a2n T n (f ) (x) = f(0)

n÷+∞

  f(0) = 0  f ∈ Cp0 \ V      T n (f ) (x)


  x, y  B (0, 1) 



n÷+∞

f(0)
.
a2n

 
   


f (y) − f (x) =



1−


y
22

         
·
2 







1−


x
22 , y


−x

·,     lim f (y) = f (x) , 
y→x

    f    

 
f (x)
22 = 1 −
x
22 +
x
22 = 1,    f         ! 
  S (0, 1)  2 (N) .

"   x ∈ B (0, 1)    #  f       x0 = 1 −
x
22 

xk = xk−1   k ∈ N,     xk = 1 −
x
22   k ∈ N  
   x = 0  
x
22 = +∞  x0 = 1,     $
"  f        #

%
 


   




1
ψi : y → max 0, −
y − yi

n

      E,     &   ψ.

  f (B)           ψ : f (B) → R+  $  
   $   #   y0 ∈ f (B)   

δ = inf ψ (y) = ψ (y0 ) ≥ 0
y∈f (B)

______________________________________________________________________________________________
1-2075-CT-WB-02-13
19



Nn


1
,    

  k
  y0 

f (B) ⊂ B yi,
n
i=1


1
1
,  
y0 − yk
<  
 

 1  n   y0 ∈ B yk ,
n
n
δ = ψ (y0 ) =

Nn



ψi (y) ≥ ψk (y) =

i=1



1

y − yk
> 0
n


 ψ (y) ≥ δ = ψ (y0 ) > 0   y ∈ f (B).
  ψ (y) > 0   y ∈ f (B),  
 
fn    


  x ∈ B, 

n


1
f (x) − fn (x) = f (x) −
ψi (f (x)) yi
ψ (f (x)) i=1

N

  
ψ (f (x)) (f (x) − fn (x)) = ψ (f (x)) f (x) −

Nn



ψi (f (x)) yi

i=1

=

Nn



ψi (f (x)) (f (x) − yi )

i=1

 
ψ (f (x))
f (x) − fn (x)


Nn



ψi (f (x))
f (x) − yi


i=1



1
1
, 

f (x) − yi
≤ 
 i ∈ {1, · · · , n}     f (x) ∈ B yi ,
n
n


1
, 
 ψi (f (x)) = 0, 

 
f (x)
  
B yi,
n

 


ψi (f (x))
f (x) − yi


 
 

 

1
ψi (f (x))
n

n
1

1
ψ (f (x))
f (x) − fn (x)

ψi (f (x)) = ψ (f (x))
n i=1
n

N

1
n


f (x) − fn (x)
≤ .
!

  
    


 F 
"
   x1 , · · · , xn , 
 "
   C

 
   
    

  

   F 
fn   

  C
C fn (x)     #
  yi   $% 
&'
    fn 

  xn ∈ C ⊂ B.

______________________________________________________________________________________________
1-2075-CT-WB-02-13
20

  

n ∈ N∗ ,





f (xn ) − fn (xn )

  


(f (xn ))n∈N∗





  

 

f xϕ(n) n∈N∗

f (B),  
x ∈ f (B) ⊂ B





 

  

xϕ(n) − x ≤ xϕ(n) − fn xϕ(n)  + fn xϕ(n) − x

 


  

≤ f xϕ(n) − fn xϕ(n)  + fn xϕ(n) − x

 

1

+ fn xϕ(n) − x
ϕ (n)

  



B

→ 0

n→+∞

  
 
   


 
       

f :B→B

1
n

lim xϕ(n) = x

n→+∞


 
   

f,


 

:



f (x) = lim f xϕ(n)
n→+∞
 



lim f xϕ(n) − fn xϕ(n) = 0,   
n→+∞


f (x) = lim fn xϕ(n) = lim xϕ(n) = x
n→+∞

 

x



   

n→+∞

 

f




B.

 

   

f    (fn )n∈N        
[a, b]          f   
        fn 
    C    
 C 0 ([a, b] , R)    
  
     
  
 C.
# u, v    
 $   
  %  αu + βv   
 α  β.
# u  −u   
 C, 

 u (x) ≥ 0  u (x) ≤ 0   x ∈ [a, b] ,  
u = 0.
&  $ C    '   C 0 ([a, b] , R) .
# f ∈ C 

     x0 ∈ [a, b] .    ε > 0, 
 
◦ 
ε
ε
g ∈ C 0 ([a, b] , R)   
 g (x) = f (x) −  
 
  B f,

2
2

/ C. "
 
 
 
 C,   f ∈



C ⊂ f ∈ C 0 ([a, b] , R) | ∀x ∈ [a, b] , f (x) > 0

! " 
  
    

(  $ 

[a, b] ,


f ∈ C 0 ([a, b] , R+,∗ ) .

 )    
  
 

f        

  m > 0     
 

g ∈ B (f, m) , 

g − f
∞ < m,   g (x) > f (x) − m > 0   x ∈ [a, b] ,
 g ∈ C 0 ([a, b] , R+,∗ ) .     C 0 ([a, b] , R+,∗ ) ⊂ C      
  $      




C.

* 




C = C 0 ([a, b] , R+,∗ ) .

______________________________________________________________________________________________
1-2075-CT-WB-02-13
21



  
  T  
       T (x) ∈ C 
x ∈ C \ {0} 
 
1
lim T (x) − x = T (x)
ω→+∞
ω

1
    T (x) − x ∈ C 
ω > 0  
  ωT x ≥ x.
ω
 
 



 
n ≥ 1.

n = 1,   
y = MT y + εMT x ≥ εMT x ≥ εMx
   ω = 1 
!  "
"   
 n ≥ 1,   

y = MT y + εMT x ≥ MT y ≥ MT (εM n x) = εM n+1 T (x) ≥ εM n+1 x
1
y − x ∈ C  C 
     −x = lim yn ∈
n→+∞
εM n
C,     x = 0   x ∈ C,   
  x ∈ C \ {0} .
   M ≤ 1.
! M > 1,   yn =

#
   

Cε = C ∩ Dε ∩ B (0, R)

$ Dε = {y ∈ E | y ≥ εx} = {y ∈ E | y − εx ∈ C}  % 
  


  % 
 C 
" "   &    y → y − εx
   C  B (0, R) . '" 
"  Cε  % 

(   " )" 
 B (0, R) ,   )" Cε  )

*+   x ∈ C \ {0} ,   )" Cε     0.
) 
ε > 0  R ≥ ε
x
,   εx ∈ Cε = C ∩ Dε ∩ B (0, R) ,  Cε  
 
1
 "   ϕε : y →
y + εx     E \ {0}  E, "  

y

 ,   Tε = T ◦ ϕε .
1
y ∈ C,   ϕε (y) ≥ εx

y ∈ Cε ⊂ C,   ϕε (y) ∈ C  ϕε (y) − εx =

y

 Tε (y) = T (ϕε (y)) ≥ εT (x) ≥ εx,   Tε (y) ∈ C ∩ Dε .
*+  


Tε (y)
=

1

T (y + ε
y
x)

T
(1 + ε
x
)

y



  " R ≥
T
(1 + ε
x
) ,  
 Tε (y) ∈ Cε = C∩Dε ∩B (0, R) .
* +   
R ≥ max (ε
x
,
T
(1 + ε
x
)) , Tε     Cε
 Cε .
-
ϕε (y)
≤ 1 + ε
x

 y ∈ Cε ,     Tε (Cε )  
 T (B (0, 1 + ε
x
))      
     .
"   Tε    

______________________________________________________________________________________________
1-2075-CT-WB-02-13
22

  
   
      yε
    0 ≤ Mε ≤ 1.



∈ Cε



Tε .   

    



=
Tε (yε )

T
(1 + ε
x
)

ε
=
.  
R > 0  
  Cεn = ∅  Tεn
n + 1 n∈N
   Cεn  Cεn  
  n ∈ N.
     (yεn )n∈N   yεn ∈ Cεn    


  

(εn )n∈N

  
 
 



Tεn

  

Mεn =
 
 



1
∈ [0, 1]

yεε





yεn

T
(1 + εn
x
) ≤
T
(1 + ε
x
)

n ∈ N.

!  [0, 1]      

  (Mεn )n∈N ,   " (Mn )n∈N





Mεϕ(n)

   
#

n∈N

!  




=

λ ∈ [0, 1] .

1 =
zεn
= Mεn
yεn
≤ Mεn
T
(1 + ε
x
)

λ > 0.
!   

T     (yεn + εn x)n∈N  $
   "
 

  "   
#


    T yεϕ(n) + εϕ(n) x
n∈N
 






T yεψ(n) + εψ(n) x
%   zεψ(n)
= Mεψ(n) T yεψ(n) + εψ(n) x
  
 

n∈N

  
#


   

!      

z



C C

n∈N

 &
 

n∈N

zεψ(n) ∈ C %

z = lim zεψ(n) = λT (z)
n→+∞

 


z
= 1

 

zεn

  
 



z∈C



T

 &
 

 &%
'%
  $




A = {s ≥ 0 | z − sz  ∈ C}     &
  R    #
    &
 C 
  (    s → z − sz  ,

    
 &
%
! 

{0} .
 A 



z ∈ C,

 


z − sz  ∈ C

*

  $
   


 
 

 


s > 0 )

A = R+ ,

 

 

z − sz  ∈ C

A

 


 
 


s ≥ 0,

1
(z − sz  ) ∈ C
s→+∞ s

−z  = lim

z  = 0  z  ∈ C,      $
  A  $
   A = [0, b]   b > 0.






z  ∈ C.

______________________________________________________________________________________________
1-2075-CT-WB-02-13
23

   

λ
λ
z − b z  = λT (z) − b μT (z  ) = λT (z − bz  ) ∈ C
μ
μ
 


b

λ
∈A
μ



λ ≤ μ.

A = {s ≥ 0 | z  − sz ∈ C}    
      {0}
  0,  
   μ ≤ λ.
      λ = μ,   
   λ.
    

 

______________________________________________________________________________________________
1-2075-CT-WB-02-13
24


Aperçu du document 2075-TC-WB-06-16-corr.pdf - page 1/24

 
2075-TC-WB-06-16-corr.pdf - page 2/24
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2075-TC-WB-06-16-corr.pdf - page 4/24
2075-TC-WB-06-16-corr.pdf - page 5/24
2075-TC-WB-06-16-corr.pdf - page 6/24
 




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