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Gateaux Derivative of Fourier Transform/Characteristic Function

Fourier Transform ConfusionFourier transform of the identity function $f(x)=x$Sampling, Fourier Transform, and Discrete Fourier TransformFourier Transform of derivative a vector functionCharacteristic Function and Inverse Fourier TransformDifferentiability of discrete-time Fourier transformWhat exactly is the Fourier transform?Functional derivatives of Fourier transformUnderstanding Fourier transform of derivative of distributionWhat is the square root of a Fourier transform?

1

$begingroup$

Suppose that $X$ is a square integrable random-variable defined on a probability space $(Omega,mathcalF,mathbbP)$. It's characteristic function/Fourier transform is defined to be
$$
mathfrakF[X](gamma)triangleq int_omega in Omega
e^-gamma X(omega)
mathbbP(domega)
.
$$

For any fixed $gamma in mathbbR$, when is the map
beginalign
L^2(Omega,mathcalF,mathbbP)rightarrow mathbbR
\
X mapsto mathfrakF[X](gamma),
endalign
Gateau differentiable (for what X)? Moreover, what is its Gateau derivative?

functional-analysis probability-theory fourier-analysis frechet-derivative gateaux-derivative

share|cite|improve this question

edited 17 hours ago

AIM_BLB

asked yesterday

AIM_BLBAIM_BLB

2,5152819

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Square integrability is not enough for your characteristic function to exist. This is rather called the moment generating function (with $-gamma$ instead of $gamma$).
  $endgroup$
  – Abdelmalek Abdesselam
  yesterday
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  On the subspace of $L^2$ on which the MFG exists, what can be said about it derivative then?
  $endgroup$
  – AIM_BLB
  yesterday
  

  
  
  
  

add a comment | 

1

$begingroup$

Suppose that $X$ is a square integrable random-variable defined on a probability space $(Omega,mathcalF,mathbbP)$. It's characteristic function/Fourier transform is defined to be
$$
mathfrakF[X](gamma)triangleq int_omega in Omega
e^-gamma X(omega)
mathbbP(domega)
.
$$

For any fixed $gamma in mathbbR$, when is the map
beginalign
L^2(Omega,mathcalF,mathbbP)rightarrow mathbbR
\
X mapsto mathfrakF[X](gamma),
endalign
Gateau differentiable (for what X)? Moreover, what is its Gateau derivative?

functional-analysis probability-theory fourier-analysis frechet-derivative gateaux-derivative

share|cite|improve this question

edited 17 hours ago

AIM_BLB

asked yesterday

AIM_BLBAIM_BLB

2,5152819

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Square integrability is not enough for your characteristic function to exist. This is rather called the moment generating function (with $-gamma$ instead of $gamma$).
  $endgroup$
  – Abdelmalek Abdesselam
  yesterday
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  On the subspace of $L^2$ on which the MFG exists, what can be said about it derivative then?
  $endgroup$
  – AIM_BLB
  yesterday
  

  
  
  
  

add a comment | 

$begingroup$

Suppose that $X$ is a square integrable random-variable defined on a probability space $(Omega,mathcalF,mathbbP)$. It's characteristic function/Fourier transform is defined to be
$$
mathfrakF[X](gamma)triangleq int_omega in Omega
e^-gamma X(omega)
mathbbP(domega)
.
$$

For any fixed $gamma in mathbbR$, when is the map
beginalign
L^2(Omega,mathcalF,mathbbP)rightarrow mathbbR
\
X mapsto mathfrakF[X](gamma),
endalign
Gateau differentiable (for what X)? Moreover, what is its Gateau derivative?

functional-analysis probability-theory fourier-analysis frechet-derivative gateaux-derivative

share|cite|improve this question

edited 17 hours ago

AIM_BLB

asked yesterday

AIM_BLBAIM_BLB

2,5152819

$endgroup$

share|cite|improve this question

edited 17 hours ago

AIM_BLB

asked yesterday

AIM_BLBAIM_BLB

2,5152819

share|cite|improve this question

edited 17 hours ago

AIM_BLB

asked yesterday

AIM_BLBAIM_BLB

2,5152819

asked yesterday

AIM_BLBAIM_BLB

2,5152819

asked yesterday

AIM_BLBAIM_BLB

2,5152819