Mixing
Monotonicity of the convolution residual in L^2
Clash Royale CLAN TAG#URR8PPP
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Consider an approximation of unity $(phi_epsilon)_epsilon $ defined such that :
$forall epsilon > 0, quad int_mathbbR^n |phi_epsilon(x)| dx = 1$
$forall R>0, quad lim_epsilon rightarrow 0 int_ > R |phi_epsilon(x)| dx = 0$
$ forall f in L^2(mathbbR^n), quad lim_epsilon rightarrow 0 || f - phi_epsilon * f ||_L^2(mathbbR^n) = 0.$
In the sequel, we assume that this family is constructed using a single function $phi: mathbbR^n rightarrow mathbbR$ via the definition:
beginequation
phi_epsilon = epsilon^-nphi(cdot/epsilon), quad quad (*)
endequation
where $phi$ is integrable with $L^1$ integral equal to $1$.
Let $f$ be a function in $L^2(mathbbR^n)$, and $Omega subset mathbbR^n$, define the mapping
beginequation
beginarraylcll
Psi_f,Omega colon & mathbbR^+ & longrightarrow & mathbbR^+ \
& epsilon & longmapsto & ||f - phi_epsilon*f||_L^2(Omega)^2
endarray
endequation
Aim: How to ensure the monotonicity of the mapping $Psi_f,Omega$ (at least in a neighbourhood of 0)? That is, what are the sufficient conditions on the function $phi$ to guarantee that
$$
forall epsilon_1 < epsilon_2 ll 1, quad
Psi_f,Omega(epsilon_1) leq Psi_f,Omega(epsilon_2).
$$
For example, if $Omega = mathbbR^n$, I can prove the monotonicity of the function $Psi_f,mathbbR^n$ by imposing the following conditions on the Fourier transform $hatphi$ of the function $phi$:
- The function $hatphi$ is bounded on $mathbbR^n$.
- $hatphi(xi) < 1 $ for $||xi|| ll 1$
- The function $hatphi$ is differentiable and for all $xi in mathbbR^n$, $xi^top hatphi'(xi) leq 0.$
Indeed, by Parseval's Theorem
$$
Psi_f,mathbbR^n(epsilon) = ||mathcalFleft(f - phi_epsilon*f right)||_L^2(mathbbR^n)^2 = ||hatf - hatphi_epsilon hatf||_L^2(mathbbR^n)^2 = || (1- hatphi_epsilon) hatf||_L^2(mathbbR^n)^2
$$
By (*), $ hatphi_epsilon = hatphi(epsilon cdot)$, then
$$
Psi_f,mathbbR^n(epsilon) = || (1- hatphi(epsilon cdot) hatf(cdot)||_L^2(mathbbR^n)^2
$$
Now, since the function $hatphi$ is bounded and differentiable, the function $Psi_f,mathbbR^n$ is also differentiable and
$$
Psi_f,mathbbR^n'(epsilon) = -2 int_mathbbR^n xi^top hatphi'(beta xi), (1-hatphi(beta xi) ),|hatf(xi)|^2 dxi
$$
Using the second and third condition on the function $hatphi$, we deduce that $Psi_f,mathbbR^n'(epsilon) geq 0$ for small values of $epsilon$. Whence the monotonicity of the mapping $Psi_f,mathbbR^n$ in a neighborhood of $0$.
Recall that there are functions $phi$ such that all the conditions stated above are fulfilled. An example is the multivariate Gaussian function
beginequation
labelexple phi Multivariate gaussian
phi(x) = frac1sqrtdet(2 pi Sigma) expleft(-frac12 x^top Sigma^-1 x right),
endequation
with a diagonal covariance matrix $Sigma = diag(sigma_1,...,sigma_n)$.
Main Problem: How can we ensure monotonicity of $Psi_f,Omega$ for a subset $Omega$ of $mathbbR^n$.
Please can someone help me ??
integration analysis fourier-transform convolution
asked Jul 14 at 14:40
Walter
63
add a comment |Â
up vote
1
down vote
favorite
Consider an approximation of unity $(phi_epsilon)_epsilon $ defined such that :
$forall epsilon > 0, quad int_mathbbR^n |phi_epsilon(x)| dx = 1$
$forall R>0, quad lim_epsilon rightarrow 0 int_ > R |phi_epsilon(x)| dx = 0$
$ forall f in L^2(mathbbR^n), quad lim_epsilon rightarrow 0 || f - phi_epsilon * f ||_L^2(mathbbR^n) = 0.$
In the sequel, we assume that this family is constructed using a single function $phi: mathbbR^n rightarrow mathbbR$ via the definition:
beginequation
phi_epsilon = epsilon^-nphi(cdot/epsilon), quad quad (*)
endequation
where $phi$ is integrable with $L^1$ integral equal to $1$.
Let $f$ be a function in $L^2(mathbbR^n)$, and $Omega subset mathbbR^n$, define the mapping
beginequation
beginarraylcll
Psi_f,Omega colon & mathbbR^+ & longrightarrow & mathbbR^+ \
& epsilon & longmapsto & ||f - phi_epsilon*f||_L^2(Omega)^2
endarray
endequation
Aim: How to ensure the monotonicity of the mapping $Psi_f,Omega$ (at least in a neighbourhood of 0)? That is, what are the sufficient conditions on the function $phi$ to guarantee that
$$
forall epsilon_1 < epsilon_2 ll 1, quad
Psi_f,Omega(epsilon_1) leq Psi_f,Omega(epsilon_2).
$$
For example, if $Omega = mathbbR^n$, I can prove the monotonicity of the function $Psi_f,mathbbR^n$ by imposing the following conditions on the Fourier transform $hatphi$ of the function $phi$:
- The function $hatphi$ is bounded on $mathbbR^n$.
- $hatphi(xi) < 1 $ for $||xi|| ll 1$
- The function $hatphi$ is differentiable and for all $xi in mathbbR^n$, $xi^top hatphi'(xi) leq 0.$
Indeed, by Parseval's Theorem
$$
Psi_f,mathbbR^n(epsilon) = ||mathcalFleft(f - phi_epsilon*f right)||_L^2(mathbbR^n)^2 = ||hatf - hatphi_epsilon hatf||_L^2(mathbbR^n)^2 = || (1- hatphi_epsilon) hatf||_L^2(mathbbR^n)^2
$$
By (*), $ hatphi_epsilon = hatphi(epsilon cdot)$, then
$$
Psi_f,mathbbR^n(epsilon) = || (1- hatphi(epsilon cdot) hatf(cdot)||_L^2(mathbbR^n)^2
$$
Now, since the function $hatphi$ is bounded and differentiable, the function $Psi_f,mathbbR^n$ is also differentiable and
$$
Psi_f,mathbbR^n'(epsilon) = -2 int_mathbbR^n xi^top hatphi'(beta xi), (1-hatphi(beta xi) ),|hatf(xi)|^2 dxi
$$
Using the second and third condition on the function $hatphi$, we deduce that $Psi_f,mathbbR^n'(epsilon) geq 0$ for small values of $epsilon$. Whence the monotonicity of the mapping $Psi_f,mathbbR^n$ in a neighborhood of $0$.
Recall that there are functions $phi$ such that all the conditions stated above are fulfilled. An example is the multivariate Gaussian function
beginequation
labelexple phi Multivariate gaussian
phi(x) = frac1sqrtdet(2 pi Sigma) expleft(-frac12 x^top Sigma^-1 x right),
endequation
with a diagonal covariance matrix $Sigma = diag(sigma_1,...,sigma_n)$.
Main Problem: How can we ensure monotonicity of $Psi_f,Omega$ for a subset $Omega$ of $mathbbR^n$.
Please can someone help me ??
integration analysis fourier-transform convolution
asked Jul 14 at 14:40
Walter
63
This is quite a good question. At the moment I don't know how to answer it, so let me ask a "counter question": Why do you need this? Do you want to use such a function for proving something else? If so, one can maybe circumvent this. In any case: Good question.
– PhoemueX
Jul 14 at 16:46
Yes, I want to use that result to prove something else.
– Walter
Jul 23 at 15:08
I want to prove the following claim :
– Walter
Jul 23 at 15:08
Claim: Let $Omega_1, Omega_2$ be a partition of $mathbbR^n$,($mathbbR^n = Omega_1 cup Omega_2$ with $Omega_1 cap Omega_2 = emptyset $), $f in L^2(mathbbR ^n)$ and $left( phi_epsilon right)_epsilon$ be an approximation of unity defined as above satisfying the monotonicity of the function $Psi_f,mathbbR ^n$. Then there exists $rho,, delta >0$ such that $ forall ,,epsilon_1 < epsilon_2$, $$ delta ||f - phi_rho epsilon_1*f||_L^2(mathbbR^n) leq ||f - phi_epsilon_1*f||_L^2(Omega_1) + ||f - phi_epsilon_2*f||_L^2(Omega_2) $$
– Walter
Jul 23 at 15:16
Obviously, if the function $Psi_f, Omega_i $ satisfies the monotonicity for $i=1$ or $i=2$,then the proof of the claim is trivial with $rho = delta=1$
– Walter
Jul 23 at 15:21
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Consider an approximation of unity $(phi_epsilon)_epsilon $ defined such that :
$forall epsilon > 0, quad int_mathbbR^n |phi_epsilon(x)| dx = 1$
$forall R>0, quad lim_epsilon rightarrow 0 int_ > R |phi_epsilon(x)| dx = 0$
$ forall f in L^2(mathbbR^n), quad lim_epsilon rightarrow 0 || f - phi_epsilon * f ||_L^2(mathbbR^n) = 0.$
In the sequel, we assume that this family is constructed using a single function $phi: mathbbR^n rightarrow mathbbR$ via the definition:
beginequation
phi_epsilon = epsilon^-nphi(cdot/epsilon), quad quad (*)
endequation
where $phi$ is integrable with $L^1$ integral equal to $1$.
Let $f$ be a function in $L^2(mathbbR^n)$, and $Omega subset mathbbR^n$, define the mapping
beginequation
beginarraylcll
Psi_f,Omega colon & mathbbR^+ & longrightarrow & mathbbR^+ \
& epsilon & longmapsto & ||f - phi_epsilon*f||_L^2(Omega)^2
endarray
endequation
Aim: How to ensure the monotonicity of the mapping $Psi_f,Omega$ (at least in a neighbourhood of 0)? That is, what are the sufficient conditions on the function $phi$ to guarantee that
$$
forall epsilon_1 < epsilon_2 ll 1, quad
Psi_f,Omega(epsilon_1) leq Psi_f,Omega(epsilon_2).
$$
For example, if $Omega = mathbbR^n$, I can prove the monotonicity of the function $Psi_f,mathbbR^n$ by imposing the following conditions on the Fourier transform $hatphi$ of the function $phi$:
- The function $hatphi$ is bounded on $mathbbR^n$.
- $hatphi(xi) < 1 $ for $||xi|| ll 1$
- The function $hatphi$ is differentiable and for all $xi in mathbbR^n$, $xi^top hatphi'(xi) leq 0.$
Indeed, by Parseval's Theorem
$$
Psi_f,mathbbR^n(epsilon) = ||mathcalFleft(f - phi_epsilon*f right)||_L^2(mathbbR^n)^2 = ||hatf - hatphi_epsilon hatf||_L^2(mathbbR^n)^2 = || (1- hatphi_epsilon) hatf||_L^2(mathbbR^n)^2
$$
By (*), $ hatphi_epsilon = hatphi(epsilon cdot)$, then
$$
Psi_f,mathbbR^n(epsilon) = || (1- hatphi(epsilon cdot) hatf(cdot)||_L^2(mathbbR^n)^2
$$
Now, since the function $hatphi$ is bounded and differentiable, the function $Psi_f,mathbbR^n$ is also differentiable and
$$
Psi_f,mathbbR^n'(epsilon) = -2 int_mathbbR^n xi^top hatphi'(beta xi), (1-hatphi(beta xi) ),|hatf(xi)|^2 dxi
$$
Using the second and third condition on the function $hatphi$, we deduce that $Psi_f,mathbbR^n'(epsilon) geq 0$ for small values of $epsilon$. Whence the monotonicity of the mapping $Psi_f,mathbbR^n$ in a neighborhood of $0$.
Recall that there are functions $phi$ such that all the conditions stated above are fulfilled. An example is the multivariate Gaussian function
beginequation
labelexple phi Multivariate gaussian
phi(x) = frac1sqrtdet(2 pi Sigma) expleft(-frac12 x^top Sigma^-1 x right),
endequation
with a diagonal covariance matrix $Sigma = diag(sigma_1,...,sigma_n)$.
Main Problem: How can we ensure monotonicity of $Psi_f,Omega$ for a subset $Omega$ of $mathbbR^n$.
Please can someone help me ??
integration analysis fourier-transform convolution
asked Jul 14 at 14:40
Walter
63
Consider an approximation of unity $(phi_epsilon)_epsilon $ defined such that :
$forall epsilon > 0, quad int_mathbbR^n |phi_epsilon(x)| dx = 1$
$forall R>0, quad lim_epsilon rightarrow 0 int_ > R |phi_epsilon(x)| dx = 0$
$ forall f in L^2(mathbbR^n), quad lim_epsilon rightarrow 0 || f - phi_epsilon * f ||_L^2(mathbbR^n) = 0.$
In the sequel, we assume that this family is constructed using a single function $phi: mathbbR^n rightarrow mathbbR$ via the definition:
beginequation
phi_epsilon = epsilon^-nphi(cdot/epsilon), quad quad (*)
endequation
where $phi$ is integrable with $L^1$ integral equal to $1$.
Let $f$ be a function in $L^2(mathbbR^n)$, and $Omega subset mathbbR^n$, define the mapping
beginequation
beginarraylcll
Psi_f,Omega colon & mathbbR^+ & longrightarrow & mathbbR^+ \
& epsilon & longmapsto & ||f - phi_epsilon*f||_L^2(Omega)^2
endarray
endequation
Aim: How to ensure the monotonicity of the mapping $Psi_f,Omega$ (at least in a neighbourhood of 0)? That is, what are the sufficient conditions on the function $phi$ to guarantee that
$$
forall epsilon_1 < epsilon_2 ll 1, quad
Psi_f,Omega(epsilon_1) leq Psi_f,Omega(epsilon_2).
$$
For example, if $Omega = mathbbR^n$, I can prove the monotonicity of the function $Psi_f,mathbbR^n$ by imposing the following conditions on the Fourier transform $hatphi$ of the function $phi$:
- The function $hatphi$ is bounded on $mathbbR^n$.
- $hatphi(xi) < 1 $ for $||xi|| ll 1$
- The function $hatphi$ is differentiable and for all $xi in mathbbR^n$, $xi^top hatphi'(xi) leq 0.$
Indeed, by Parseval's Theorem
$$
Psi_f,mathbbR^n(epsilon) = ||mathcalFleft(f - phi_epsilon*f right)||_L^2(mathbbR^n)^2 = ||hatf - hatphi_epsilon hatf||_L^2(mathbbR^n)^2 = || (1- hatphi_epsilon) hatf||_L^2(mathbbR^n)^2
$$
By (*), $ hatphi_epsilon = hatphi(epsilon cdot)$, then
$$
Psi_f,mathbbR^n(epsilon) = || (1- hatphi(epsilon cdot) hatf(cdot)||_L^2(mathbbR^n)^2
$$
Now, since the function $hatphi$ is bounded and differentiable, the function $Psi_f,mathbbR^n$ is also differentiable and
$$
Psi_f,mathbbR^n'(epsilon) = -2 int_mathbbR^n xi^top hatphi'(beta xi), (1-hatphi(beta xi) ),|hatf(xi)|^2 dxi
$$
Using the second and third condition on the function $hatphi$, we deduce that $Psi_f,mathbbR^n'(epsilon) geq 0$ for small values of $epsilon$. Whence the monotonicity of the mapping $Psi_f,mathbbR^n$ in a neighborhood of $0$.
Recall that there are functions $phi$ such that all the conditions stated above are fulfilled. An example is the multivariate Gaussian function
beginequation
labelexple phi Multivariate gaussian
phi(x) = frac1sqrtdet(2 pi Sigma) expleft(-frac12 x^top Sigma^-1 x right),
endequation
with a diagonal covariance matrix $Sigma = diag(sigma_1,...,sigma_n)$.
Main Problem: How can we ensure monotonicity of $Psi_f,Omega$ for a subset $Omega$ of $mathbbR^n$.
Please can someone help me ??
integration analysis fourier-transform convolution
asked Jul 14 at 14:40
Walter
63
edited Jul 23 at 15:27
asked Jul 14 at 14:40
Walter
63
asked Jul 14 at 14:40
Walter
63
asked Jul 14 at 14:40
Walter
63
63
This is quite a good question. At the moment I don't know how to answer it, so let me ask a "counter question": Why do you need this? Do you want to use such a function for proving something else? If so, one can maybe circumvent this. In any case: Good question.
– PhoemueX
Jul 14 at 16:46
Yes, I want to use that result to prove something else.
– Walter
Jul 23 at 15:08
I want to prove the following claim :
– Walter
Jul 23 at 15:08
Claim: Let $Omega_1, Omega_2$ be a partition of $mathbbR^n$,($mathbbR^n = Omega_1 cup Omega_2$ with $Omega_1 cap Omega_2 = emptyset $), $f in L^2(mathbbR ^n)$ and $left( phi_epsilon right)_epsilon$ be an approximation of unity defined as above satisfying the monotonicity of the function $Psi_f,mathbbR ^n$. Then there exists $rho,, delta >0$ such that $ forall ,,epsilon_1 < epsilon_2$, $$ delta ||f - phi_rho epsilon_1*f||_L^2(mathbbR^n) leq ||f - phi_epsilon_1*f||_L^2(Omega_1) + ||f - phi_epsilon_2*f||_L^2(Omega_2) $$
– Walter
Jul 23 at 15:16
Obviously, if the function $Psi_f, Omega_i $ satisfies the monotonicity for $i=1$ or $i=2$,then the proof of the claim is trivial with $rho = delta=1$
– Walter
Jul 23 at 15:21
add a comment |Â
This is quite a good question. At the moment I don't know how to answer it, so let me ask a "counter question": Why do you need this? Do you want to use such a function for proving something else? If so, one can maybe circumvent this. In any case: Good question.
– PhoemueX
Jul 14 at 16:46
Yes, I want to use that result to prove something else.
– Walter
Jul 23 at 15:08
I want to prove the following claim :
– Walter
Jul 23 at 15:08
Claim: Let $Omega_1, Omega_2$ be a partition of $mathbbR^n$,($mathbbR^n = Omega_1 cup Omega_2$ with $Omega_1 cap Omega_2 = emptyset $), $f in L^2(mathbbR ^n)$ and $left( phi_epsilon right)_epsilon$ be an approximation of unity defined as above satisfying the monotonicity of the function $Psi_f,mathbbR ^n$. Then there exists $rho,, delta >0$ such that $ forall ,,epsilon_1 < epsilon_2$, $$ delta ||f - phi_rho epsilon_1*f||_L^2(mathbbR^n) leq ||f - phi_epsilon_1*f||_L^2(Omega_1) + ||f - phi_epsilon_2*f||_L^2(Omega_2) $$
– Walter
Jul 23 at 15:16
Obviously, if the function $Psi_f, Omega_i $ satisfies the monotonicity for $i=1$ or $i=2$,then the proof of the claim is trivial with $rho = delta=1$
– Walter
Jul 23 at 15:21
This is quite a good question. At the moment I don't know how to answer it, so let me ask a "counter question": Why do you need this? Do you want to use such a function for proving something else? If so, one can maybe circumvent this. In any case: Good question.
– PhoemueX
Jul 14 at 16:46
This is quite a good question. At the moment I don't know how to answer it, so let me ask a "counter question": Why do you need this? Do you want to use such a function for proving something else? If so, one can maybe circumvent this. In any case: Good question.
– PhoemueX
Jul 14 at 16:46
Yes, I want to use that result to prove something else.
– Walter
Jul 23 at 15:08
Yes, I want to use that result to prove something else.
– Walter
Jul 23 at 15:08
I want to prove the following claim :
– Walter
Jul 23 at 15:08
I want to prove the following claim :
– Walter
Jul 23 at 15:08
Claim: Let $Omega_1, Omega_2$ be a partition of $mathbbR^n$,($mathbbR^n = Omega_1 cup Omega_2$ with $Omega_1 cap Omega_2 = emptyset $), $f in L^2(mathbbR ^n)$ and $left( phi_epsilon right)_epsilon$ be an approximation of unity defined as above satisfying the monotonicity of the function $Psi_f,mathbbR ^n$. Then there exists $rho,, delta >0$ such that $ forall ,,epsilon_1 < epsilon_2$, $$ delta ||f - phi_rho epsilon_1*f||_L^2(mathbbR^n) leq ||f - phi_epsilon_1*f||_L^2(Omega_1) + ||f - phi_epsilon_2*f||_L^2(Omega_2) $$
– Walter
Jul 23 at 15:16
Claim: Let $Omega_1, Omega_2$ be a partition of $mathbbR^n$,($mathbbR^n = Omega_1 cup Omega_2$ with $Omega_1 cap Omega_2 = emptyset $), $f in L^2(mathbbR ^n)$ and $left( phi_epsilon right)_epsilon$ be an approximation of unity defined as above satisfying the monotonicity of the function $Psi_f,mathbbR ^n$. Then there exists $rho,, delta >0$ such that $ forall ,,epsilon_1 < epsilon_2$, $$ delta ||f - phi_rho epsilon_1*f||_L^2(mathbbR^n) leq ||f - phi_epsilon_1*f||_L^2(Omega_1) + ||f - phi_epsilon_2*f||_L^2(Omega_2) $$
– Walter
Jul 23 at 15:16
Obviously, if the function $Psi_f, Omega_i $ satisfies the monotonicity for $i=1$ or $i=2$,then the proof of the claim is trivial with $rho = delta=1$
– Walter
Jul 23 at 15:21
Obviously, if the function $Psi_f, Omega_i $ satisfies the monotonicity for $i=1$ or $i=2$,then the proof of the claim is trivial with $rho = delta=1$
– Walter
Jul 23 at 15:21
add a comment |Â
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This is quite a good question. At the moment I don't know how to answer it, so let me ask a "counter question": Why do you need this? Do you want to use such a function for proving something else? If so, one can maybe circumvent this. In any case: Good question.
– PhoemueX
Jul 14 at 16:46
Yes, I want to use that result to prove something else.
– Walter
Jul 23 at 15:08
I want to prove the following claim :
– Walter
Jul 23 at 15:08
Claim: Let $Omega_1, Omega_2$ be a partition of $mathbbR^n$,($mathbbR^n = Omega_1 cup Omega_2$ with $Omega_1 cap Omega_2 = emptyset $), $f in L^2(mathbbR ^n)$ and $left( phi_epsilon right)_epsilon$ be an approximation of unity defined as above satisfying the monotonicity of the function $Psi_f,mathbbR ^n$. Then there exists $rho,, delta >0$ such that $ forall ,,epsilon_1 < epsilon_2$, $$ delta ||f - phi_rho epsilon_1*f||_L^2(mathbbR^n) leq ||f - phi_epsilon_1*f||_L^2(Omega_1) + ||f - phi_epsilon_2*f||_L^2(Omega_2) $$
– Walter
Jul 23 at 15:16
Obviously, if the function $Psi_f, Omega_i $ satisfies the monotonicity for $i=1$ or $i=2$,then the proof of the claim is trivial with $rho = delta=1$
– Walter
Jul 23 at 15:21