Maximum for Fourier transform results in an error
up vote
1
down vote
favorite
assume for a function $varphi in L^1$, that $varphi geq 0$ and $int_mathbb{R}varphi(x) dx = 1$. Then the following should hold:
$$|hat{varphi}(omega)| leq int_mathbb{R} |varphi(x)||e^{-ixomega}|dx =int_mathbb{R}varphi(x) dx = 1 = hat{varphi}(0)
$$
which means, that $hat{varphi}$ has a maximum at $omega =0$.
Therefore
$$
|0|=|hat{varphi}'(0)| = |int_mathbb{R}xvarphi(x) dx |
$$
Now assume $varphi(x)= begin{cases}1 & x in [0,1],\
0 & otherwise
end{cases}$ which means all assumptions are fullfiled.
But now $|hat{varphi}'(0)| = frac{1}{2}$
Where is my error?
Thanks for any help
Matthias
fourier-analysis fourier-transform
edited Nov 15 at 21:01
Bernard
116k637108
asked Nov 15 at 20:36
Matthias Lauber
162
add a comment |
up vote
1
down vote
favorite
assume for a function $varphi in L^1$, that $varphi geq 0$ and $int_mathbb{R}varphi(x) dx = 1$. Then the following should hold:
$$|hat{varphi}(omega)| leq int_mathbb{R} |varphi(x)||e^{-ixomega}|dx =int_mathbb{R}varphi(x) dx = 1 = hat{varphi}(0)
$$
which means, that $hat{varphi}$ has a maximum at $omega =0$.
Therefore
$$
|0|=|hat{varphi}'(0)| = |int_mathbb{R}xvarphi(x) dx |
$$
Now assume $varphi(x)= begin{cases}1 & x in [0,1],\
0 & otherwise
end{cases}$ which means all assumptions are fullfiled.
But now $|hat{varphi}'(0)| = frac{1}{2}$
Where is my error?
Thanks for any help
Matthias
fourier-analysis fourier-transform
edited Nov 15 at 21:01
Bernard
116k637108
asked Nov 15 at 20:36
Matthias Lauber
162
2
You're forgetting that $hat phi$ is complex valued.
– ε-δ
Nov 15 at 22:09
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
assume for a function $varphi in L^1$, that $varphi geq 0$ and $int_mathbb{R}varphi(x) dx = 1$. Then the following should hold:
$$|hat{varphi}(omega)| leq int_mathbb{R} |varphi(x)||e^{-ixomega}|dx =int_mathbb{R}varphi(x) dx = 1 = hat{varphi}(0)
$$
which means, that $hat{varphi}$ has a maximum at $omega =0$.
Therefore
$$
|0|=|hat{varphi}'(0)| = |int_mathbb{R}xvarphi(x) dx |
$$
Now assume $varphi(x)= begin{cases}1 & x in [0,1],\
0 & otherwise
end{cases}$ which means all assumptions are fullfiled.
But now $|hat{varphi}'(0)| = frac{1}{2}$
Where is my error?
Thanks for any help
Matthias
fourier-analysis fourier-transform
edited Nov 15 at 21:01
Bernard
116k637108
asked Nov 15 at 20:36
Matthias Lauber
162
assume for a function $varphi in L^1$, that $varphi geq 0$ and $int_mathbb{R}varphi(x) dx = 1$. Then the following should hold:
$$|hat{varphi}(omega)| leq int_mathbb{R} |varphi(x)||e^{-ixomega}|dx =int_mathbb{R}varphi(x) dx = 1 = hat{varphi}(0)
$$
which means, that $hat{varphi}$ has a maximum at $omega =0$.
Therefore
$$
|0|=|hat{varphi}'(0)| = |int_mathbb{R}xvarphi(x) dx |
$$
Now assume $varphi(x)= begin{cases}1 & x in [0,1],\
0 & otherwise
end{cases}$ which means all assumptions are fullfiled.
But now $|hat{varphi}'(0)| = frac{1}{2}$
Where is my error?
Thanks for any help
Matthias
fourier-analysis fourier-transform
fourier-analysis fourier-transform
edited Nov 15 at 21:01
Bernard
116k637108
asked Nov 15 at 20:36
Matthias Lauber
162
edited Nov 15 at 21:01
Bernard
116k637108
asked Nov 15 at 20:36
Matthias Lauber
162
edited Nov 15 at 21:01
Bernard
116k637108
edited Nov 15 at 21:01
Bernard
116k637108
edited Nov 15 at 21:01
Bernard
116k637108
116k637108
asked Nov 15 at 20:36
Matthias Lauber
162
asked Nov 15 at 20:36
Matthias Lauber
162
asked Nov 15 at 20:36
Matthias Lauber
162
162
2
You're forgetting that $hat phi$ is complex valued.
– ε-δ
Nov 15 at 22:09
add a comment |
2
You're forgetting that $hat phi$ is complex valued.
– ε-δ
Nov 15 at 22:09
2
2
You're forgetting that $hat phi$ is complex valued.
– ε-δ
Nov 15 at 22:09
You're forgetting that $hat phi$ is complex valued.
– ε-δ
Nov 15 at 22:09
add a comment |
1 Answer
1
active
oldest
votes
up vote
0
down vote
It is $|hatvarphi$| that attains its maximum at $omega=0$, not $hatvarphi$. In fact, since $varphi$ is complex valued, it does not make sense to talk about its maximum.
$$
varphi(omega)=frac{1-e^{-iomega}}{i,omega}
$$
and
$$
|varphi(omega)|^2=frac{2(1-cosomega)}{omega^2}=1-frac{omega^2}{12}+dots
$$
answered Nov 26 at 17:04
Julián Aguirre
66.9k23994
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
It is $|hatvarphi$| that attains its maximum at $omega=0$, not $hatvarphi$. In fact, since $varphi$ is complex valued, it does not make sense to talk about its maximum.
$$
varphi(omega)=frac{1-e^{-iomega}}{i,omega}
$$
and
$$
|varphi(omega)|^2=frac{2(1-cosomega)}{omega^2}=1-frac{omega^2}{12}+dots
$$
answered Nov 26 at 17:04
Julián Aguirre
66.9k23994
add a comment |
up vote
0
down vote
It is $|hatvarphi$| that attains its maximum at $omega=0$, not $hatvarphi$. In fact, since $varphi$ is complex valued, it does not make sense to talk about its maximum.
$$
varphi(omega)=frac{1-e^{-iomega}}{i,omega}
$$
and
$$
|varphi(omega)|^2=frac{2(1-cosomega)}{omega^2}=1-frac{omega^2}{12}+dots
$$
answered Nov 26 at 17:04
Julián Aguirre
66.9k23994
add a comment |
up vote
0
down vote
up vote
0
down vote
It is $|hatvarphi$| that attains its maximum at $omega=0$, not $hatvarphi$. In fact, since $varphi$ is complex valued, it does not make sense to talk about its maximum.
$$
varphi(omega)=frac{1-e^{-iomega}}{i,omega}
$$
and
$$
|varphi(omega)|^2=frac{2(1-cosomega)}{omega^2}=1-frac{omega^2}{12}+dots
$$
answered Nov 26 at 17:04
Julián Aguirre
66.9k23994
It is $|hatvarphi$| that attains its maximum at $omega=0$, not $hatvarphi$. In fact, since $varphi$ is complex valued, it does not make sense to talk about its maximum.
$$
varphi(omega)=frac{1-e^{-iomega}}{i,omega}
$$
and
$$
|varphi(omega)|^2=frac{2(1-cosomega)}{omega^2}=1-frac{omega^2}{12}+dots
$$
answered Nov 26 at 17:04
Julián Aguirre
66.9k23994
answered Nov 26 at 17:04
Julián Aguirre
66.9k23994
answered Nov 26 at 17:04
Julián Aguirre
66.9k23994
answered Nov 26 at 17:04
Julián Aguirre
66.9k23994
66.9k23994
add a comment |
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3000288%2fmaximum-for-fourier-transform-results-in-an-error%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
2
You're forgetting that $hat phi$ is complex valued.
– ε-δ
Nov 15 at 22:09