Weak derivative
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I really need help with this exercise:
Let $f in L_2 (mathbb{R})$. Show the equivalence of the following statements:
(a) $f in H_1 (mathbb{R})$.
(b) The function $xi mapsto xi hat{f}(xi) in L_2 (mathbb{R})$.
(c) $lim_{h to 0} (f (x + h)-f (x)) / h$ exists in $L_2 (R)$.
I think I succeed in showing that (a) and (c) are equivalent, but I really don't know how to show that they are equivalent to (b).
functional-analysis fourier-transform weak-derivatives
edited Dec 12 '18 at 13:53
Daniele Tampieri
2,3272922
asked Dec 12 '18 at 13:35
Francesca BallatoreFrancesca Ballatore
426
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add a comment |
$begingroup$
I really need help with this exercise:
Let $f in L_2 (mathbb{R})$. Show the equivalence of the following statements:
(a) $f in H_1 (mathbb{R})$.
(b) The function $xi mapsto xi hat{f}(xi) in L_2 (mathbb{R})$.
(c) $lim_{h to 0} (f (x + h)-f (x)) / h$ exists in $L_2 (R)$.
I think I succeed in showing that (a) and (c) are equivalent, but I really don't know how to show that they are equivalent to (b).
functional-analysis fourier-transform weak-derivatives
edited Dec 12 '18 at 13:53
Daniele Tampieri
2,3272922
asked Dec 12 '18 at 13:35
Francesca BallatoreFrancesca Ballatore
426
$endgroup$
$begingroup$
Hi Francesca, and welcome to the Math.SE: if you want to produce nicely formatted mathematical text for your questions, have a look at the Math Jax reference. Also, I would like to point out that, while being a customary notation, it should be pointed out that $hat{f}(xi)$ is the Fourier transform of $f(x)$
$endgroup$
– Daniele Tampieri
Dec 12 '18 at 13:49
add a comment |
$begingroup$
I really need help with this exercise:
Let $f in L_2 (mathbb{R})$. Show the equivalence of the following statements:
(a) $f in H_1 (mathbb{R})$.
(b) The function $xi mapsto xi hat{f}(xi) in L_2 (mathbb{R})$.
(c) $lim_{h to 0} (f (x + h)-f (x)) / h$ exists in $L_2 (R)$.
I think I succeed in showing that (a) and (c) are equivalent, but I really don't know how to show that they are equivalent to (b).
functional-analysis fourier-transform weak-derivatives
edited Dec 12 '18 at 13:53
Daniele Tampieri
2,3272922
asked Dec 12 '18 at 13:35
Francesca BallatoreFrancesca Ballatore
426
$endgroup$
I really need help with this exercise:
Let $f in L_2 (mathbb{R})$. Show the equivalence of the following statements:
(a) $f in H_1 (mathbb{R})$.
(b) The function $xi mapsto xi hat{f}(xi) in L_2 (mathbb{R})$.
(c) $lim_{h to 0} (f (x + h)-f (x)) / h$ exists in $L_2 (R)$.
I think I succeed in showing that (a) and (c) are equivalent, but I really don't know how to show that they are equivalent to (b).
functional-analysis fourier-transform weak-derivatives
functional-analysis fourier-transform weak-derivatives
edited Dec 12 '18 at 13:53
Daniele Tampieri
2,3272922
asked Dec 12 '18 at 13:35
Francesca BallatoreFrancesca Ballatore
426
edited Dec 12 '18 at 13:53
Daniele Tampieri
2,3272922
asked Dec 12 '18 at 13:35
Francesca BallatoreFrancesca Ballatore
426
edited Dec 12 '18 at 13:53
Daniele Tampieri
2,3272922
edited Dec 12 '18 at 13:53
Daniele Tampieri
2,3272922
edited Dec 12 '18 at 13:53
Daniele Tampieri
2,3272922
2,3272922
asked Dec 12 '18 at 13:35
Francesca BallatoreFrancesca Ballatore
426
asked Dec 12 '18 at 13:35
Francesca BallatoreFrancesca Ballatore
426
asked Dec 12 '18 at 13:35
Francesca BallatoreFrancesca Ballatore
426
426
$begingroup$
Hi Francesca, and welcome to the Math.SE: if you want to produce nicely formatted mathematical text for your questions, have a look at the Math Jax reference. Also, I would like to point out that, while being a customary notation, it should be pointed out that $hat{f}(xi)$ is the Fourier transform of $f(x)$
$endgroup$
– Daniele Tampieri
Dec 12 '18 at 13:49
add a comment |
$begingroup$
Hi Francesca, and welcome to the Math.SE: if you want to produce nicely formatted mathematical text for your questions, have a look at the Math Jax reference. Also, I would like to point out that, while being a customary notation, it should be pointed out that $hat{f}(xi)$ is the Fourier transform of $f(x)$
$endgroup$
– Daniele Tampieri
Dec 12 '18 at 13:49
$begingroup$
Hi Francesca, and welcome to the Math.SE: if you want to produce nicely formatted mathematical text for your questions, have a look at the Math Jax reference. Also, I would like to point out that, while being a customary notation, it should be pointed out that $hat{f}(xi)$ is the Fourier transform of $f(x)$
$endgroup$
– Daniele Tampieri
Dec 12 '18 at 13:49
$begingroup$
Hi Francesca, and welcome to the Math.SE: if you want to produce nicely formatted mathematical text for your questions, have a look at the Math Jax reference. Also, I would like to point out that, while being a customary notation, it should be pointed out that $hat{f}(xi)$ is the Fourier transform of $f(x)$
$endgroup$
– Daniele Tampieri
Dec 12 '18 at 13:49
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Hint: The fourier transform of $i partial_x f(x)$ is $xi hat{f}(xi)$ (Think about why by pulling the derivative inside the Fourier integral, then try to justify that.). Since you know that the Fourier transform is a bijective and unitary map $L^2(mathbb{R}) to L^2(mathbb{R})$, you will be able to see that $partial_x f(x) in L^2$ and $xi hat{f}(xi) in L^2$ are equivalent.
Always be careful when proving things for the Fourier transform in $L^2$ because the Fourier integral $int f(x) e^{ikx}$ is only defined for some functions in a dense set in $L^2$.
answered Dec 12 '18 at 14:05
LukeLuke
760216
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add a comment |
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1 Answer
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active
oldest
votes
1 Answer
1
active
oldest
votes
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oldest
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active
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votes
$begingroup$
Hint: The fourier transform of $i partial_x f(x)$ is $xi hat{f}(xi)$ (Think about why by pulling the derivative inside the Fourier integral, then try to justify that.). Since you know that the Fourier transform is a bijective and unitary map $L^2(mathbb{R}) to L^2(mathbb{R})$, you will be able to see that $partial_x f(x) in L^2$ and $xi hat{f}(xi) in L^2$ are equivalent.
Always be careful when proving things for the Fourier transform in $L^2$ because the Fourier integral $int f(x) e^{ikx}$ is only defined for some functions in a dense set in $L^2$.
answered Dec 12 '18 at 14:05
LukeLuke
760216
$endgroup$
add a comment |
$begingroup$
Hint: The fourier transform of $i partial_x f(x)$ is $xi hat{f}(xi)$ (Think about why by pulling the derivative inside the Fourier integral, then try to justify that.). Since you know that the Fourier transform is a bijective and unitary map $L^2(mathbb{R}) to L^2(mathbb{R})$, you will be able to see that $partial_x f(x) in L^2$ and $xi hat{f}(xi) in L^2$ are equivalent.
Always be careful when proving things for the Fourier transform in $L^2$ because the Fourier integral $int f(x) e^{ikx}$ is only defined for some functions in a dense set in $L^2$.
answered Dec 12 '18 at 14:05
LukeLuke
760216
$endgroup$
add a comment |
$begingroup$
Hint: The fourier transform of $i partial_x f(x)$ is $xi hat{f}(xi)$ (Think about why by pulling the derivative inside the Fourier integral, then try to justify that.). Since you know that the Fourier transform is a bijective and unitary map $L^2(mathbb{R}) to L^2(mathbb{R})$, you will be able to see that $partial_x f(x) in L^2$ and $xi hat{f}(xi) in L^2$ are equivalent.
Always be careful when proving things for the Fourier transform in $L^2$ because the Fourier integral $int f(x) e^{ikx}$ is only defined for some functions in a dense set in $L^2$.
answered Dec 12 '18 at 14:05
LukeLuke
760216
$endgroup$
Hint: The fourier transform of $i partial_x f(x)$ is $xi hat{f}(xi)$ (Think about why by pulling the derivative inside the Fourier integral, then try to justify that.). Since you know that the Fourier transform is a bijective and unitary map $L^2(mathbb{R}) to L^2(mathbb{R})$, you will be able to see that $partial_x f(x) in L^2$ and $xi hat{f}(xi) in L^2$ are equivalent.
Always be careful when proving things for the Fourier transform in $L^2$ because the Fourier integral $int f(x) e^{ikx}$ is only defined for some functions in a dense set in $L^2$.
answered Dec 12 '18 at 14:05
LukeLuke
760216
answered Dec 12 '18 at 14:05
LukeLuke
760216
answered Dec 12 '18 at 14:05
LukeLuke
760216
answered Dec 12 '18 at 14:05
LukeLuke
760216
760216
add a comment |
add a comment |
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$begingroup$
Hi Francesca, and welcome to the Math.SE: if you want to produce nicely formatted mathematical text for your questions, have a look at the Math Jax reference. Also, I would like to point out that, while being a customary notation, it should be pointed out that $hat{f}(xi)$ is the Fourier transform of $f(x)$
$endgroup$
– Daniele Tampieri
Dec 12 '18 at 13:49