Mixing
Fourier transform of f on the unit sphere
My question is: do we have the right to compute the fourier transform of a function over the unit sphere? To be more precise, let $fin L^1(mathbb{R} ^n)$, $ngeq1$. Is the integral
$$int_{S^{n-1}}f(t)e^{-ilangle omega|trangle} dsigma(t), $$
make any sense? With of course $S^{n-1}$ denotes the unit sphere and $dsigma$ is the measure over the sphere.
fourier-analysis fourier-transform
asked Nov 20 '18 at 12:31
Mathsy
61
add a comment |
My question is: do we have the right to compute the fourier transform of a function over the unit sphere? To be more precise, let $fin L^1(mathbb{R} ^n)$, $ngeq1$. Is the integral
$$int_{S^{n-1}}f(t)e^{-ilangle omega|trangle} dsigma(t), $$
make any sense? With of course $S^{n-1}$ denotes the unit sphere and $dsigma$ is the measure over the sphere.
fourier-analysis fourier-transform
asked Nov 20 '18 at 12:31
Mathsy
61
3
You have the right to do anything you want. The question is whether it has some utility. There are constructions like this that are useful. Read about the Radon transform, and read about Harmonic analysis on Lie groups. Without some explanation of why you are interested, any answer would just be a shot in the dark.
– Charlie Frohman
Nov 20 '18 at 12:43
Thank you for your reply. Actually, I'm interested in harmonic analysis and uncertainty principles associated to many operator integrals
– Mathsy
Nov 20 '18 at 13:21
@CharlieFrohman: I totally agree with the incipit of your comment. That is why I deeply dislike the use of the adjective "legit" in mathematics.
– Giuseppe Negro
Nov 20 '18 at 15:51
The keyword to search for is "Restriction problem for the Fourier transform". Here's a Terry Tao blog post on the subject: terrytao.wordpress.com/2010/12/28/…
– Giuseppe Negro
Nov 23 '18 at 8:32
add a comment |
My question is: do we have the right to compute the fourier transform of a function over the unit sphere? To be more precise, let $fin L^1(mathbb{R} ^n)$, $ngeq1$. Is the integral
$$int_{S^{n-1}}f(t)e^{-ilangle omega|trangle} dsigma(t), $$
make any sense? With of course $S^{n-1}$ denotes the unit sphere and $dsigma$ is the measure over the sphere.
fourier-analysis fourier-transform
asked Nov 20 '18 at 12:31
Mathsy
61
My question is: do we have the right to compute the fourier transform of a function over the unit sphere? To be more precise, let $fin L^1(mathbb{R} ^n)$, $ngeq1$. Is the integral
$$int_{S^{n-1}}f(t)e^{-ilangle omega|trangle} dsigma(t), $$
make any sense? With of course $S^{n-1}$ denotes the unit sphere and $dsigma$ is the measure over the sphere.
fourier-analysis fourier-transform
fourier-analysis fourier-transform
asked Nov 20 '18 at 12:31
Mathsy
61
asked Nov 20 '18 at 12:31
Mathsy
61
asked Nov 20 '18 at 12:31
Mathsy
61
asked Nov 20 '18 at 12:31
Mathsy
61
asked Nov 20 '18 at 12:31
Mathsy
61
61
3
You have the right to do anything you want. The question is whether it has some utility. There are constructions like this that are useful. Read about the Radon transform, and read about Harmonic analysis on Lie groups. Without some explanation of why you are interested, any answer would just be a shot in the dark.
– Charlie Frohman
Nov 20 '18 at 12:43
Thank you for your reply. Actually, I'm interested in harmonic analysis and uncertainty principles associated to many operator integrals
– Mathsy
Nov 20 '18 at 13:21
@CharlieFrohman: I totally agree with the incipit of your comment. That is why I deeply dislike the use of the adjective "legit" in mathematics.
– Giuseppe Negro
Nov 20 '18 at 15:51
The keyword to search for is "Restriction problem for the Fourier transform". Here's a Terry Tao blog post on the subject: terrytao.wordpress.com/2010/12/28/…
– Giuseppe Negro
Nov 23 '18 at 8:32
add a comment |
3
You have the right to do anything you want. The question is whether it has some utility. There are constructions like this that are useful. Read about the Radon transform, and read about Harmonic analysis on Lie groups. Without some explanation of why you are interested, any answer would just be a shot in the dark.
– Charlie Frohman
Nov 20 '18 at 12:43
Thank you for your reply. Actually, I'm interested in harmonic analysis and uncertainty principles associated to many operator integrals
– Mathsy
Nov 20 '18 at 13:21
@CharlieFrohman: I totally agree with the incipit of your comment. That is why I deeply dislike the use of the adjective "legit" in mathematics.
– Giuseppe Negro
Nov 20 '18 at 15:51
The keyword to search for is "Restriction problem for the Fourier transform". Here's a Terry Tao blog post on the subject: terrytao.wordpress.com/2010/12/28/…
– Giuseppe Negro
Nov 23 '18 at 8:32
3
3
You have the right to do anything you want. The question is whether it has some utility. There are constructions like this that are useful. Read about the Radon transform, and read about Harmonic analysis on Lie groups. Without some explanation of why you are interested, any answer would just be a shot in the dark.
– Charlie Frohman
Nov 20 '18 at 12:43
You have the right to do anything you want. The question is whether it has some utility. There are constructions like this that are useful. Read about the Radon transform, and read about Harmonic analysis on Lie groups. Without some explanation of why you are interested, any answer would just be a shot in the dark.
– Charlie Frohman
Nov 20 '18 at 12:43
Thank you for your reply. Actually, I'm interested in harmonic analysis and uncertainty principles associated to many operator integrals
– Mathsy
Nov 20 '18 at 13:21
Thank you for your reply. Actually, I'm interested in harmonic analysis and uncertainty principles associated to many operator integrals
– Mathsy
Nov 20 '18 at 13:21
@CharlieFrohman: I totally agree with the incipit of your comment. That is why I deeply dislike the use of the adjective "legit" in mathematics.
– Giuseppe Negro
Nov 20 '18 at 15:51
@CharlieFrohman: I totally agree with the incipit of your comment. That is why I deeply dislike the use of the adjective "legit" in mathematics.
– Giuseppe Negro
Nov 20 '18 at 15:51
The keyword to search for is "Restriction problem for the Fourier transform". Here's a Terry Tao blog post on the subject: terrytao.wordpress.com/2010/12/28/…
– Giuseppe Negro
Nov 23 '18 at 8:32
The keyword to search for is "Restriction problem for the Fourier transform". Here's a Terry Tao blog post on the subject: terrytao.wordpress.com/2010/12/28/…
– Giuseppe Negro
Nov 23 '18 at 8:32
add a comment |
1 Answer
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It does not make sense as is. An element of $L^1(mathbb{R}^n)$ is not a function. It is an equivalence class of functions modulo equality away from (Lebesgue) measure zero sets. The sphere $S^{n-1}$ has measure zero, so it does not make sense to evaluate your $f$ on it. You need some regularity hypothesis on $f$, e.g., being continuous or being in a Sobolev space with a trace theorem to $S^{n-1}$.
answered Nov 20 '18 at 15:45
Abdelmalek Abdesselam
406210
add a comment |
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1 Answer
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It does not make sense as is. An element of $L^1(mathbb{R}^n)$ is not a function. It is an equivalence class of functions modulo equality away from (Lebesgue) measure zero sets. The sphere $S^{n-1}$ has measure zero, so it does not make sense to evaluate your $f$ on it. You need some regularity hypothesis on $f$, e.g., being continuous or being in a Sobolev space with a trace theorem to $S^{n-1}$.
answered Nov 20 '18 at 15:45
Abdelmalek Abdesselam
406210
add a comment |
It does not make sense as is. An element of $L^1(mathbb{R}^n)$ is not a function. It is an equivalence class of functions modulo equality away from (Lebesgue) measure zero sets. The sphere $S^{n-1}$ has measure zero, so it does not make sense to evaluate your $f$ on it. You need some regularity hypothesis on $f$, e.g., being continuous or being in a Sobolev space with a trace theorem to $S^{n-1}$.
answered Nov 20 '18 at 15:45
Abdelmalek Abdesselam
406210
add a comment |
It does not make sense as is. An element of $L^1(mathbb{R}^n)$ is not a function. It is an equivalence class of functions modulo equality away from (Lebesgue) measure zero sets. The sphere $S^{n-1}$ has measure zero, so it does not make sense to evaluate your $f$ on it. You need some regularity hypothesis on $f$, e.g., being continuous or being in a Sobolev space with a trace theorem to $S^{n-1}$.
answered Nov 20 '18 at 15:45
Abdelmalek Abdesselam
406210
It does not make sense as is. An element of $L^1(mathbb{R}^n)$ is not a function. It is an equivalence class of functions modulo equality away from (Lebesgue) measure zero sets. The sphere $S^{n-1}$ has measure zero, so it does not make sense to evaluate your $f$ on it. You need some regularity hypothesis on $f$, e.g., being continuous or being in a Sobolev space with a trace theorem to $S^{n-1}$.
answered Nov 20 '18 at 15:45
Abdelmalek Abdesselam
406210
answered Nov 20 '18 at 15:45
Abdelmalek Abdesselam
406210
answered Nov 20 '18 at 15:45
Abdelmalek Abdesselam
406210
answered Nov 20 '18 at 15:45
Abdelmalek Abdesselam
406210
406210
add a comment |
add a comment |
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You have the right to do anything you want. The question is whether it has some utility. There are constructions like this that are useful. Read about the Radon transform, and read about Harmonic analysis on Lie groups. Without some explanation of why you are interested, any answer would just be a shot in the dark.
– Charlie Frohman
Nov 20 '18 at 12:43
Thank you for your reply. Actually, I'm interested in harmonic analysis and uncertainty principles associated to many operator integrals
– Mathsy
Nov 20 '18 at 13:21
@CharlieFrohman: I totally agree with the incipit of your comment. That is why I deeply dislike the use of the adjective "legit" in mathematics.
– Giuseppe Negro
Nov 20 '18 at 15:51
The keyword to search for is "Restriction problem for the Fourier transform". Here's a Terry Tao blog post on the subject: terrytao.wordpress.com/2010/12/28/…
– Giuseppe Negro
Nov 23 '18 at 8:32