Expansion of function in polar coordinates
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I'd like to expand a function in polar coordinates to something that splits radius and angle
$f(r,theta)=sum_i A_i(r)B_i(theta)$
I've found some hints on the internet by the name of polar Fourier transform, but I didn't find a Wikipedia page or a good explanation. What is the name for such a decomposition and how to find a basic description?
Is using Bessel function the only way to find such a factor decomposition?
Is it possible (or useful) to write this operation in complex number representation for the 2D coordinates?
fourier-series
edited Jan 27 '15 at 14:59
Arjang
5,57362363
asked Jan 27 '15 at 14:43
GerenukGerenuk
1,0481920
$endgroup$
add a comment |
$begingroup$
I'd like to expand a function in polar coordinates to something that splits radius and angle
$f(r,theta)=sum_i A_i(r)B_i(theta)$
I've found some hints on the internet by the name of polar Fourier transform, but I didn't find a Wikipedia page or a good explanation. What is the name for such a decomposition and how to find a basic description?
Is using Bessel function the only way to find such a factor decomposition?
Is it possible (or useful) to write this operation in complex number representation for the 2D coordinates?
fourier-series
edited Jan 27 '15 at 14:59
Arjang
5,57362363
asked Jan 27 '15 at 14:43
GerenukGerenuk
1,0481920
$endgroup$
$begingroup$
I like how this is explained in the book on harmonic analysis by Stein and Weiss, chapter on spherical harmonics.
$endgroup$
– Giuseppe Negro
Oct 30 '18 at 10:02
add a comment |
$begingroup$
I'd like to expand a function in polar coordinates to something that splits radius and angle
$f(r,theta)=sum_i A_i(r)B_i(theta)$
I've found some hints on the internet by the name of polar Fourier transform, but I didn't find a Wikipedia page or a good explanation. What is the name for such a decomposition and how to find a basic description?
Is using Bessel function the only way to find such a factor decomposition?
Is it possible (or useful) to write this operation in complex number representation for the 2D coordinates?
fourier-series
edited Jan 27 '15 at 14:59
Arjang
5,57362363
asked Jan 27 '15 at 14:43
GerenukGerenuk
1,0481920
$endgroup$
I'd like to expand a function in polar coordinates to something that splits radius and angle
$f(r,theta)=sum_i A_i(r)B_i(theta)$
I've found some hints on the internet by the name of polar Fourier transform, but I didn't find a Wikipedia page or a good explanation. What is the name for such a decomposition and how to find a basic description?
Is using Bessel function the only way to find such a factor decomposition?
Is it possible (or useful) to write this operation in complex number representation for the 2D coordinates?
fourier-series
fourier-series
edited Jan 27 '15 at 14:59
Arjang
5,57362363
asked Jan 27 '15 at 14:43
GerenukGerenuk
1,0481920
edited Jan 27 '15 at 14:59
Arjang
5,57362363
asked Jan 27 '15 at 14:43
GerenukGerenuk
1,0481920
edited Jan 27 '15 at 14:59
Arjang
5,57362363
edited Jan 27 '15 at 14:59
Arjang
5,57362363
edited Jan 27 '15 at 14:59
Arjang
5,57362363
5,57362363
asked Jan 27 '15 at 14:43
GerenukGerenuk
1,0481920
asked Jan 27 '15 at 14:43
GerenukGerenuk
1,0481920
asked Jan 27 '15 at 14:43
GerenukGerenuk
1,0481920
1,0481920
$begingroup$
I like how this is explained in the book on harmonic analysis by Stein and Weiss, chapter on spherical harmonics.
$endgroup$
– Giuseppe Negro
Oct 30 '18 at 10:02
add a comment |
$begingroup$
I like how this is explained in the book on harmonic analysis by Stein and Weiss, chapter on spherical harmonics.
$endgroup$
– Giuseppe Negro
Oct 30 '18 at 10:02
$begingroup$
I like how this is explained in the book on harmonic analysis by Stein and Weiss, chapter on spherical harmonics.
$endgroup$
– Giuseppe Negro
Oct 30 '18 at 10:02
$begingroup$
I like how this is explained in the book on harmonic analysis by Stein and Weiss, chapter on spherical harmonics.
$endgroup$
– Giuseppe Negro
Oct 30 '18 at 10:02
add a comment |
1 Answer
1
active
oldest
votes
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Consider the 2D Fourier transform in rectangular coordinates,
$$
F(u,v) = int_{-infty}^{infty} left[ int_{-infty}^{infty} f(x,y)e^{2ipi ux}dx right]e^{2ipi vy}dy
$$
One can take 1D FT in X direction then in Y direction. But for Polar coordinates the expression changes,
Let , $ux+vy = rho r cos (phi-theta)$, then
$$
F(u,v) = int_{0}^{2pi} int_{0}^{infty} f(r,theta)e^{2ipirho r cos(phi- theta)}rdr dtheta
$$
Now the function $f(r,theta) $ has seperability in polar coordinates if it can be written in the form ,
$$f(r,theta)= f_r(r)f_{theta}(theta)$$
Then the function $f(r,theta)$ is a circularly symmetric with $f_{theta}(theta)=1$ , examples include like that of a cylinder etc. Read more about Hankel transform, 2D Polar Fourier transform and functions of this type.
edited Nov 6 '17 at 9:36
Faraad Armwood
7,5342619
answered Aug 2 '15 at 18:37
Syed Alam AbbasSyed Alam Abbas
333410
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add a comment |
protected by user26857 Nov 4 '15 at 22:03
Thank you for your interest in this question.
Because it has attracted low-quality or spam answers that had to be removed, posting an answer now requires 10 reputation on this site (the association bonus does not count).
Would you like to answer one of these unanswered questions instead?
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Consider the 2D Fourier transform in rectangular coordinates,
$$
F(u,v) = int_{-infty}^{infty} left[ int_{-infty}^{infty} f(x,y)e^{2ipi ux}dx right]e^{2ipi vy}dy
$$
One can take 1D FT in X direction then in Y direction. But for Polar coordinates the expression changes,
Let , $ux+vy = rho r cos (phi-theta)$, then
$$
F(u,v) = int_{0}^{2pi} int_{0}^{infty} f(r,theta)e^{2ipirho r cos(phi- theta)}rdr dtheta
$$
Now the function $f(r,theta) $ has seperability in polar coordinates if it can be written in the form ,
$$f(r,theta)= f_r(r)f_{theta}(theta)$$
Then the function $f(r,theta)$ is a circularly symmetric with $f_{theta}(theta)=1$ , examples include like that of a cylinder etc. Read more about Hankel transform, 2D Polar Fourier transform and functions of this type.
edited Nov 6 '17 at 9:36
Faraad Armwood
7,5342619
answered Aug 2 '15 at 18:37
Syed Alam AbbasSyed Alam Abbas
333410
$endgroup$
add a comment |
$begingroup$
Consider the 2D Fourier transform in rectangular coordinates,
$$
F(u,v) = int_{-infty}^{infty} left[ int_{-infty}^{infty} f(x,y)e^{2ipi ux}dx right]e^{2ipi vy}dy
$$
One can take 1D FT in X direction then in Y direction. But for Polar coordinates the expression changes,
Let , $ux+vy = rho r cos (phi-theta)$, then
$$
F(u,v) = int_{0}^{2pi} int_{0}^{infty} f(r,theta)e^{2ipirho r cos(phi- theta)}rdr dtheta
$$
Now the function $f(r,theta) $ has seperability in polar coordinates if it can be written in the form ,
$$f(r,theta)= f_r(r)f_{theta}(theta)$$
Then the function $f(r,theta)$ is a circularly symmetric with $f_{theta}(theta)=1$ , examples include like that of a cylinder etc. Read more about Hankel transform, 2D Polar Fourier transform and functions of this type.
edited Nov 6 '17 at 9:36
Faraad Armwood
7,5342619
answered Aug 2 '15 at 18:37
Syed Alam AbbasSyed Alam Abbas
333410
$endgroup$
add a comment |
$begingroup$
Consider the 2D Fourier transform in rectangular coordinates,
$$
F(u,v) = int_{-infty}^{infty} left[ int_{-infty}^{infty} f(x,y)e^{2ipi ux}dx right]e^{2ipi vy}dy
$$
One can take 1D FT in X direction then in Y direction. But for Polar coordinates the expression changes,
Let , $ux+vy = rho r cos (phi-theta)$, then
$$
F(u,v) = int_{0}^{2pi} int_{0}^{infty} f(r,theta)e^{2ipirho r cos(phi- theta)}rdr dtheta
$$
Now the function $f(r,theta) $ has seperability in polar coordinates if it can be written in the form ,
$$f(r,theta)= f_r(r)f_{theta}(theta)$$
Then the function $f(r,theta)$ is a circularly symmetric with $f_{theta}(theta)=1$ , examples include like that of a cylinder etc. Read more about Hankel transform, 2D Polar Fourier transform and functions of this type.
edited Nov 6 '17 at 9:36
Faraad Armwood
7,5342619
answered Aug 2 '15 at 18:37
Syed Alam AbbasSyed Alam Abbas
333410
$endgroup$
Consider the 2D Fourier transform in rectangular coordinates,
$$
F(u,v) = int_{-infty}^{infty} left[ int_{-infty}^{infty} f(x,y)e^{2ipi ux}dx right]e^{2ipi vy}dy
$$
One can take 1D FT in X direction then in Y direction. But for Polar coordinates the expression changes,
Let , $ux+vy = rho r cos (phi-theta)$, then
$$
F(u,v) = int_{0}^{2pi} int_{0}^{infty} f(r,theta)e^{2ipirho r cos(phi- theta)}rdr dtheta
$$
Now the function $f(r,theta) $ has seperability in polar coordinates if it can be written in the form ,
$$f(r,theta)= f_r(r)f_{theta}(theta)$$
Then the function $f(r,theta)$ is a circularly symmetric with $f_{theta}(theta)=1$ , examples include like that of a cylinder etc. Read more about Hankel transform, 2D Polar Fourier transform and functions of this type.
edited Nov 6 '17 at 9:36
Faraad Armwood
7,5342619
answered Aug 2 '15 at 18:37
Syed Alam AbbasSyed Alam Abbas
333410
edited Nov 6 '17 at 9:36
Faraad Armwood
7,5342619
edited Nov 6 '17 at 9:36
Faraad Armwood
7,5342619
edited Nov 6 '17 at 9:36
Faraad Armwood
7,5342619
7,5342619
answered Aug 2 '15 at 18:37
Syed Alam AbbasSyed Alam Abbas
333410
answered Aug 2 '15 at 18:37
Syed Alam AbbasSyed Alam Abbas
333410
answered Aug 2 '15 at 18:37
Syed Alam AbbasSyed Alam Abbas
333410
333410
add a comment |
add a comment |
protected by user26857 Nov 4 '15 at 22:03
Thank you for your interest in this question.
Because it has attracted low-quality or spam answers that had to be removed, posting an answer now requires 10 reputation on this site (the association bonus does not count).
Would you like to answer one of these unanswered questions instead?
$begingroup$
I like how this is explained in the book on harmonic analysis by Stein and Weiss, chapter on spherical harmonics.
$endgroup$
– Giuseppe Negro
Oct 30 '18 at 10:02