Abstract

For functions that are best described with spherical coordinates, the three-dimensional Fourier transform can be written in spherical coordinates as a combination of spherical Hankel transforms and spherical harmonic series. However, to be as useful as its Cartesian counterpart, a spherical version of the Fourier operational toolset is required for the standard operations of shift, multiplication, convolution, etc. This paper derives the spherical version of the standard Fourier operation toolset. In particular, convolution in various forms is discussed in detail as this has important consequences for filtering. It is shown that standard multiplication and convolution rules do apply as long as the correct definition of convolution is applied.

© 2010 Optical Society of America

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References

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  1. R. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, 1999).
  2. K. Howell, “Fourier transforms,” in The Transforms and Applications Handbook (CRC, 2000), pp. 2.1–2.159.
  3. G. S. Chirikjian and A. B. Kyatkin, Engineering Applications of Noncommutative Harmonic Analysis (CRC Press, 2000).
    [CrossRef]
  4. M. Xu and L. V. Wang, “Time-domain reconstruction for thermoacoustic tomography in a spherical geometry,” IEEE Trans. Med. Imaging 21, 814–822 (2002).
    [CrossRef] [PubMed]
  5. A. Averbuch, R. R. Coifman, D. L. Donoho, M. Elad, and M. Israeli, “Fast and accurate polar Fourier transform,” J. Appl. Comput. Harmonic Anal. 21, 145–167 (2006).
    [CrossRef]
  6. M. Slaney and A. Kak, Principles of Computerized Tomographic Imaging (SIAM, 1988).
  7. N. Baddour, “Fourier diffraction theorem for diffusion-based thermal tomography,” J. Phys. A 39, 14379–14395 (2006).
    [CrossRef]
  8. N. Baddour, “Operational and convolution properties of two-dimensional Fourier transforms in polar coordinates,” J. Opt. Soc. Am. A 26, 1767–1777 (2009).
    [CrossRef]
  9. J. R. Driscoll and D. M. Healy, “Computing Fourier transforms and convolutions on the 2-sphere,” Adv. Appl. Math. 15, 202–250 (1994).
    [CrossRef]
  10. R. Piessens, “The Hankel transform,” in The Transforms and Applications Handbook (CRC, 2000), pp. 9.1–9.30.
    [CrossRef]
  11. “Spherical harmonics,” Wikipedia, the Free Encyclopedia.
  12. J. C. Slater, Quantum Theory of Atomic Structure, Vol. I of International Series in Pure and Applied Physics (McGraw-Hill, New York, 1960).
  13. “Slater integrals,” Wikipedia, the Free Encyclopedia.
  14. G. Arfken and H. Weber, Mathematical Methods for Physicists (Elsevier Academic, 2005).
  15. E. W. Weisstein, “Spherical Harmonics,” Wolfram MathWorld.
  16. R. Basri and D. Jacobs, “Lambertian reflectance and linear subspaces,” in Proceedings of the Eighth IEEE International Conference on Computer Vision, 2001 (ICCV 2001) (2001), Vol. 2, pp. 383–390.
    [CrossRef]
  17. R. Mehrem, J. T. Londergan, and M. H. Macfarlane, “Analytic expressions for integrals of products of spherical Bessel functions,” J. Phys. A 24, 1435–1453 (1991).
    [CrossRef]
  18. V. Fabrikant, “Computation of infinite integrals involving three Bessel functions by introduction of new formalism,” Z. Angew. Math. Mech. 83, 363–374 (2003).
    [CrossRef]
  19. A. D. Jackson and L. C. Maximon, “Integrals of products of Bessel functions,” SIAM J. Math. Anal. 3, 446–460 (1972).
    [CrossRef]
  20. R. Ramamoorthi and P. Hanrahan, “A signal-processing framework for reflection,” ACM Trans. Graphics 23, 1004–1042 (2004).
    [CrossRef]
  21. T. Inui, Group Theory and Its Applications in Physics (Springer-Verlag, 1990).
    [CrossRef]
  22. R. Ramamoorthi and P. Hanrahan, “An efficient representation for irradiance environment maps,” in Proceedings of the 28th Annual Conference on Computer Graphics and Interactive Techniques (ACM, 2001), pp. 497–500.

2009 (1)

2006 (2)

A. Averbuch, R. R. Coifman, D. L. Donoho, M. Elad, and M. Israeli, “Fast and accurate polar Fourier transform,” J. Appl. Comput. Harmonic Anal. 21, 145–167 (2006).
[CrossRef]

N. Baddour, “Fourier diffraction theorem for diffusion-based thermal tomography,” J. Phys. A 39, 14379–14395 (2006).
[CrossRef]

2005 (1)

G. Arfken and H. Weber, Mathematical Methods for Physicists (Elsevier Academic, 2005).

2004 (1)

R. Ramamoorthi and P. Hanrahan, “A signal-processing framework for reflection,” ACM Trans. Graphics 23, 1004–1042 (2004).
[CrossRef]

2003 (1)

V. Fabrikant, “Computation of infinite integrals involving three Bessel functions by introduction of new formalism,” Z. Angew. Math. Mech. 83, 363–374 (2003).
[CrossRef]

2002 (1)

M. Xu and L. V. Wang, “Time-domain reconstruction for thermoacoustic tomography in a spherical geometry,” IEEE Trans. Med. Imaging 21, 814–822 (2002).
[CrossRef] [PubMed]

2001 (2)

R. Basri and D. Jacobs, “Lambertian reflectance and linear subspaces,” in Proceedings of the Eighth IEEE International Conference on Computer Vision, 2001 (ICCV 2001) (2001), Vol. 2, pp. 383–390.
[CrossRef]

R. Ramamoorthi and P. Hanrahan, “An efficient representation for irradiance environment maps,” in Proceedings of the 28th Annual Conference on Computer Graphics and Interactive Techniques (ACM, 2001), pp. 497–500.

2000 (3)

R. Piessens, “The Hankel transform,” in The Transforms and Applications Handbook (CRC, 2000), pp. 9.1–9.30.
[CrossRef]

K. Howell, “Fourier transforms,” in The Transforms and Applications Handbook (CRC, 2000), pp. 2.1–2.159.

G. S. Chirikjian and A. B. Kyatkin, Engineering Applications of Noncommutative Harmonic Analysis (CRC Press, 2000).
[CrossRef]

1999 (1)

R. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, 1999).

1994 (1)

J. R. Driscoll and D. M. Healy, “Computing Fourier transforms and convolutions on the 2-sphere,” Adv. Appl. Math. 15, 202–250 (1994).
[CrossRef]

1991 (1)

R. Mehrem, J. T. Londergan, and M. H. Macfarlane, “Analytic expressions for integrals of products of spherical Bessel functions,” J. Phys. A 24, 1435–1453 (1991).
[CrossRef]

1990 (1)

T. Inui, Group Theory and Its Applications in Physics (Springer-Verlag, 1990).
[CrossRef]

1988 (1)

M. Slaney and A. Kak, Principles of Computerized Tomographic Imaging (SIAM, 1988).

1972 (1)

A. D. Jackson and L. C. Maximon, “Integrals of products of Bessel functions,” SIAM J. Math. Anal. 3, 446–460 (1972).
[CrossRef]

1960 (1)

J. C. Slater, Quantum Theory of Atomic Structure, Vol. I of International Series in Pure and Applied Physics (McGraw-Hill, New York, 1960).

Arfken, G.

G. Arfken and H. Weber, Mathematical Methods for Physicists (Elsevier Academic, 2005).

Averbuch, A.

A. Averbuch, R. R. Coifman, D. L. Donoho, M. Elad, and M. Israeli, “Fast and accurate polar Fourier transform,” J. Appl. Comput. Harmonic Anal. 21, 145–167 (2006).
[CrossRef]

Baddour, N.

N. Baddour, “Operational and convolution properties of two-dimensional Fourier transforms in polar coordinates,” J. Opt. Soc. Am. A 26, 1767–1777 (2009).
[CrossRef]

N. Baddour, “Fourier diffraction theorem for diffusion-based thermal tomography,” J. Phys. A 39, 14379–14395 (2006).
[CrossRef]

Basri, R.

R. Basri and D. Jacobs, “Lambertian reflectance and linear subspaces,” in Proceedings of the Eighth IEEE International Conference on Computer Vision, 2001 (ICCV 2001) (2001), Vol. 2, pp. 383–390.
[CrossRef]

Bracewell, R.

R. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, 1999).

Chirikjian, G. S.

G. S. Chirikjian and A. B. Kyatkin, Engineering Applications of Noncommutative Harmonic Analysis (CRC Press, 2000).
[CrossRef]

Coifman, R. R.

A. Averbuch, R. R. Coifman, D. L. Donoho, M. Elad, and M. Israeli, “Fast and accurate polar Fourier transform,” J. Appl. Comput. Harmonic Anal. 21, 145–167 (2006).
[CrossRef]

Donoho, D. L.

A. Averbuch, R. R. Coifman, D. L. Donoho, M. Elad, and M. Israeli, “Fast and accurate polar Fourier transform,” J. Appl. Comput. Harmonic Anal. 21, 145–167 (2006).
[CrossRef]

Driscoll, J. R.

J. R. Driscoll and D. M. Healy, “Computing Fourier transforms and convolutions on the 2-sphere,” Adv. Appl. Math. 15, 202–250 (1994).
[CrossRef]

Elad, M.

A. Averbuch, R. R. Coifman, D. L. Donoho, M. Elad, and M. Israeli, “Fast and accurate polar Fourier transform,” J. Appl. Comput. Harmonic Anal. 21, 145–167 (2006).
[CrossRef]

Fabrikant, V.

V. Fabrikant, “Computation of infinite integrals involving three Bessel functions by introduction of new formalism,” Z. Angew. Math. Mech. 83, 363–374 (2003).
[CrossRef]

Hanrahan, P.

R. Ramamoorthi and P. Hanrahan, “A signal-processing framework for reflection,” ACM Trans. Graphics 23, 1004–1042 (2004).
[CrossRef]

R. Ramamoorthi and P. Hanrahan, “An efficient representation for irradiance environment maps,” in Proceedings of the 28th Annual Conference on Computer Graphics and Interactive Techniques (ACM, 2001), pp. 497–500.

Healy, D. M.

J. R. Driscoll and D. M. Healy, “Computing Fourier transforms and convolutions on the 2-sphere,” Adv. Appl. Math. 15, 202–250 (1994).
[CrossRef]

Howell, K.

K. Howell, “Fourier transforms,” in The Transforms and Applications Handbook (CRC, 2000), pp. 2.1–2.159.

Inui, T.

T. Inui, Group Theory and Its Applications in Physics (Springer-Verlag, 1990).
[CrossRef]

Israeli, M.

A. Averbuch, R. R. Coifman, D. L. Donoho, M. Elad, and M. Israeli, “Fast and accurate polar Fourier transform,” J. Appl. Comput. Harmonic Anal. 21, 145–167 (2006).
[CrossRef]

Jackson, A. D.

A. D. Jackson and L. C. Maximon, “Integrals of products of Bessel functions,” SIAM J. Math. Anal. 3, 446–460 (1972).
[CrossRef]

Jacobs, D.

R. Basri and D. Jacobs, “Lambertian reflectance and linear subspaces,” in Proceedings of the Eighth IEEE International Conference on Computer Vision, 2001 (ICCV 2001) (2001), Vol. 2, pp. 383–390.
[CrossRef]

Kak, A.

M. Slaney and A. Kak, Principles of Computerized Tomographic Imaging (SIAM, 1988).

Kyatkin, A. B.

G. S. Chirikjian and A. B. Kyatkin, Engineering Applications of Noncommutative Harmonic Analysis (CRC Press, 2000).
[CrossRef]

Londergan, J. T.

R. Mehrem, J. T. Londergan, and M. H. Macfarlane, “Analytic expressions for integrals of products of spherical Bessel functions,” J. Phys. A 24, 1435–1453 (1991).
[CrossRef]

Macfarlane, M. H.

R. Mehrem, J. T. Londergan, and M. H. Macfarlane, “Analytic expressions for integrals of products of spherical Bessel functions,” J. Phys. A 24, 1435–1453 (1991).
[CrossRef]

Maximon, L. C.

A. D. Jackson and L. C. Maximon, “Integrals of products of Bessel functions,” SIAM J. Math. Anal. 3, 446–460 (1972).
[CrossRef]

Mehrem, R.

R. Mehrem, J. T. Londergan, and M. H. Macfarlane, “Analytic expressions for integrals of products of spherical Bessel functions,” J. Phys. A 24, 1435–1453 (1991).
[CrossRef]

Piessens, R.

R. Piessens, “The Hankel transform,” in The Transforms and Applications Handbook (CRC, 2000), pp. 9.1–9.30.
[CrossRef]

Ramamoorthi, R.

R. Ramamoorthi and P. Hanrahan, “A signal-processing framework for reflection,” ACM Trans. Graphics 23, 1004–1042 (2004).
[CrossRef]

R. Ramamoorthi and P. Hanrahan, “An efficient representation for irradiance environment maps,” in Proceedings of the 28th Annual Conference on Computer Graphics and Interactive Techniques (ACM, 2001), pp. 497–500.

Slaney, M.

M. Slaney and A. Kak, Principles of Computerized Tomographic Imaging (SIAM, 1988).

Slater, J. C.

J. C. Slater, Quantum Theory of Atomic Structure, Vol. I of International Series in Pure and Applied Physics (McGraw-Hill, New York, 1960).

Wang, L. V.

M. Xu and L. V. Wang, “Time-domain reconstruction for thermoacoustic tomography in a spherical geometry,” IEEE Trans. Med. Imaging 21, 814–822 (2002).
[CrossRef] [PubMed]

Weber, H.

G. Arfken and H. Weber, Mathematical Methods for Physicists (Elsevier Academic, 2005).

Weisstein, E. W.

E. W. Weisstein, “Spherical Harmonics,” Wolfram MathWorld.

Xu, M.

M. Xu and L. V. Wang, “Time-domain reconstruction for thermoacoustic tomography in a spherical geometry,” IEEE Trans. Med. Imaging 21, 814–822 (2002).
[CrossRef] [PubMed]

ACM Trans. Graphics (1)

R. Ramamoorthi and P. Hanrahan, “A signal-processing framework for reflection,” ACM Trans. Graphics 23, 1004–1042 (2004).
[CrossRef]

Adv. Appl. Math. (1)

J. R. Driscoll and D. M. Healy, “Computing Fourier transforms and convolutions on the 2-sphere,” Adv. Appl. Math. 15, 202–250 (1994).
[CrossRef]

IEEE Trans. Med. Imaging (1)

M. Xu and L. V. Wang, “Time-domain reconstruction for thermoacoustic tomography in a spherical geometry,” IEEE Trans. Med. Imaging 21, 814–822 (2002).
[CrossRef] [PubMed]

J. Appl. Comput. Harmonic Anal. (1)

A. Averbuch, R. R. Coifman, D. L. Donoho, M. Elad, and M. Israeli, “Fast and accurate polar Fourier transform,” J. Appl. Comput. Harmonic Anal. 21, 145–167 (2006).
[CrossRef]

J. Opt. Soc. Am. A (1)

J. Phys. A (2)

N. Baddour, “Fourier diffraction theorem for diffusion-based thermal tomography,” J. Phys. A 39, 14379–14395 (2006).
[CrossRef]

R. Mehrem, J. T. Londergan, and M. H. Macfarlane, “Analytic expressions for integrals of products of spherical Bessel functions,” J. Phys. A 24, 1435–1453 (1991).
[CrossRef]

SIAM J. Math. Anal. (1)

A. D. Jackson and L. C. Maximon, “Integrals of products of Bessel functions,” SIAM J. Math. Anal. 3, 446–460 (1972).
[CrossRef]

Z. Angew. Math. Mech. (1)

V. Fabrikant, “Computation of infinite integrals involving three Bessel functions by introduction of new formalism,” Z. Angew. Math. Mech. 83, 363–374 (2003).
[CrossRef]

Other (13)

M. Slaney and A. Kak, Principles of Computerized Tomographic Imaging (SIAM, 1988).

R. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, 1999).

K. Howell, “Fourier transforms,” in The Transforms and Applications Handbook (CRC, 2000), pp. 2.1–2.159.

G. S. Chirikjian and A. B. Kyatkin, Engineering Applications of Noncommutative Harmonic Analysis (CRC Press, 2000).
[CrossRef]

R. Piessens, “The Hankel transform,” in The Transforms and Applications Handbook (CRC, 2000), pp. 9.1–9.30.
[CrossRef]

“Spherical harmonics,” Wikipedia, the Free Encyclopedia.

J. C. Slater, Quantum Theory of Atomic Structure, Vol. I of International Series in Pure and Applied Physics (McGraw-Hill, New York, 1960).

“Slater integrals,” Wikipedia, the Free Encyclopedia.

G. Arfken and H. Weber, Mathematical Methods for Physicists (Elsevier Academic, 2005).

E. W. Weisstein, “Spherical Harmonics,” Wolfram MathWorld.

R. Basri and D. Jacobs, “Lambertian reflectance and linear subspaces,” in Proceedings of the Eighth IEEE International Conference on Computer Vision, 2001 (ICCV 2001) (2001), Vol. 2, pp. 383–390.
[CrossRef]

T. Inui, Group Theory and Its Applications in Physics (Springer-Verlag, 1990).
[CrossRef]

R. Ramamoorthi and P. Hanrahan, “An efficient representation for irradiance environment maps,” in Proceedings of the 28th Annual Conference on Computer Graphics and Interactive Techniques (ACM, 2001), pp. 497–500.

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Tables (1)

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Table 1 3D Spherical Polar Fourier Toolset a

Equations (90)

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j n ( z ) = π 2 z J n + 1 / 2 ( z ) .
F ̑ n ( ρ ) = S n { f ( r ) } = 0 f ( r ) j n ( ρ r ) r 2 d r .
f ( r ) = 2 π 0 F ̑ n ( ρ ) j n ( ρ r ) ρ 2 d ρ .
Y l m ( ψ , θ ) = ( 2 l + 1 ) ( l m ) ! 4 π ( l + m ) ! P l m ( cos   ψ ) e i m θ ,
0 2 π 0 π Y l m Y l m ¯   sin   ψ d ψ d θ = δ l l δ m m .
F ( ω ) = F ( ω x , ω y , ω z ) = f ( r ) e i ( ω r ) d r .
F ( ω ) = F ( ρ , ψ ω , θ ω ) = 0 2 π 0 π 0 f ( r , ψ r , θ r ) e i ω r r 2   sin   ψ r d r d ψ r d θ r .
e i ω r = 4 π l = 0 k = l l ( i ) l j l ( ρ r ) Y l k ( ψ r , θ r ) ¯ Y l k ( ψ ω , θ ω ) .
f ( r ) = f ( r , ψ r , θ r ) = l = 0 k = l l f l k ( r ) Y l k ( ψ r , θ r ) ,
f l k ( r ) = 0 2 π 0 π f ( r , ψ r , θ r ) Y l k ( ψ r , θ r ) ¯ sin   ψ r d ψ r d θ r .
F ( ρ , ψ ω , θ ω ) = l = 0 k = l l 4 π ( i ) l { 0 f l k ( r ) j l ( ρ r ) r 2 d r } Y l k ( ψ ω , θ ω ) = l = 0 k = l l 4 π ( i ) l F ̑ l k ( ρ ) Y l k ( ψ ω , θ ω ) ,
f ( r ) = 1 ( 2 π ) 3 F ( ω ) e i ( ω r ) d ω ,
F ( ω ) = F ( ρ , ψ ω , θ ω ) = l = 0 k = l l F l k ( ρ ) Y l k ( ψ ω , θ ω ) ,
F l k ( ρ ) = 0 2 π 0 π F ( ρ , ψ ω , θ ω ) Y l k ( ψ ω , θ ω ) ¯ sin   ψ ω d ψ ω d θ ω .
f ( r ) = f ( r , ψ r , θ r ) = l = 0 k = l l 1 4 π ( i ) l { 2 π 0 F l k ( ρ ) j l ( ρ r ) ρ 2 d ρ } Y l k ( ψ r , θ r ) ,
F l k ( ρ ) = 4 π ( i ) l F ̑ l k ( ρ ) = 4 π ( i ) l S l { f l k ( r ) } ,
f ( r ) = δ ( r r 0 ) = 1 r 2   sin   ψ r δ ( r r 0 ) δ ( ψ r ψ r 0 ) δ ( θ r θ r 0 ) .
f l k ( r ) = 1 r 2 δ ( r r 0 ) Y l k ( ψ r 0 , θ r 0 ) ¯ ,
F ( ω ) = F ( ρ , ψ ω , θ ω ) = l = 0 k = l l 4 π ( i ) l j l ( ρ r 0 ) Y l k ( ψ r 0 , θ r 0 ) ¯ Y l k ( ψ ω , θ ω ) = e i ω r 0 .
F l k ( ρ ) = 4 π ( i ) l j l ( ρ r 0 ) Y l k ( ψ r 0 , θ r 0 ) ¯ .
f l k ( r ) = 4 π ( i ) l j l ( ρ 0 r ) Y l k ( ψ ω 0 , θ ω 0 ) ¯ .
F ( ω ) = l = 0 k = l l ( 4 π ) 2 π 2 ρ 2 δ ( ρ ρ 0 ) Y l k ( ψ ω 0 , θ ω 0 ) ¯ Y l k ( ψ ω , θ ω ) .
F l k ( ρ ) = ( 2 π ) 3 1 ρ 2 δ ( ρ ρ 0 ) Y l k ( ψ ω 0 , θ ω 0 ) ¯ .
H ( ω ) = f ( r ) g ( r ) e i ω r d r = 0 0 2 π 0 π L f l k ( r ) Y l k ( ψ r , θ r ) L f l k ( r ) Y l k ( ψ r , θ r ) L 4 π ( i ) l j l ( ρ r ) Y l k ( ψ r , θ r ) ¯ Y l k ( ψ ω , θ ω ) sin   ψ r d ψ r d θ r r 2 d r ,
0 2 π 0 π Y l k ( ψ r , θ r ) Y l k ( ψ r , θ r ) Y l k ( ψ r , θ r ) ¯ sin   ψ r d ψ r d θ r .
c l ( l , k , l , k ) = 0 2 π 0 π Y l k ( ψ r , θ r ) ¯ Y l k ( ψ r , θ r ) Y l k k ( ψ r , θ r ) sin   ψ r d ψ r d θ r ,
H ( ω ) = l = 0 k = l l 4 π ( i ) l { 0 l = 0 k = l l f l k ( r ) l = | l l | l + l c l ( l , k , l , k ) g l k k ( r ) j l ( ρ r ) r 2 d r } Y l k ( ψ ω , θ ω ) .
h l k ( r ) = l = 0 k = l l f l k ( r ) l = | l l | l + l c l ( l , k , l , k ) g l k k ( r ) .
f l k ( r ) g l k ( r ) l = 0 k = l l f l k ( r ) l = | l l | l + l c l ( l , k , l , k ) g l k k ( r ) ,
( f g ) l k = f l k g l k .
f ( r r 0 ) = F 1 { e i ω r 0 F ( ω ) } .
f ( r r 0 ) = 1 ( 2 π ) 3 0 0 2 π 0 π l = 0 k = l l 4 π ( i ) l { 0 f l k ( u ) j l ( ρ u ) u 2 d u } Y l k ( ψ ω , θ ω ) l = 0 k = l l 4 π ( i ) l j l ( ρ r 0 ) Y l k ( ψ r 0 , θ r 0 ) ¯ Y l k ( ψ ω , θ ω ) l = 0 k = l l 4 π ( i ) l j l ( ρ r ) Y l k ( ψ ω , θ ω ) ¯ Y l k ( ψ r , θ r ) ρ 2   sin   ψ ω d ψ ω d θ ω d ρ .
S l l , l ( u , r , r 0 ) = 0 j l ( ρ u ) j l ( ρ r 0 ) j l ( ρ r ) ρ 2 d ρ .
f ( r r 0 ) = l = 0 k = l l 8 ( i ) l Y l k ( ψ r , θ r ) l = 0 k = l l ( i ) l Y l k ( ψ r 0 , θ r 0 ) ¯ l = | l l | l + l ( i ) l c l ( l , k , l , k ) 0 f l k k ( u ) S l l , l ( u , r , r 0 ) u 2 d u .
[ f ( r r 0 ) ] l k = l = 0 k = l l 8 ( i ) l l Y l k ( ψ r 0 , θ r 0 ) ¯ l = | l l | l + l ( i ) l c l ( l , k , l , k ) 0 f l k k ( u ) S l l , l ( u , r , r 0 ) u 2 d u .
H l k ( ρ ) = [ 4 π ( i ) l j l ( ρ r 0 ) Y l k ( ψ r 0 , θ r 0 ) ¯ ] F l k ( ρ ) ,
H l k ( ρ ) = l = 0 k = l l 4 π ( i ) l j l ( ρ r 0 ) Y l k ( ψ r 0 , θ r 0 ) ¯ l = | l l | l + l c l ( l , k , l , k ) F l k k ( ρ ) ,
h ( r ) = f ( r ) g ( r ) = g ( r 0 ) f ( r r 0 ) d r 0 .
h ( r ) = 0 0 2 π 0 π l = 0 k = l l g l k ( r 0 ) Y l k ( ψ r 0 , θ r 0 ) l = 0 k = l l 8 ( i ) l Y l k ( ψ r , θ r ) l = 0 k = l l ( i ) l Y l k ( ψ r 0 , θ r 0 ) ¯ l = | l l | l + l ( i ) l c l ( l , k , l , k ) 0 f l k k ( u ) S l l , l ( u , r , r 0 ) u 2 d u   sin   ψ r 0 d ψ r 0 d θ r 0 r 0 2 d r 0 .
h ( r ) = l = 0 k = l l ( i ) l 4 π { 2 π 0 H l k ( ρ ) j l ( ρ r ) ρ 2 d ρ } Y l k ( ψ r , θ r ) = l = 0 k = l l h l k ( r ) Y l k ( ψ r , θ r ) ,
H l k ( ρ ) = l = 0 k = l l G l k ( ρ ) l = | l l | l + l c l ( l , k , l , k ) F l k k ( ρ ) = G l k ( ρ ) F l k ( ρ ) .
h ( r ) = f ( r ) g ( r ) = g ( r 0 ) f ( r r 0 ) d r 0 ,
0 g ( r 0 ) f ( r r 0 ) d r 0 ,
h ( r ) = f ( r ) g ( r ) = g ( r 0 ) 8 1 4 π l = 0 k = l l ( i ) l Y l k ( ψ r 0 , θ r 0 ) ¯ l = | l | l ( i ) l c l ( 0 , 0 , l , k ) 0 f l k ( u ) S l 0 , l ( u , r , r 0 ) u 2 d u d r 0 .
h ( r ) = 0 0 2 π 0 π g ( r 0 ) 8 1 4 π 1 4 π c 0 ( 0 , 0 , 0 , 0 ) 0 f 0 0 ( u ) S 0 0 , 0 ( u , r , r 0 ) u 2 d u   sin   ψ r 0 d ψ r 0 d θ r 0 r 0 2 d r 0 .
h ( r ) = f ( r ) g ( r ) = 0 g ( r 0 ) 8 0 f ( u ) S 0 0 , 0 ( u , r , r 0 ) u 2 d u r 0 2 d r 0 ,
f ( r ) g ( r ) = f ( r ) g ( r ) = 0 g ( r 0 ) Φ 3 D ( r r 0 ) r 0 d r 0 ,
Φ 3 D ( r r 0 ) = 0 f ( u ) 8 S 0 0 , 0 ( u , r , r 0 ) u 2 d u = 0 2 π 0 π f ( r r 0 ) sin   ψ r 0 d ψ r 0 d θ r 0 .
h ( r ) = f ( r ) g ( r ) H ( ρ ) = F ( ρ ) G ( ρ ) .
Y l m ( R α , β , γ ( ψ , θ ) ) = m = l l D m m l ( α , β , γ ) Y l m ( ψ , θ ) .
D m m l ( α , β , γ ) = d m m l ( α ) e i m β e i m γ ,
d m m l ( α ) = 0 2 π 0 π Y l m ( R y ( α ) ( ψ , θ ) ) Y l m ( ψ , θ ) ¯ sin ( ψ ) d ψ d θ .
R α , β = R α , β , 0 = R z ( β ) R y ( α ) ,
D m m l ( α , β ) = D m m l ( α , β , 0 ) = d m m l ( α ) e i m β .
β = 0 2 π α = 0 π ( D m n l ( α , β ) ) ¯ ( D m n l ( α , β ) ) sin   α d α d β = 4 π 2 l + 1 δ l l δ m m .
α = 0 π ( d m n l ( α ) ) ¯ ( d m n l ( α ) ) sin   α d α = 2 2 l + 1 δ l l .
h ( r ) = f ( r ) ( ψ , θ ) g ( r ) = 1 ( 2 π ) 2 0 2 π 0 π f ( r , ψ 0 , θ 0 ) g ( r , ψ ψ 0 , θ θ 0 ) sin   ψ 0 d ψ 0 d θ 0 .
g ( r , ψ ψ 0 , θ θ 0 ) = l = 0 k = l l g l k ( r ) Y l k ( ψ ψ 0 , θ θ 0 ) ,
g ( r , ψ ψ 0 , θ θ 0 ) = l = 0 k = l l γ l k ( r ) Y l k ( ψ , θ ) .
γ l k ( r ) = 0 2 π 0 π { l = 0 k = l l g l k ( r ) Y l k ( ψ ψ 0 , θ θ 0 ) } Y l k ( ψ , θ ) ¯ sin   ψ d ψ d θ = l = 0 k = l l g l k ( r ) 0 2 π 0 π Y l k ( ψ ψ 0 , θ θ 0 ) Y l k ( ψ , θ ) ¯ sin   ψ d ψ d θ .
0 2 π 0 π Y l k ( ψ ψ 0 , θ θ 0 ) Y l k ( ψ , θ ) ¯ sin   ψ d ψ d θ = 0 2 π 0 π k = l l D k k l ( ψ 0 , θ 0 ) Y l k ( ψ , θ ) Y l k ( ψ , θ ) ¯ sin   ψ d ψ d θ .
0 2 π 0 π Y l k ( ψ ψ 0 , θ θ 0 ) Y l k ( ψ , θ ) ¯ sin   ψ d ψ d θ = δ l l D k k l ( ψ 0 , θ 0 ) .
γ l k ( r ) = l = 0 k = l l g l k ( r ) δ l l D k k l ( ψ 0 , θ 0 ) = k = l l g l k ( r ) D k k l ( ψ 0 , θ 0 ) ,
g ( r , ψ ψ 0 , θ θ 0 ) = l = 0 k = l l { k = l l g l k ( r ) D k k l ( ψ 0 , θ 0 ) } Y l k ( ψ , θ ) .
f ( r ) ( ψ , θ ) g ( r ) = 1 ( 2 π ) 2 0 2 π 0 π f ( r , ψ 0 , θ 0 ) g ( r , ψ ψ 0 , θ θ 0 ) sin   ψ 0 d ψ 0 d θ 0 = 1 ( 2 π ) 2 0 2 π 0 π l = 0 k = l l f l k ( r ) Y l k ( ψ 0 , θ 0 ) [ l = 0 k = l l k = l l g l k ( r ) D k k l ( ψ 0 , θ 0 ) Y l k ( ψ , θ ) ] sin   ψ 0 d ψ 0 d θ 0 .
0 2 π 0 π Y l k ( ψ 0 , θ 0 ) D k k l ( ψ 0 , θ 0 ) sin   ψ 0 d ψ 0 d θ 0 = 0 2 π 0 π Y l k ( ψ 0 , θ 0 ) 0 2 π 0 π Y l k ( ψ ψ 0 , θ θ 0 ) Y l k ( ψ , θ ) ¯ sin   ψ d ψ d θ   sin   ψ 0 d ψ 0 d θ 0 .
exp ( i k θ 0 ) exp ( i k ( θ θ 0 ) ) exp ( i k θ ) = exp ( i ( k k ) θ 0 ) exp ( i ( k k ) θ ) .
0 2 π 0 2 π exp ( i ( k k ) θ 0 ) exp ( i ( k k ) θ ) d θ 0 d θ = ( 2 π ) 2 δ k k δ k k .
f ( r ) ( ψ , θ ) g ( r ) = 1 ( 2 π ) 2 l = 0 k = l l f l k ( r ) l = 0 k = l l k = l l g l k ( r ) ( 2 π ) 2 δ l l δ k k δ k k Y l k ( ψ , θ ) = l = 0 k = l l f l k ( r ) g l k ( r ) Y l k ( ψ , θ ) .
h ( r ) = f ( r ) ( ψ , θ ) g ( r ) = l = 0 k = l l h l k ( r ) Y l k ( ψ , θ ) = l = 0 k = l l f l k ( r ) g l k ( r ) Y l k ( ψ , θ ) ,
h l k ( r ) = f l k ( r ) g l k ( r ) .
Λ ( R ) f ( Ω ) = f ( R 1 Ω ) .
( f R g ) ( r , Ω ) = f ( r , R η ) Λ ( R ) d R g ( r , Ω ) = f ( r , R η ) g ( r , R 1 Ω ) d R ,
f ( r , R η ) = f ( r , α , β ) = l = 0 k = l l f l k ( r ) Y l k ( α , β ) .
g ( r , R 1 Ω ) = g ( r , R α , β , γ 1 ( ψ , θ ) ) = l = 0 k = l l k = l l g l k ( r ) D k k l ( α , β , γ ) Y l k ( ψ , θ ) = l = 0 k = l l k = l l g l k ( r ) e i k γ D k k l ( α , β ) Y l k ( ψ , θ ) .
( f R g ) ( r , Ω ) = f ( r , R η ) g ( r , R 1 Ω ) d R = 0 2 π 0 π 0 2 π l = 0 k = l l f l k ( r ) Y l k ( α , β ) l = 0 k = l l k = l l g l k ( r ) e i k γ D k k l ( α , β ) Y l k ( ψ , θ ) d γ   sin   α d α d β .
D 0 k l ( α , β ) = 4 π 2 l + 1 Y l k ( α , π ) ¯ e i k β .
D 0 k l ( α , β ) = 4 π 2 l + 1 Y l k ( α , β ) ¯ e i k π = ( 1 ) k 4 π 2 l + 1 Y l k ( α , β ) ¯ .
( f R g ) ( r , Ω ) = 0 2 π 0 π l = 0 k = l l f l k ( r ) Y l k ( α , β ) l = 0 k = l l g l 0 ( r ) ( 1 ) k 4 π 2 l + 1 Y l k ( α , β ) ¯ sin   α d α d β Y l k ( ψ , θ ) .
( f R g ) ( r , Ω ) = l = 0 k = l l f l k ( r ) g l 0 ( r ) ( 1 ) k 4 π 2 l + 1 Y l k ( ψ , θ ) .
( f R g ) l k = f l k ( r ) g l 0 ( r ) ( 1 ) k 4 π 2 l + 1 ,
( f ( ψ , θ ) g ) l k = f l k ( r ) g l k ( r ) .
f ( r ) ¯ = l = 0 k = l l f l k ( r ) ¯ ( 1 ) k Y l k ( ψ , θ ) = l = 0 k = l l f l k ( r ) ¯ ( 1 ) k Y l k ( ψ , θ ) .
c l ( 0 , 0 , l , k ) = 1 4 π 0 2 π 0 π Y l k ( ψ r , θ r ) Y l k ( ψ r , θ r ) sin   ψ r d ψ r d θ r = 1 4 π ( 1 ) k δ l l .
f g r = 0 = g ( r 0 ) f ( r 0 ) d r 0 = 8 4 π l = 0 k = l l ( 1 ) l ( 1 ) k 1 4 π 0 0 g l k ( r 0 ) j l ( ρ r 0 ) r 0 2 d r 0 0 f l k ( u ) j l ( ρ u ) u 2 d u ρ 2 d ρ .
g ( r ) f ( r ) ¯ d r = 1 ( 2 π ) 3 l = 0 k = l l 0 G l k ( ρ ) F l k ( ρ ) ¯ ρ 2 d ρ .
f ( r ) g ( r ) = l = 0 k = l l { l = 0 k = l l f l k ( r ) l = | l l | l + l c l ( l , k , l , k ) g l k k ( r ) } Y l k ( ψ r , θ r ) .
g ( r ) f ( r ) ¯ d r = l = 0 k = l l 0 g l k ( r ) f l k ( r ) ¯ r 2 d r .
l = 0 k = l l 0 g l k ( r ) f l k ( r ) ¯ r 2 d r = 1 ( 2 π ) 3 l = 0 k = l l 0 G l k ( ρ ) F l k ( ρ ) ¯ ρ 2 d ρ .
l = 0 k = l l 0 | f l k ( r ) | 2 r 2 d r = 1 ( 2 π ) 3 l = 0 k = l l 0 | F l k ( ρ ) | 2 ρ 2 d ρ .

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