Abstract

The fractional Fourier transform (FRFT) for quasi-periodic Bloch functions is studied. An isomorphism between square-integrable functions on the real line and quasi-periodic Bloch functions is used to extend existing work on the fractional Fourier transform for the former functions to the latter. The properties of the FRFT for quasi-periodic Bloch functions are discussed, and various numerical examples are presented.

© 2001 Optical Society of America

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References

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  1. V. Namias, “The fractional order Fourier transform and its application to quantum mechanics,” J. Inst. Math. Appl. 25, 241–265 (1980).
    [CrossRef]
  2. A. C. McBride, F. H. Kerr, “On Namias’s fractional Fourier transforms,” IMA J. Appl. Math. 39, 159–175 (1987).
    [CrossRef]
  3. D. Mustard, “The fractional Fourier transform and the Wigner distribution,” J. Aust. Math. Soc. B Appl. Math. 38, 209–219 (1996).
    [CrossRef]
  4. H. M. Ozaktas, D. Mendlovic, “Fractional Fourier transformations and their optical implementation. II,” J. Opt. Soc. Am. A 10, 2522–2531 (1993).
    [CrossRef]
  5. H. M. Ozaktas, B. Barshan, D. Mendlovic, L. Onural, “Convolution, filtering, and multiplexing in fractional domains and their relation to chirp and wavelet transforms,” J. Opt. Soc. Am. A 11, 547–559 (1994).
    [CrossRef]
  6. A. W. Lohmann, B. H. Soffer, “Relationships between the Radon–Wigner and fractional Fourier transforms,” J. Opt. Soc. Am. A 11, 1798–1801 (1994).
    [CrossRef]
  7. H. M. Ozakta, M. F. Erden, “Relationships among ray optical, Gaussian bearn, and fractional Fourier transform descriptions of first-order optical systems,” Opt. Commun. 143, 75–86 (1997).
    [CrossRef]
  8. D. Mendlovic, H. M. Ozaktas, A. W. Lohmann, “Fractional correlation,” Appl. Opt. 34, 303–309 (1995).
    [CrossRef] [PubMed]
  9. D. Dragoman, “Fractional Wigner distribution function,” J. Opt. Soc. Am. A 13, 474–478 (1996).
    [CrossRef]
  10. H. M. Ozaktas, M. A. Kutay, D. Mendlovic, “Introduction to the fractional Fourier transform and its applications,” Adv. Imaging Electron. Phys. 106, 239–291 (1999).
    [CrossRef]
  11. S. Chountasis, A. Vourdas, C. Bendjaballah, “Fractional Fourier operators and generalized Wigner functions,” Phys. Rev. A 60, 3467–3473 (1999).
    [CrossRef]
  12. L. B. Almeida, “The fractional Fourier transform and time-frequency representations,” IEEE Trans. Signal Process. 42, 3084–3091 (1994).
    [CrossRef]
  13. S. C. Pei, M. H. Yeh, C. C. Tseng, “Discrete fractional Fourier transform based on orthogonal projections,” IEEE Trans. Signal Process. 47, 1335–1348 (1999).
    [CrossRef]
  14. M. A. Man’ko, “Fractional Fourier transform in information processing, tomography of optical signal, and Green function of harmonic oscillator,” J. Russ. Laser Res. 20, 226–238 (1999).
    [CrossRef]
  15. M. A. Man’ko, “Quasidistributions tomography, and fractional Fourier transform in signal analysis,” J. Russ. Laser Res. 21, 411–437 (2000).
    [CrossRef]
  16. G. Cariolaro, T. Erseghe, P. Kraniauskas, “Multiplicity of fractional Fourier transforms and their relationships,” IEEE Trans. Signal Process. 48, 227–241 (2000).
    [CrossRef]
  17. L. S. Schulman, Techniques and Applications of Path Integration (Wiley, New York, 1981).
  18. M. Reed, B. Simon, Analysis of Operators, Vol. IV of Methods of Modern Mathematical Physics (Academic, London, 1978).
  19. J. Zak, “Finite translations in solid-state physics,” Phys. Rev. Lett. 19, 1385–1387 (1967).
    [CrossRef]
  20. J. Zak, “Dynamics of electronics in solids in external fields,” Phys. Rev. 168, 686–695 (1968).
    [CrossRef]
  21. M. Boon, J. Zak, “Amplitudes on von Neumann lattices,” J. Math. Phys. 22, 1090–1099 (1981).
    [CrossRef]
  22. M. Boon, J. Zak, I. J. Zucker, “Rational von Neumann lattices,” J. Math. Phys. 24, 316–323 (1983).
    [CrossRef]
  23. A. J. E. M. Janssen, “Bargmann transform, Zak transform, and coherent states,” J. Math. Phys. 23, 720–731 (1982).
    [CrossRef]
  24. A. J. E. M. Janssen, “The Zak transform: a signal transform for sampled time-continuous signals,” Philips J. Res. 43, 23–69 (1988).
  25. I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals Series and Products (Academic, London, 2000).
  26. D. Mumford, Tata Lectures on Theta (Birkhauser, Boston, Mass., 1983), Vols. 1 and 2.
  27. N. J. Vilenkin, A. V. Klimyk, Representation of Lie Groups and Special Functions (Kluwer Academic, Dordrecht, The Netherlands, 1972), Vol. 3.
  28. A. M. Perelomov, Generalized Coherent States and Their Applications (Springer-Verlag, Berlin, 1986).
  29. Y. S. Kim, M. E. Noz, Phase Space Picture of Quantum Mechanics (World Scientific, Singapore, 1991).
  30. C. Bendjaballah, Introduction to Photon Communication (Springer-Verlag, Heidelberg, 1995).

2000 (2)

M. A. Man’ko, “Quasidistributions tomography, and fractional Fourier transform in signal analysis,” J. Russ. Laser Res. 21, 411–437 (2000).
[CrossRef]

G. Cariolaro, T. Erseghe, P. Kraniauskas, “Multiplicity of fractional Fourier transforms and their relationships,” IEEE Trans. Signal Process. 48, 227–241 (2000).
[CrossRef]

1999 (4)

H. M. Ozaktas, M. A. Kutay, D. Mendlovic, “Introduction to the fractional Fourier transform and its applications,” Adv. Imaging Electron. Phys. 106, 239–291 (1999).
[CrossRef]

S. Chountasis, A. Vourdas, C. Bendjaballah, “Fractional Fourier operators and generalized Wigner functions,” Phys. Rev. A 60, 3467–3473 (1999).
[CrossRef]

S. C. Pei, M. H. Yeh, C. C. Tseng, “Discrete fractional Fourier transform based on orthogonal projections,” IEEE Trans. Signal Process. 47, 1335–1348 (1999).
[CrossRef]

M. A. Man’ko, “Fractional Fourier transform in information processing, tomography of optical signal, and Green function of harmonic oscillator,” J. Russ. Laser Res. 20, 226–238 (1999).
[CrossRef]

1997 (1)

H. M. Ozakta, M. F. Erden, “Relationships among ray optical, Gaussian bearn, and fractional Fourier transform descriptions of first-order optical systems,” Opt. Commun. 143, 75–86 (1997).
[CrossRef]

1996 (2)

D. Mustard, “The fractional Fourier transform and the Wigner distribution,” J. Aust. Math. Soc. B Appl. Math. 38, 209–219 (1996).
[CrossRef]

D. Dragoman, “Fractional Wigner distribution function,” J. Opt. Soc. Am. A 13, 474–478 (1996).
[CrossRef]

1995 (1)

1994 (3)

1993 (1)

1988 (1)

A. J. E. M. Janssen, “The Zak transform: a signal transform for sampled time-continuous signals,” Philips J. Res. 43, 23–69 (1988).

1987 (1)

A. C. McBride, F. H. Kerr, “On Namias’s fractional Fourier transforms,” IMA J. Appl. Math. 39, 159–175 (1987).
[CrossRef]

1983 (1)

M. Boon, J. Zak, I. J. Zucker, “Rational von Neumann lattices,” J. Math. Phys. 24, 316–323 (1983).
[CrossRef]

1982 (1)

A. J. E. M. Janssen, “Bargmann transform, Zak transform, and coherent states,” J. Math. Phys. 23, 720–731 (1982).
[CrossRef]

1981 (1)

M. Boon, J. Zak, “Amplitudes on von Neumann lattices,” J. Math. Phys. 22, 1090–1099 (1981).
[CrossRef]

1980 (1)

V. Namias, “The fractional order Fourier transform and its application to quantum mechanics,” J. Inst. Math. Appl. 25, 241–265 (1980).
[CrossRef]

1968 (1)

J. Zak, “Dynamics of electronics in solids in external fields,” Phys. Rev. 168, 686–695 (1968).
[CrossRef]

1967 (1)

J. Zak, “Finite translations in solid-state physics,” Phys. Rev. Lett. 19, 1385–1387 (1967).
[CrossRef]

Almeida, L. B.

L. B. Almeida, “The fractional Fourier transform and time-frequency representations,” IEEE Trans. Signal Process. 42, 3084–3091 (1994).
[CrossRef]

Barshan, B.

Bendjaballah, C.

S. Chountasis, A. Vourdas, C. Bendjaballah, “Fractional Fourier operators and generalized Wigner functions,” Phys. Rev. A 60, 3467–3473 (1999).
[CrossRef]

C. Bendjaballah, Introduction to Photon Communication (Springer-Verlag, Heidelberg, 1995).

Boon, M.

M. Boon, J. Zak, I. J. Zucker, “Rational von Neumann lattices,” J. Math. Phys. 24, 316–323 (1983).
[CrossRef]

M. Boon, J. Zak, “Amplitudes on von Neumann lattices,” J. Math. Phys. 22, 1090–1099 (1981).
[CrossRef]

Cariolaro, G.

G. Cariolaro, T. Erseghe, P. Kraniauskas, “Multiplicity of fractional Fourier transforms and their relationships,” IEEE Trans. Signal Process. 48, 227–241 (2000).
[CrossRef]

Chountasis, S.

S. Chountasis, A. Vourdas, C. Bendjaballah, “Fractional Fourier operators and generalized Wigner functions,” Phys. Rev. A 60, 3467–3473 (1999).
[CrossRef]

Dragoman, D.

Erden, M. F.

H. M. Ozakta, M. F. Erden, “Relationships among ray optical, Gaussian bearn, and fractional Fourier transform descriptions of first-order optical systems,” Opt. Commun. 143, 75–86 (1997).
[CrossRef]

Erseghe, T.

G. Cariolaro, T. Erseghe, P. Kraniauskas, “Multiplicity of fractional Fourier transforms and their relationships,” IEEE Trans. Signal Process. 48, 227–241 (2000).
[CrossRef]

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals Series and Products (Academic, London, 2000).

Janssen, A. J. E. M.

A. J. E. M. Janssen, “The Zak transform: a signal transform for sampled time-continuous signals,” Philips J. Res. 43, 23–69 (1988).

A. J. E. M. Janssen, “Bargmann transform, Zak transform, and coherent states,” J. Math. Phys. 23, 720–731 (1982).
[CrossRef]

Kerr, F. H.

A. C. McBride, F. H. Kerr, “On Namias’s fractional Fourier transforms,” IMA J. Appl. Math. 39, 159–175 (1987).
[CrossRef]

Kim, Y. S.

Y. S. Kim, M. E. Noz, Phase Space Picture of Quantum Mechanics (World Scientific, Singapore, 1991).

Klimyk, A. V.

N. J. Vilenkin, A. V. Klimyk, Representation of Lie Groups and Special Functions (Kluwer Academic, Dordrecht, The Netherlands, 1972), Vol. 3.

Kraniauskas, P.

G. Cariolaro, T. Erseghe, P. Kraniauskas, “Multiplicity of fractional Fourier transforms and their relationships,” IEEE Trans. Signal Process. 48, 227–241 (2000).
[CrossRef]

Kutay, M. A.

H. M. Ozaktas, M. A. Kutay, D. Mendlovic, “Introduction to the fractional Fourier transform and its applications,” Adv. Imaging Electron. Phys. 106, 239–291 (1999).
[CrossRef]

Lohmann, A. W.

Man’ko, M. A.

M. A. Man’ko, “Quasidistributions tomography, and fractional Fourier transform in signal analysis,” J. Russ. Laser Res. 21, 411–437 (2000).
[CrossRef]

M. A. Man’ko, “Fractional Fourier transform in information processing, tomography of optical signal, and Green function of harmonic oscillator,” J. Russ. Laser Res. 20, 226–238 (1999).
[CrossRef]

McBride, A. C.

A. C. McBride, F. H. Kerr, “On Namias’s fractional Fourier transforms,” IMA J. Appl. Math. 39, 159–175 (1987).
[CrossRef]

Mendlovic, D.

Mumford, D.

D. Mumford, Tata Lectures on Theta (Birkhauser, Boston, Mass., 1983), Vols. 1 and 2.

Mustard, D.

D. Mustard, “The fractional Fourier transform and the Wigner distribution,” J. Aust. Math. Soc. B Appl. Math. 38, 209–219 (1996).
[CrossRef]

Namias, V.

V. Namias, “The fractional order Fourier transform and its application to quantum mechanics,” J. Inst. Math. Appl. 25, 241–265 (1980).
[CrossRef]

Noz, M. E.

Y. S. Kim, M. E. Noz, Phase Space Picture of Quantum Mechanics (World Scientific, Singapore, 1991).

Onural, L.

Ozakta, H. M.

H. M. Ozakta, M. F. Erden, “Relationships among ray optical, Gaussian bearn, and fractional Fourier transform descriptions of first-order optical systems,” Opt. Commun. 143, 75–86 (1997).
[CrossRef]

Ozaktas, H. M.

Pei, S. C.

S. C. Pei, M. H. Yeh, C. C. Tseng, “Discrete fractional Fourier transform based on orthogonal projections,” IEEE Trans. Signal Process. 47, 1335–1348 (1999).
[CrossRef]

Perelomov, A. M.

A. M. Perelomov, Generalized Coherent States and Their Applications (Springer-Verlag, Berlin, 1986).

Reed, M.

M. Reed, B. Simon, Analysis of Operators, Vol. IV of Methods of Modern Mathematical Physics (Academic, London, 1978).

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals Series and Products (Academic, London, 2000).

Schulman, L. S.

L. S. Schulman, Techniques and Applications of Path Integration (Wiley, New York, 1981).

Simon, B.

M. Reed, B. Simon, Analysis of Operators, Vol. IV of Methods of Modern Mathematical Physics (Academic, London, 1978).

Soffer, B. H.

Tseng, C. C.

S. C. Pei, M. H. Yeh, C. C. Tseng, “Discrete fractional Fourier transform based on orthogonal projections,” IEEE Trans. Signal Process. 47, 1335–1348 (1999).
[CrossRef]

Vilenkin, N. J.

N. J. Vilenkin, A. V. Klimyk, Representation of Lie Groups and Special Functions (Kluwer Academic, Dordrecht, The Netherlands, 1972), Vol. 3.

Vourdas, A.

S. Chountasis, A. Vourdas, C. Bendjaballah, “Fractional Fourier operators and generalized Wigner functions,” Phys. Rev. A 60, 3467–3473 (1999).
[CrossRef]

Yeh, M. H.

S. C. Pei, M. H. Yeh, C. C. Tseng, “Discrete fractional Fourier transform based on orthogonal projections,” IEEE Trans. Signal Process. 47, 1335–1348 (1999).
[CrossRef]

Zak, J.

M. Boon, J. Zak, I. J. Zucker, “Rational von Neumann lattices,” J. Math. Phys. 24, 316–323 (1983).
[CrossRef]

M. Boon, J. Zak, “Amplitudes on von Neumann lattices,” J. Math. Phys. 22, 1090–1099 (1981).
[CrossRef]

J. Zak, “Dynamics of electronics in solids in external fields,” Phys. Rev. 168, 686–695 (1968).
[CrossRef]

J. Zak, “Finite translations in solid-state physics,” Phys. Rev. Lett. 19, 1385–1387 (1967).
[CrossRef]

Zucker, I. J.

M. Boon, J. Zak, I. J. Zucker, “Rational von Neumann lattices,” J. Math. Phys. 24, 316–323 (1983).
[CrossRef]

Adv. Imaging Electron. Phys. (1)

H. M. Ozaktas, M. A. Kutay, D. Mendlovic, “Introduction to the fractional Fourier transform and its applications,” Adv. Imaging Electron. Phys. 106, 239–291 (1999).
[CrossRef]

Appl. Opt. (1)

IEEE Trans. Signal Process (2)

S. C. Pei, M. H. Yeh, C. C. Tseng, “Discrete fractional Fourier transform based on orthogonal projections,” IEEE Trans. Signal Process. 47, 1335–1348 (1999).
[CrossRef]

G. Cariolaro, T. Erseghe, P. Kraniauskas, “Multiplicity of fractional Fourier transforms and their relationships,” IEEE Trans. Signal Process. 48, 227–241 (2000).
[CrossRef]

IEEE Trans. Signal Process. (1)

L. B. Almeida, “The fractional Fourier transform and time-frequency representations,” IEEE Trans. Signal Process. 42, 3084–3091 (1994).
[CrossRef]

IMA J. Appl. Math. (1)

A. C. McBride, F. H. Kerr, “On Namias’s fractional Fourier transforms,” IMA J. Appl. Math. 39, 159–175 (1987).
[CrossRef]

J. Aust. Math. Soc. B Appl. Math. (1)

D. Mustard, “The fractional Fourier transform and the Wigner distribution,” J. Aust. Math. Soc. B Appl. Math. 38, 209–219 (1996).
[CrossRef]

J. Inst. Math. Appl. (1)

V. Namias, “The fractional order Fourier transform and its application to quantum mechanics,” J. Inst. Math. Appl. 25, 241–265 (1980).
[CrossRef]

J. Math. Phys. (3)

M. Boon, J. Zak, “Amplitudes on von Neumann lattices,” J. Math. Phys. 22, 1090–1099 (1981).
[CrossRef]

M. Boon, J. Zak, I. J. Zucker, “Rational von Neumann lattices,” J. Math. Phys. 24, 316–323 (1983).
[CrossRef]

A. J. E. M. Janssen, “Bargmann transform, Zak transform, and coherent states,” J. Math. Phys. 23, 720–731 (1982).
[CrossRef]

J. Opt. Soc. Am. A (4)

J. Russ. Laser Res. (2)

M. A. Man’ko, “Fractional Fourier transform in information processing, tomography of optical signal, and Green function of harmonic oscillator,” J. Russ. Laser Res. 20, 226–238 (1999).
[CrossRef]

M. A. Man’ko, “Quasidistributions tomography, and fractional Fourier transform in signal analysis,” J. Russ. Laser Res. 21, 411–437 (2000).
[CrossRef]

Opt. Commun. (1)

H. M. Ozakta, M. F. Erden, “Relationships among ray optical, Gaussian bearn, and fractional Fourier transform descriptions of first-order optical systems,” Opt. Commun. 143, 75–86 (1997).
[CrossRef]

Philips J. Res. (1)

A. J. E. M. Janssen, “The Zak transform: a signal transform for sampled time-continuous signals,” Philips J. Res. 43, 23–69 (1988).

Phys. Rev. (1)

J. Zak, “Dynamics of electronics in solids in external fields,” Phys. Rev. 168, 686–695 (1968).
[CrossRef]

Phys. Rev. A (1)

S. Chountasis, A. Vourdas, C. Bendjaballah, “Fractional Fourier operators and generalized Wigner functions,” Phys. Rev. A 60, 3467–3473 (1999).
[CrossRef]

Phys. Rev. Lett. (1)

J. Zak, “Finite translations in solid-state physics,” Phys. Rev. Lett. 19, 1385–1387 (1967).
[CrossRef]

Other (8)

L. S. Schulman, Techniques and Applications of Path Integration (Wiley, New York, 1981).

M. Reed, B. Simon, Analysis of Operators, Vol. IV of Methods of Modern Mathematical Physics (Academic, London, 1978).

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals Series and Products (Academic, London, 2000).

D. Mumford, Tata Lectures on Theta (Birkhauser, Boston, Mass., 1983), Vols. 1 and 2.

N. J. Vilenkin, A. V. Klimyk, Representation of Lie Groups and Special Functions (Kluwer Academic, Dordrecht, The Netherlands, 1972), Vol. 3.

A. M. Perelomov, Generalized Coherent States and Their Applications (Springer-Verlag, Berlin, 1986).

Y. S. Kim, M. E. Noz, Phase Space Picture of Quantum Mechanics (World Scientific, Singapore, 1991).

C. Bendjaballah, Introduction to Photon Communication (Springer-Verlag, Heidelberg, 1995).

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Figures (4)

Fig. 1
Fig. 1

Real and imaginary parts of function S(ϕ, θ; A=1).

Fig. 2
Fig. 2

Real and imaginary parts of function G2(ϕ, θ).

Fig. 3
Fig. 3

Real and imaginary parts of the FRFT of function S(ϕ, θ; A=1) for α=π/2.

Fig. 4
Fig. 4

Real and imaginary parts of the FRFT of function S(ϕ, θ; A=1) for α=π/6.

Equations (76)

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-+|f(x)|2dx<.
[-x2+V(x)]F(x)=EF(x),
F(x+L)=exp(iθ)F(x).
0L|F(x)|2dx<,
-+|f(x)|2dx=1.
F(ϕ+2π, θ)=F(ϕ, θ)exp(iθ),
F(ϕ, θ)=n=-f(x=ϕ+2πn)exp(-iθn),
f(x=ϕ+2πn)=12π 02πF(ϕ, θ)exp(iθn)dθ.
(F1, F2)=12π 02π02πF1(ϕ, θ)[F2(ϕ, θ)]*dϕdθ,
f˜(y)=-+ f(x)exp(-iyx)dx,
F(ϕ, θ)=12π k=-f˜y=k+θ2πexpik+θ2πϕ.
F(ϕ, θ)=k=-Fk(θ)expik+θ2πϕ,
Δ(ϕ-ϕ0, θ-θ0)=n=-δ(ϕ+2πn-ϕ0)×exp[-i(θ-θ0)n],
F(ϕ0, θ0)=12π 02π02πF(ϕ, θ)Δ*×(ϕ-ϕ0, θ-θ0)dϕdθ.
s(x; A)=π-1/4 exp-x22+2Ax-AAR,
S(ϕ, θ; A)=π-1/4 exp-ϕ22+2Aϕ-AARΘ3×-θ2+iπ(ϕ-2A); i2π,
Θ3[u; τ]=n=- exp(iπτn2+i2nu).
umn=(2m-1) π2+(2n-1) π2τ,
S(ϕ0, θ0; Amn)=0,
Amn=2-1/2ϕ0-π(2n-1)+im-12+θ02π.
12π 02π02π S(ϕ, θ; Amn)Δ*(ϕ-ϕ0, θ-θ0)dϕdθ=0.
gm(x)=(π2mm!)-1/2 exp-x22Hm(x),
Gm(ϕ, θ)=n=- gm(ϕ+2πn)exp(-iθn).
F(ϕ, θ)=m=-cmGm(ϕ, θ),
cm=12π 02π02πF(ϕ, θ)Gm*(ϕ, θ)dϕdθ.
S(ϕ, θ; A)=exp-|A|22m=0Am(m!)-1/2Gm(ϕ, θ).
xϕ+i2πθ,-ix-iϕ.
a=2-1/2(x+ip)b=2-1/2(ϕ+i2πθ+ϕ),
a=2-1/2(x-ip)b=2-1/2(ϕ+i2πθ-ϕ),
bGm(ϕ, θ)=m1/2Gm-1(ϕ, θ),
bGm(ϕ, θ)=(m+1)1/2Gm+1(ϕ, θ);
bb=12[(ϕ+i2πθ)2-ϕ2-1],
bbGm(ϕ, θ)=mGm(ϕ, θ),
bS(ϕ, θ; A)=AS(ϕ, θ; A).
f˜α(y)=(Fαf)(y),Fαexp(iαaa),
f˜α(y)=-+kα(y, x)f(x)dx,
kα(y, x)=m=0 exp(imα)gm(y)gm*(x)
=1+i cot α2π1/2×exp-i x2+y22 tan α+i xysin α,
F˜α(ϕ, θ)=FαF(ϕ, θ),Fαexp(iαbb),
α/2π=p/q,
Fp/qGm+nq(ϕ, θ)=expi2πmpqGm+nq(ϕ, θ).
F˜αF˜β=F˜α+β.
F˜α(ϕ, θ)=12π 02π02πKα(ϕ, θ; ϕ, θ)×F(ϕ, θ)dθdϕ,
Kα(ϕ, θ; ϕ, θ)=m=0 exp(imα)Gm(ϕ, θ)Gm*(ϕ, θ).
Θ(ui; τij)=ni=- expiπniτijnj+2iniui,
Θ(ui+2πmi; τij)=Θ(ui; τij).
Kα(ϕ, θ; ϕ, θ)=1+i cot α2π1/2Θ(ui; τij)×exp-i ϕ2+ϕ22 tan α+i ϕϕsin α,
u1=-πϕ cot α+πϕsin α-θ2,
u2=-πϕ cot α+πϕsin α+θ2,
τ11=τ22=-2π cot α,τ12=τ21=2πsin α.
Kπ/2(ϕ, θ; ϕ, θ)=(2π)-1/2Θ(ui; τij)exp(iϕϕ),
u1=πϕ-θ2,u2=πϕ+θ/2,
τ11=τ22=0,τ12=τ21=2π.
12π n=-02π02πkα(y, ϕ+2πn)×exp(iθn)F(ϕ, θ)dθdϕ.
F˜α(ϕ, θ)=12π n,n=-02π02πkα(ϕ+2πn, ϕ+2πn)exp(iθn-iθn)F(ϕ, θ)dθdϕ=12π 02π02πKα(ϕ, θ; ϕ, θ)×F(ϕ, θ)dθdϕ.
s˜α(y; A)=π-1/4 exp-y22+2Ay exp(iα)-iA2 exp(iα)sin α-AAR.
S˜α(ϕ, θ; A)=π-1/4 exp-ϕ22+2A exp(iα)ϕ-AAR-iA2×exp(iα)sin αΘ3-θ2+iπ[ϕ-2A×exp(iα)]; i2π.
Fα[xf(x)]=cos αyf˜α(y)-iy sin αf˜α(y).
Fα[ϕF(ϕ, θ)]=cos αϕF˜α(ϕ, θ)-i(ϕ+i2πθ)sin αF˜α(ϕ, θ).
Fα[f(x+γ)]=f˜α(y+γ cos α)×exp-iγ24 sin(2α)-iyγ sin α.
Fα[F(ϕ+γ, θ)]=exp-iγ24 sin(2α)-iγϕ sin α×F˜α(ϕ+γ cos α, θ+2πγ sin α).
F(ϕ, θ; R)=R1/2n=-f(x=ϕ+2πnR)exp(-iθnR),
f(x=ϕ+2πnR)=R1/22π 02π/RF(ϕ, θ; R)exp(iθnR)dθ.
F(ϕ+2πR, θ)=F(ϕ, θ)exp(iRθ).
F˜(α, β; R)=R1/2n=-f˜(y=α+2πnR)exp(-iβnR).
F(ϕ, θ)=12π expiθϕ2πn=-f˜y=θ+2πn2π×exp(inϕ).
F(ϕ, θ)=12πR1/2 expiθϕ2πF˜θ2π,-2πϕ; R=12π.
Fα[f(cx)]=1+i cot αc2+i cot α1/2×expi y2 cot α2 cos2 αcos2 β-1f˜βy sin βc sin α,
Fα[F(cϕ, θ)]=R-1/21+i cot αc2+i cot α1/2×expi (ϕ+i2πθ)22 cot αcos2 αcos2 β-1×F˜βsin βc sin αϕ, c sin αsin βθ; R=sin βc sin α.
12π n,n=-02π02π kα(ϕ+2πn, ϕ+2πn)×exp[i(θn-θn)]ϕF(ϕ, θ)dθdϕ=n,n=-02π exp(-iθn)kα(ϕ+2πn,ϕ+2πn)ϕf(ϕ+2πn)dϕ=n=- exp(-iθn)-+kα(ϕ+2πn, x)xf(x)dx.
n=- exp(-iθn)[cos αϕf˜α(ϕ+2πn)-i(ϕ+2πn)sin αf˜α(ϕ+2πn)].
12π n,n=-02π02πkα(ϕ+2πn, ϕ+2πn)×exp[i(θn-θn)]F(ϕ+γ, θ)dθdϕ=n,n=-02π exp(-iθn)kα(ϕ+2πn, ϕ+2πn)×f(ϕ+γ+2πn)dϕ=n,n=-02πexp(-iθn)×expi 2γ(ϕ+γ+2πn)-γ22×cot α-iγ(ϕ+2πn)sin αkα(ϕ+2πn, ϕ+γ+2πn)f(ϕ+γ+2πn)d(ϕ+γ)=n=- exp(-iθn)exp-iγ22 cot α-iγ(ϕ+2πn)sin α×-+ exp(iγx cot α)kα(ϕ+2πn, x)f(x)dx.
Fα[exp(iγx)f(x)] =f˜α(y+γ sin α)expi γ24 sin(2α)+iyγ cos α.
n=- exp(-iθn)exp-i γ22 cot α-iγ(ϕ+2πn)sin α×expi γ24 cos2 α sin(2α)+i(ϕ+2πn)γ cot α cos α×f˜α(ϕ+2πn+γ cot α sin α).
12π n,n=-02π02πkα(ϕ+2πn, ϕ+2πn)×exp[i(θn-θn)]F(cϕ, θ)dθdϕ=n,n=-02π exp(-iθn)kα(ϕ+2πn, ϕ+2πn)×f(cϕ+2πcn)dϕ=n=- exp(-iθn)-+kα(ϕ+2πn, x)f(cx)dx.
n=- exp(-iθn)1+i cot αc2+i cot α1/2×expi (ϕ+2πn)22 cot αcos2 αcos2 β-1×f˜β(ϕ+2πn)sin βc sin α.

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