Abstract

The concept of self-Fourier functions, i.e., functions that equal their Fourier transform, is almost always associated with specific functions, the most well known being the Gaussian and the Dirac delta comb. We show that there exists an infinite number of distinct families of these functions, and we provide an algorithm for both generating and characterizing their distinct classes. This formalism allows us to show the existence of these families of functions without actually evaluating any Fourier or other transform-type integrals, a task often challenging and frequently not even possible.

© 2006 Optical Society of America

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References

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  1. M. Coffey, "Self-reciprocal Fourier functions," J. Opt. Soc. Am. A 11, 2453-2455 (1994).
    [CrossRef]
  2. A. Lohmann and D. Mendlovic, "Self-Fourier objects and other self-transform objects," J. Opt. Soc. Am. A 9, 2009-2012 (1992).
    [CrossRef]
  3. J. Glimm and A. Jaffe, Quantum Physics (Springer, 1981).
  4. S. Lipson, "Self-Fourier objects and other self-transform objects: comment," J. Opt. Soc. Am. A 10, 2088-2089 (1993).
    [CrossRef]
  5. E. Titchmarsh, Introduction to the Theory of Fourier Integrals (Clarendon, 1937).
  6. H. Edwards, Riemann's Zeta Function (Dover, 2001).
  7. K. Nishi, "Generalized comb function: a new self-Fourier function," in Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing (IEEE, 2004), Vol. 2, pp. 573-576 (2004).
  8. C. Corcoran and K. Pasch, "Self-Fourier functions and coherent laser combination," J. Phys. A 37, L461-L469 (2004).
    [CrossRef]
  9. C. Corcoran and K. Pasch, "Modal analysis of a self-Fourier laser cavity," J. Opt. A, Pure Appl. Opt. 7, L1-L7 (2005).
    [CrossRef]
  10. G. Cincotti, F. Gori, and M. Santarsiero, "Generalized self-Fourier functions," J. Phys. A 25, L1191-L1194 (1992).
    [CrossRef]
  11. M. Caola, "Self-Fourier functions," J. Phys. A 24, L1143-L1144 (1991).
    [CrossRef]
  12. I. Stakgold, Green's Functions and Boundary Value Problems (Wiley-Interscience, 1997).
  13. H. Dym and H. McKean, Fourier Series and Integrals (Academic, 1972).
  14. B. Gelbaum and J. Olmsted, Counterexamples in Analysis (Dover, 2003).
  15. M. Lighthill, Introduction to Fourier Analysis (Cambridge U. Press, 1958).
  16. T. Alieva and A. Barbe, "Self-fractional Fourier functions and selection of modes," J. Phys. A 30, L2111-L215 (1997).
    [CrossRef]

2005

C. Corcoran and K. Pasch, "Modal analysis of a self-Fourier laser cavity," J. Opt. A, Pure Appl. Opt. 7, L1-L7 (2005).
[CrossRef]

2004

C. Corcoran and K. Pasch, "Self-Fourier functions and coherent laser combination," J. Phys. A 37, L461-L469 (2004).
[CrossRef]

1997

T. Alieva and A. Barbe, "Self-fractional Fourier functions and selection of modes," J. Phys. A 30, L2111-L215 (1997).
[CrossRef]

1994

1993

1992

A. Lohmann and D. Mendlovic, "Self-Fourier objects and other self-transform objects," J. Opt. Soc. Am. A 9, 2009-2012 (1992).
[CrossRef]

G. Cincotti, F. Gori, and M. Santarsiero, "Generalized self-Fourier functions," J. Phys. A 25, L1191-L1194 (1992).
[CrossRef]

1991

M. Caola, "Self-Fourier functions," J. Phys. A 24, L1143-L1144 (1991).
[CrossRef]

Alieva, T.

T. Alieva and A. Barbe, "Self-fractional Fourier functions and selection of modes," J. Phys. A 30, L2111-L215 (1997).
[CrossRef]

Barbe, A.

T. Alieva and A. Barbe, "Self-fractional Fourier functions and selection of modes," J. Phys. A 30, L2111-L215 (1997).
[CrossRef]

Caola, M.

M. Caola, "Self-Fourier functions," J. Phys. A 24, L1143-L1144 (1991).
[CrossRef]

Cincotti, G.

G. Cincotti, F. Gori, and M. Santarsiero, "Generalized self-Fourier functions," J. Phys. A 25, L1191-L1194 (1992).
[CrossRef]

Coffey, M.

Corcoran, C.

C. Corcoran and K. Pasch, "Modal analysis of a self-Fourier laser cavity," J. Opt. A, Pure Appl. Opt. 7, L1-L7 (2005).
[CrossRef]

C. Corcoran and K. Pasch, "Self-Fourier functions and coherent laser combination," J. Phys. A 37, L461-L469 (2004).
[CrossRef]

Dym, H.

H. Dym and H. McKean, Fourier Series and Integrals (Academic, 1972).

Edwards, H.

H. Edwards, Riemann's Zeta Function (Dover, 2001).

Gelbaum, B.

B. Gelbaum and J. Olmsted, Counterexamples in Analysis (Dover, 2003).

Glimm, J.

J. Glimm and A. Jaffe, Quantum Physics (Springer, 1981).

Gori, F.

G. Cincotti, F. Gori, and M. Santarsiero, "Generalized self-Fourier functions," J. Phys. A 25, L1191-L1194 (1992).
[CrossRef]

Jaffe, A.

J. Glimm and A. Jaffe, Quantum Physics (Springer, 1981).

Lighthill, M.

M. Lighthill, Introduction to Fourier Analysis (Cambridge U. Press, 1958).

Lipson, S.

Lohmann, A.

McKean, H.

H. Dym and H. McKean, Fourier Series and Integrals (Academic, 1972).

Mendlovic, D.

Nishi, K.

K. Nishi, "Generalized comb function: a new self-Fourier function," in Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing (IEEE, 2004), Vol. 2, pp. 573-576 (2004).

Olmsted, J.

B. Gelbaum and J. Olmsted, Counterexamples in Analysis (Dover, 2003).

Pasch, K.

C. Corcoran and K. Pasch, "Modal analysis of a self-Fourier laser cavity," J. Opt. A, Pure Appl. Opt. 7, L1-L7 (2005).
[CrossRef]

C. Corcoran and K. Pasch, "Self-Fourier functions and coherent laser combination," J. Phys. A 37, L461-L469 (2004).
[CrossRef]

Santarsiero, M.

G. Cincotti, F. Gori, and M. Santarsiero, "Generalized self-Fourier functions," J. Phys. A 25, L1191-L1194 (1992).
[CrossRef]

Stakgold, I.

I. Stakgold, Green's Functions and Boundary Value Problems (Wiley-Interscience, 1997).

Titchmarsh, E.

E. Titchmarsh, Introduction to the Theory of Fourier Integrals (Clarendon, 1937).

J. Opt. A, Pure Appl. Opt.

C. Corcoran and K. Pasch, "Modal analysis of a self-Fourier laser cavity," J. Opt. A, Pure Appl. Opt. 7, L1-L7 (2005).
[CrossRef]

J. Opt. Soc. Am. A

J. Phys. A

C. Corcoran and K. Pasch, "Self-Fourier functions and coherent laser combination," J. Phys. A 37, L461-L469 (2004).
[CrossRef]

T. Alieva and A. Barbe, "Self-fractional Fourier functions and selection of modes," J. Phys. A 30, L2111-L215 (1997).
[CrossRef]

G. Cincotti, F. Gori, and M. Santarsiero, "Generalized self-Fourier functions," J. Phys. A 25, L1191-L1194 (1992).
[CrossRef]

M. Caola, "Self-Fourier functions," J. Phys. A 24, L1143-L1144 (1991).
[CrossRef]

Other

I. Stakgold, Green's Functions and Boundary Value Problems (Wiley-Interscience, 1997).

H. Dym and H. McKean, Fourier Series and Integrals (Academic, 1972).

B. Gelbaum and J. Olmsted, Counterexamples in Analysis (Dover, 2003).

M. Lighthill, Introduction to Fourier Analysis (Cambridge U. Press, 1958).

E. Titchmarsh, Introduction to the Theory of Fourier Integrals (Clarendon, 1937).

H. Edwards, Riemann's Zeta Function (Dover, 2001).

K. Nishi, "Generalized comb function: a new self-Fourier function," in Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing (IEEE, 2004), Vol. 2, pp. 573-576 (2004).

J. Glimm and A. Jaffe, Quantum Physics (Springer, 1981).

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Equations (52)

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F { f ( x ) } = f ̂ ( ω ) f ( x ) exp ( i ω x ) d x
F 1 { f ̂ ( ω ) } = f ( x ) 1 2 π f ̂ ( ω ) exp ( i ω x ) d ω .
f ( x ) = g ( x ) + g ( x ) + g ̂ ( x ) + g ̂ ( x ) ,
f ̂ ( x ) = f ( ω ) exp ( i ω x ) d ω = f ( x ) .
f ( ω ) exp ( i ω x ) d ω = f ̂ ( x ) = λ f ( x ) .
F 2 { f ( x ) } = f ( x ) .
F 4 { f ( x ) } = F 2 { F 2 { f ( x ) } } = F 2 { f ( x ) } = f ( x ) .
F 4 ( f ) = λ 4 f .
λ 4 = 1 λ = ± 1 or ± i .
f ̂ = λ f f ̂ ̂ = f ( x ) = λ f ̂ λ 2 f ( x ) = f ( x )
f ( x ) = α g ( x ) + β g ̂ ( x ) ,
f ̂ ( x ) = α g ̂ ( x ) + β g ( x ) = α g ̂ ( x ) + β g ( x ) = β α [ α g ( x ) ] + α β [ β g ̂ ( x ) ] .
α β = β α α = ± β
f ( x ) = α [ g ( x ) ± g ̂ ( x ) ] for λ = ± 1 .
f ( x ) = g ( x ) + g ̂ ( x ) + g ( x ) + g ̂ ( x ) = [ g ( x ) + g ( x ) ] + [ g ̂ ( x ) + g ̂ ( x ) ] = g e ( x ) + g ̂ e ( x )
f ( x ) = g ( x ) + g ̂ ( x ) + g ( x ) + g ̂ ( x )
g ( x ) = g e ( x ) + g o ( x )
f ( x ) = g e ( x ) + g ̂ e ( x )
f ( x ) = g ( x ) + g ̂ ( x ) ,
L n f ( x ) f ( n ) ( x ) ± x n f ( x ) ,
f ( n ) ( x ) d n f ( x ) d x n
L n f ̂ = f ( n ) ̂ ± x n f ̂ = ( i ω ) n f ̂ ± ( i ) n f ̂ ( n ) = ± ( i ) n ( f ̂ ( n ) ± ω n f ̂ ) ,
F { L n f } = ± ( i ) n L n { F f } .
L m n f x m f ( n ) ± ( x n f ) ( m ) ,
L 0 n = L n .
L m n f ̂ = x m f ( n ) ̂ ± ( x n f ) ( m ) ̂ = ± ( i ) m ( f ( n ) ̂ ( m ) ± ω m x n f ̂ ) = ± ( i ) m + n [ ( ω n f ̂ ) ( m ) ± ω m f ̂ ( n ) ] ,
F { L m n f } = ± ( i ) m + n L m n { F f } .
L m n f ̂ = c L m n f ̂ = c λ L m n f .
c λ = 1 .
g ( x ) = x m f ( n ) ( x )
g ̂ ( x ) = c ( x n f ( x ) ) ( m ) .
a 1 L m 1 n 1 + a 2 L m 2 n 2 + + a k L m k n k = j = 1 k a j L m j n j ,
( i ) m 1 + n 1 = ( i ) m 2 + n 2 = = ( i ) m i + n i .
L m n f ( x ) + λ L 00 f ( x ) = L m n f ( x ) + λ f ( x )
L m n = x m d n d x n ± d m d x m x n
L f ( x ) = L f ̂ ( x ) = ± ( i ) m + n L f ̂ ( x ) = 0 L f ̂ ( x ) = 0 .
f ( x ) d x < .
x m d n f d x n ± d m d x m ( x n f ) = 0 ,
f ( ± ) = d f d x ( ± ) = = d l f d x l ( ± ) = 0 , l = max ( m , n ) 1
L 01 f ( x ) = f ( x ) + x f ( x ) = 0 ,
f ( ) = f ( + ) = 0 .
L 02 f ( x ) = f ( x ) + x 2 f ( x ) = 0 ,
f ( ) = f ( + ) = 0 .
D 2 ( x ) = A x J ± 1 4 ( x 2 2 ) ,
L f = q ( x ) ,
L f ( x ) = x m f ( n ) ( x ) ± [ x n f ( x ) ] ( m ) + λ f ( x ) .
x m f ( n ) ( x ) ± [ x n f ( x ) ] ( m ) = λ f ( x ) ,
x m f ̂ ( n ) ( x ) ± [ x n f ̂ ( x ) ] ( m ) = λ ( i ) m + n f ̂ ( x ) ,
L 02 G ( x ) = G ( x ) x 2 G ( x ) = λ n G ( x ) ,
G ( ) = G ( + ) = 0 ,
G n ( x ) = A n H n ( x ) exp ( x 2 2 ) ,
G ̂ n ( x ) = ( i ) n G n ( x ) .

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