Abstract

When a continuous-signal field is sampled at a finite number N of equidistant sensor points, the N resulting data values can yield information on at most N oscillator mode components, whose coefficients should in turn restore the sampled signal. We compare the fidelity of the mode analysis and synthesis in the orthonormal basis of N-point Kravchuk functions with those in the basis of sampled Hermite–Gauss functions. The scale between the two bases is calibrated on the ground state of the field. We conclude that mode analysis is better approximated in the nonorthogonal sampled Hermite–Gauss basis, while signal restoration in the Kravchuk basis is exact.

© 2009 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. D. G. Luenberger, Optimization by Vector Space Methods (Wiley, 1969).
  2. J.-B. Martens, “The Hermite transform--theory,” IEEE Trans. Acoust., Speech, Signal Process. 38, 1607-1618 (1990).
    [CrossRef]
  3. N. Tanguy, P. Vilbé, and L. C. Calvez, “Optimum choice of free parameter in orthonormal approximations,” IEEE Trans. Autom. Control 40, 1811-1813 (1995).
    [CrossRef]
  4. A. C. den Brinker and H. J. W. Belt, “Optimal free parameters in orthonormal approximations,” IEEE Trans. Signal Process. 46, 2081-2087 (1998).
    [CrossRef]
  5. A. C. den Brinker and B. E. Sarroukh, “Modulated Hermite series expansion and time-bandwidth product,” Signal Process. 80, 243-250 (2000).
    [CrossRef]
  6. T. Alieva and M. J. Bastiaans, “Mode analysis through the fractional transforms in optics,” Opt. Lett. 24, 1206-1208 (1999).
    [CrossRef]
  7. T. Alieva and K. B. Wolf, “Finite mode analysis through harmonic waveguides,” J. Opt. Soc. Am. A 17, 1482-1484 (2000).
    [CrossRef]
  8. N. M. Atakishiyev and K. B. Wolf, “Fractional Fourier-Kravchuk transform,” J. Opt. Soc. Am. A 14, 1467-1477 (1997).
    [CrossRef]
  9. N. M. Atakishiyev, L. E. Vicent, and K. B. Wolf, “Continuous vs. discrete fractional Fourier transforms,” J. Comput. Appl. Math. 107, 73-95 (1999).
    [CrossRef]
  10. N. M. Atakishiyev, G. S. Pogosyan, and K. B. Wolf, “Finite models of the oscillator,” Phys. Part. Nucl. 36, 521-555 (2005).
  11. B. E. Sarroukh, “Signal analysis representation tools,” Ph. D. thesis (Technische Universiteit Eindhoven, The Netherlands, 2001).
  12. K. B. Wolf and G. Krötzsch, “Geometry and dynamics in the fractional discrete Fourier transform,” J. Opt. Soc. Am. A 24, 651-658 (2007).
    [CrossRef]
  13. L. C. Biedenharn and J. D. Louck, “Angular momentum in quantum mechanics,” in Encyclopedia of Mathematics and Its Applications, G.-C.Rota, ed. (Addison-Wesley, 1981); Sec. 3.6.
  14. M. Krawtchouk, “Sur une généralization des polinômes d'Hermite,” C. R. Acad. Sci. Paris 189, 620-622 (1929).
  15. N. M. Atakishiyev, G. S. Pogosyan, and K. B. Wolf, “Contraction of the finite one-dimensional oscillator,” Int. J. Mod. Phys. A 18, 317-327 (2003).
    [CrossRef]
  16. N. M. Atakishiyev, G. S. Pogosyan, L. E. Vicent, and K. B. Wolf, “Finite two-dimensional oscillator. I: The Cartesian model,” J. Phys. A 34, 9381-9398 (2001).
    [CrossRef]
  17. K. B. Wolf and T. Alieva, “Rotation and gyration of finite two-dimensional modes,” J. Opt. Soc. Am. A 25, 365-370 (2008).
    [CrossRef]
  18. N. M. Atakishiyev, G. S. Pogosyan, L. E. Vicent, and K. B. Wolf, “Finite two-dimensional oscillator. II: The radial model,” J. Phys. A 34, 9399-9415 (2001).
    [CrossRef]
  19. N. M. Atakishiyev, A. U. Klimyk, and K. B. Wolf, “Finite q-oscillator,” J. Phys. A 37, 5569-5587 (2004).
    [CrossRef]

2008

2007

2005

N. M. Atakishiyev, G. S. Pogosyan, and K. B. Wolf, “Finite models of the oscillator,” Phys. Part. Nucl. 36, 521-555 (2005).

2004

N. M. Atakishiyev, A. U. Klimyk, and K. B. Wolf, “Finite q-oscillator,” J. Phys. A 37, 5569-5587 (2004).
[CrossRef]

2003

N. M. Atakishiyev, G. S. Pogosyan, and K. B. Wolf, “Contraction of the finite one-dimensional oscillator,” Int. J. Mod. Phys. A 18, 317-327 (2003).
[CrossRef]

2001

N. M. Atakishiyev, G. S. Pogosyan, L. E. Vicent, and K. B. Wolf, “Finite two-dimensional oscillator. I: The Cartesian model,” J. Phys. A 34, 9381-9398 (2001).
[CrossRef]

N. M. Atakishiyev, G. S. Pogosyan, L. E. Vicent, and K. B. Wolf, “Finite two-dimensional oscillator. II: The radial model,” J. Phys. A 34, 9399-9415 (2001).
[CrossRef]

2000

A. C. den Brinker and B. E. Sarroukh, “Modulated Hermite series expansion and time-bandwidth product,” Signal Process. 80, 243-250 (2000).
[CrossRef]

T. Alieva and K. B. Wolf, “Finite mode analysis through harmonic waveguides,” J. Opt. Soc. Am. A 17, 1482-1484 (2000).
[CrossRef]

1999

N. M. Atakishiyev, L. E. Vicent, and K. B. Wolf, “Continuous vs. discrete fractional Fourier transforms,” J. Comput. Appl. Math. 107, 73-95 (1999).
[CrossRef]

T. Alieva and M. J. Bastiaans, “Mode analysis through the fractional transforms in optics,” Opt. Lett. 24, 1206-1208 (1999).
[CrossRef]

1998

A. C. den Brinker and H. J. W. Belt, “Optimal free parameters in orthonormal approximations,” IEEE Trans. Signal Process. 46, 2081-2087 (1998).
[CrossRef]

1997

1995

N. Tanguy, P. Vilbé, and L. C. Calvez, “Optimum choice of free parameter in orthonormal approximations,” IEEE Trans. Autom. Control 40, 1811-1813 (1995).
[CrossRef]

1990

J.-B. Martens, “The Hermite transform--theory,” IEEE Trans. Acoust., Speech, Signal Process. 38, 1607-1618 (1990).
[CrossRef]

1929

M. Krawtchouk, “Sur une généralization des polinômes d'Hermite,” C. R. Acad. Sci. Paris 189, 620-622 (1929).

Alieva, T.

Atakishiyev, N. M.

N. M. Atakishiyev, G. S. Pogosyan, and K. B. Wolf, “Finite models of the oscillator,” Phys. Part. Nucl. 36, 521-555 (2005).

N. M. Atakishiyev, A. U. Klimyk, and K. B. Wolf, “Finite q-oscillator,” J. Phys. A 37, 5569-5587 (2004).
[CrossRef]

N. M. Atakishiyev, G. S. Pogosyan, and K. B. Wolf, “Contraction of the finite one-dimensional oscillator,” Int. J. Mod. Phys. A 18, 317-327 (2003).
[CrossRef]

N. M. Atakishiyev, G. S. Pogosyan, L. E. Vicent, and K. B. Wolf, “Finite two-dimensional oscillator. II: The radial model,” J. Phys. A 34, 9399-9415 (2001).
[CrossRef]

N. M. Atakishiyev, G. S. Pogosyan, L. E. Vicent, and K. B. Wolf, “Finite two-dimensional oscillator. I: The Cartesian model,” J. Phys. A 34, 9381-9398 (2001).
[CrossRef]

N. M. Atakishiyev, L. E. Vicent, and K. B. Wolf, “Continuous vs. discrete fractional Fourier transforms,” J. Comput. Appl. Math. 107, 73-95 (1999).
[CrossRef]

N. M. Atakishiyev and K. B. Wolf, “Fractional Fourier-Kravchuk transform,” J. Opt. Soc. Am. A 14, 1467-1477 (1997).
[CrossRef]

Bastiaans, M. J.

Belt, H. J. W.

A. C. den Brinker and H. J. W. Belt, “Optimal free parameters in orthonormal approximations,” IEEE Trans. Signal Process. 46, 2081-2087 (1998).
[CrossRef]

Biedenharn, L. C.

L. C. Biedenharn and J. D. Louck, “Angular momentum in quantum mechanics,” in Encyclopedia of Mathematics and Its Applications, G.-C.Rota, ed. (Addison-Wesley, 1981); Sec. 3.6.

Calvez, L. C.

N. Tanguy, P. Vilbé, and L. C. Calvez, “Optimum choice of free parameter in orthonormal approximations,” IEEE Trans. Autom. Control 40, 1811-1813 (1995).
[CrossRef]

den Brinker, A. C.

A. C. den Brinker and B. E. Sarroukh, “Modulated Hermite series expansion and time-bandwidth product,” Signal Process. 80, 243-250 (2000).
[CrossRef]

A. C. den Brinker and H. J. W. Belt, “Optimal free parameters in orthonormal approximations,” IEEE Trans. Signal Process. 46, 2081-2087 (1998).
[CrossRef]

Klimyk, A. U.

N. M. Atakishiyev, A. U. Klimyk, and K. B. Wolf, “Finite q-oscillator,” J. Phys. A 37, 5569-5587 (2004).
[CrossRef]

Krawtchouk, M.

M. Krawtchouk, “Sur une généralization des polinômes d'Hermite,” C. R. Acad. Sci. Paris 189, 620-622 (1929).

Krötzsch, G.

Louck, J. D.

L. C. Biedenharn and J. D. Louck, “Angular momentum in quantum mechanics,” in Encyclopedia of Mathematics and Its Applications, G.-C.Rota, ed. (Addison-Wesley, 1981); Sec. 3.6.

Luenberger, D. G.

D. G. Luenberger, Optimization by Vector Space Methods (Wiley, 1969).

Martens, J.-B.

J.-B. Martens, “The Hermite transform--theory,” IEEE Trans. Acoust., Speech, Signal Process. 38, 1607-1618 (1990).
[CrossRef]

Pogosyan, G. S.

N. M. Atakishiyev, G. S. Pogosyan, and K. B. Wolf, “Finite models of the oscillator,” Phys. Part. Nucl. 36, 521-555 (2005).

N. M. Atakishiyev, G. S. Pogosyan, and K. B. Wolf, “Contraction of the finite one-dimensional oscillator,” Int. J. Mod. Phys. A 18, 317-327 (2003).
[CrossRef]

N. M. Atakishiyev, G. S. Pogosyan, L. E. Vicent, and K. B. Wolf, “Finite two-dimensional oscillator. I: The Cartesian model,” J. Phys. A 34, 9381-9398 (2001).
[CrossRef]

N. M. Atakishiyev, G. S. Pogosyan, L. E. Vicent, and K. B. Wolf, “Finite two-dimensional oscillator. II: The radial model,” J. Phys. A 34, 9399-9415 (2001).
[CrossRef]

Sarroukh, B. E.

A. C. den Brinker and B. E. Sarroukh, “Modulated Hermite series expansion and time-bandwidth product,” Signal Process. 80, 243-250 (2000).
[CrossRef]

B. E. Sarroukh, “Signal analysis representation tools,” Ph. D. thesis (Technische Universiteit Eindhoven, The Netherlands, 2001).

Tanguy, N.

N. Tanguy, P. Vilbé, and L. C. Calvez, “Optimum choice of free parameter in orthonormal approximations,” IEEE Trans. Autom. Control 40, 1811-1813 (1995).
[CrossRef]

Vicent, L. E.

N. M. Atakishiyev, G. S. Pogosyan, L. E. Vicent, and K. B. Wolf, “Finite two-dimensional oscillator. II: The radial model,” J. Phys. A 34, 9399-9415 (2001).
[CrossRef]

N. M. Atakishiyev, G. S. Pogosyan, L. E. Vicent, and K. B. Wolf, “Finite two-dimensional oscillator. I: The Cartesian model,” J. Phys. A 34, 9381-9398 (2001).
[CrossRef]

N. M. Atakishiyev, L. E. Vicent, and K. B. Wolf, “Continuous vs. discrete fractional Fourier transforms,” J. Comput. Appl. Math. 107, 73-95 (1999).
[CrossRef]

Vilbé, P.

N. Tanguy, P. Vilbé, and L. C. Calvez, “Optimum choice of free parameter in orthonormal approximations,” IEEE Trans. Autom. Control 40, 1811-1813 (1995).
[CrossRef]

Wolf, K. B.

K. B. Wolf and T. Alieva, “Rotation and gyration of finite two-dimensional modes,” J. Opt. Soc. Am. A 25, 365-370 (2008).
[CrossRef]

K. B. Wolf and G. Krötzsch, “Geometry and dynamics in the fractional discrete Fourier transform,” J. Opt. Soc. Am. A 24, 651-658 (2007).
[CrossRef]

N. M. Atakishiyev, G. S. Pogosyan, and K. B. Wolf, “Finite models of the oscillator,” Phys. Part. Nucl. 36, 521-555 (2005).

N. M. Atakishiyev, A. U. Klimyk, and K. B. Wolf, “Finite q-oscillator,” J. Phys. A 37, 5569-5587 (2004).
[CrossRef]

N. M. Atakishiyev, G. S. Pogosyan, and K. B. Wolf, “Contraction of the finite one-dimensional oscillator,” Int. J. Mod. Phys. A 18, 317-327 (2003).
[CrossRef]

N. M. Atakishiyev, G. S. Pogosyan, L. E. Vicent, and K. B. Wolf, “Finite two-dimensional oscillator. I: The Cartesian model,” J. Phys. A 34, 9381-9398 (2001).
[CrossRef]

N. M. Atakishiyev, G. S. Pogosyan, L. E. Vicent, and K. B. Wolf, “Finite two-dimensional oscillator. II: The radial model,” J. Phys. A 34, 9399-9415 (2001).
[CrossRef]

T. Alieva and K. B. Wolf, “Finite mode analysis through harmonic waveguides,” J. Opt. Soc. Am. A 17, 1482-1484 (2000).
[CrossRef]

N. M. Atakishiyev, L. E. Vicent, and K. B. Wolf, “Continuous vs. discrete fractional Fourier transforms,” J. Comput. Appl. Math. 107, 73-95 (1999).
[CrossRef]

N. M. Atakishiyev and K. B. Wolf, “Fractional Fourier-Kravchuk transform,” J. Opt. Soc. Am. A 14, 1467-1477 (1997).
[CrossRef]

C. R. Acad. Sci. Paris

M. Krawtchouk, “Sur une généralization des polinômes d'Hermite,” C. R. Acad. Sci. Paris 189, 620-622 (1929).

IEEE Trans. Acoust., Speech, Signal Process.

J.-B. Martens, “The Hermite transform--theory,” IEEE Trans. Acoust., Speech, Signal Process. 38, 1607-1618 (1990).
[CrossRef]

IEEE Trans. Autom. Control

N. Tanguy, P. Vilbé, and L. C. Calvez, “Optimum choice of free parameter in orthonormal approximations,” IEEE Trans. Autom. Control 40, 1811-1813 (1995).
[CrossRef]

IEEE Trans. Signal Process.

A. C. den Brinker and H. J. W. Belt, “Optimal free parameters in orthonormal approximations,” IEEE Trans. Signal Process. 46, 2081-2087 (1998).
[CrossRef]

Int. J. Mod. Phys. A

N. M. Atakishiyev, G. S. Pogosyan, and K. B. Wolf, “Contraction of the finite one-dimensional oscillator,” Int. J. Mod. Phys. A 18, 317-327 (2003).
[CrossRef]

J. Comput. Appl. Math.

N. M. Atakishiyev, L. E. Vicent, and K. B. Wolf, “Continuous vs. discrete fractional Fourier transforms,” J. Comput. Appl. Math. 107, 73-95 (1999).
[CrossRef]

J. Opt. Soc. Am. A

J. Phys. A

N. M. Atakishiyev, G. S. Pogosyan, L. E. Vicent, and K. B. Wolf, “Finite two-dimensional oscillator. II: The radial model,” J. Phys. A 34, 9399-9415 (2001).
[CrossRef]

N. M. Atakishiyev, A. U. Klimyk, and K. B. Wolf, “Finite q-oscillator,” J. Phys. A 37, 5569-5587 (2004).
[CrossRef]

N. M. Atakishiyev, G. S. Pogosyan, L. E. Vicent, and K. B. Wolf, “Finite two-dimensional oscillator. I: The Cartesian model,” J. Phys. A 34, 9381-9398 (2001).
[CrossRef]

Opt. Lett.

Phys. Part. Nucl.

N. M. Atakishiyev, G. S. Pogosyan, and K. B. Wolf, “Finite models of the oscillator,” Phys. Part. Nucl. 36, 521-555 (2005).

Signal Process.

A. C. den Brinker and B. E. Sarroukh, “Modulated Hermite series expansion and time-bandwidth product,” Signal Process. 80, 243-250 (2000).
[CrossRef]

Other

D. G. Luenberger, Optimization by Vector Space Methods (Wiley, 1969).

B. E. Sarroukh, “Signal analysis representation tools,” Ph. D. thesis (Technische Universiteit Eindhoven, The Netherlands, 2001).

L. C. Biedenharn and J. D. Louck, “Angular momentum in quantum mechanics,” in Encyclopedia of Mathematics and Its Applications, G.-C.Rota, ed. (Addison-Wesley, 1981); Sec. 3.6.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (8)

Fig. 1
Fig. 1

Left column, Kravchuk basis functions, Φ n j ( m ) in Eq. (3), for j = 5 (marked with dots on the N = 11 points); dotted curves show their analytic continuation to m [ 5 , 5 ] . Right column, s-HG functions Ψ n ( s ¯ ) ( m ) in Eq. (2); dots indicate the sampling points for an optimal scale factor s ¯ = 0.446410 . From top to bottom are the vectors of 11 basis states n 0 10 . Note that as a result of Eq. (5), the higher Kravchuk modes 6 n 10 have the same absolute values as the lower modes 0 n 4 .

Fig. 2
Fig. 2

Scale calibration of the ground s-HG vector to the ground Kravchuk vector. Top, for j = 7 ( N = 15 points); bottom, for j = 15 ( N = 31 points). Large dots mark the values of the Kravchuk function Φ 0 j ( m ) . Solid curves show the s-HG function Ψ 0 ( x m ) ; Small-dots mark the sampling points x m = s m , for scale factors s around the optimal value s ¯ where the s-HG and Kravchuk functions are closest. For j = 7 , this is s ¯ = 0.378138 , and for j = 15 it is s ¯ = 0.258253 . The dashed curve (barely visible in the top figure near the center and edges of the interval) is the analytic continuation of the ground Kravchuk function.

Fig. 3
Fig. 3

Density plot of the norm of the difference vector D n ( s ) in Eq. (9) for j = 15 ( N = 31 points), over the ranges n 0 30 and 0 < s < 1 . We adopt the optimum value at the n = 0 ground state, s ¯ j = 0.258253 .

Fig. 4
Fig. 4

Density plot for the elements of the 31 × 31 unitary Kravchuk transform matrix K n , m j = Φ n j ( m ) in Eq. (3) for j = 15 . Each column represents one Kravchuk vector n 0 30 ; gray is zero; light and dark pixels correspond to positive and negative values. Owing to the symmetry properties (5), the left half of the figure reflects the right half with a sign alternation. (Adapted from Fig. 7 of [12]).

Fig. 5
Fig. 5

Left, density plot for the elements of the s-HG matrix S n , m Ψ n ( s ¯ ) ( m ) in Eq. (13); as in Fig. 4, each column represents a sampled HG mode. Here j = 15 , m 15 15 , n 0 30 , and det S = 2.33458 × 10 41 ; the maximum element of S is S 0 , 0 = 0.3817 . Right, the dual basis given by the elements of the inverse matrix ( S 1 ) n , m ; the maximum elements are S 19 , ± 1 1 = 6.054 × 10 7 . (Adapted from Fig. 1 of [12]).

Fig. 6
Fig. 6

True mode coefficients { F ¯ n F ¯ 0 } n = 0 30 , from Eq. (16) for N = 31 , of the rectangle signal that is nonzero for m [ 5 , 5 ] , indicated by small dots and joined by solid lines for visibility. Superposed are the Kravchuk mode coefficients { F n K F 0 K } n = 0 30 from Eq. (12), indicated by large dots joined by dotted lines. (Their common value 1 for n = 0 is outside the graph.) Top inset, true mode coefficients up to n = 300 ; bottom inset, sampled HG mode coefficients { F n S F 0 S } n = 0 30 of the same data set from Eq. (13); notice the very large coefficients, 3.5 × 10 5 for n 20 .

Fig. 7
Fig. 7

True mode coefficients { F ¯ n F ¯ 0 } n = 0 30 of the rectangle function from Eq. (16), as in Fig. 6, compared with the s-HG mode coefficients { f n f 0 } n = 0 30 obtained as from Eq. (14). Inset, norms of the difference vectors Δ ( n ) in Eq. (16).

Fig. 8
Fig. 8

Successive approximations to the finite rectangle signal on N = 31 points ( j = 15 ) by truncated sums of modes 0 n M for M = 0 , 4, 8, 12, 18, 22, 24, 26, 28, and 30. Left column, the true modes { F ¯ n } n = 0 M from Eq. (16) synthesized with HG functions; dots mark the values at the sensor points m j j . Middle column, the s-HG mode coefficients { f n } n = 0 M from Eq. (14), also synthesized with the continuous HG functions. Right column, the Kravchuk mode coefficients { F n K } n = 0 M synthesized with Kravchuk functions at the sensor points and their analytical continuation. The signal restoration is exact for M + 1 = N = 31 .

Equations (21)

Equations on this page are rendered with MathJax. Learn more.

( F , G ) N m = j j F ( m ) * G ( m ) ,
Ψ n ( s ) ( m ) A n ( s ) exp ( 1 2 s 2 m 2 ) H n ( s m ) , n 0 N 1 ,
Φ n j ( m ) d n j , m j ( 1 2 π ) = ( 1 ) n 2 j ( 2 j n ) ( 2 j j + m ) K n ( j + m ; 1 2 , 2 j ) ,
K n ( j + m ; 1 2 , 2 j ) = F 1 2 ( n , 2 j m ; 2 j ; 2 ) = K j + m ( n ; 1 2 , 2 j ) ,
Φ n j ( m ) = ( 1 ) n Φ n j ( m ) = ( 1 ) n j Φ 2 j n j ( m ) = ( 1 ) n j m Φ j + m j ( n j ) .
lim j ( 1 ) n + j j 1 4 Φ n j ( m ) = Ψ n ( x ) for x = m j .
Φ 0 j ( m ) = 1 2 j ( 2 j j + m ) , m j j .
D n ( s ) ( m ) Ψ n j ( m ) Ψ n ( s ) ( m ) , n 0 2 j .
D n j min s D n ( s ) = D n ( s ¯ ) .
s ¯ n j , best value of the scale s ,
D n ( s ¯ ) , norm of difference vector at s ¯ n j ,
θ n j , angle between the vectors at s ¯ n j .
j ( N ) ̱ 5 ( 11 ) ̱ 7 ( 15 ) ̱ 9 ( 19 ) ̱ 11 ( 23 ) ̱ 13 ( 27 ) ̱ 15 ( 31 ) ̱ n = 0 : 0.446410 0.019418 1.113 ° 0.378138 0.015702 0.899 ° 0.333515 0.012382 0.709 ° 0.301632 0.010027 0.574 ° 0.277429 0.008383 0.480 ° 0.258253 0.007196 0.412 ° n = 1 : 0.459067 0.038087 2.182 ° 0.387106 0.034659 1.986 ° 0.339919 0.028435 1.629 ° 0.306390 0.023206 1.330 ° 0.281117 0.019358 1.109 ° 0.261214 0.016547 0.948 ° n = 2 : 0.465404 0.061012 3.496 ° 0.393058 0.064049 3.670 ° 0.344594 0.055706 3.192 ° 0.309960 0.046456 2.662 ° 0.283890 0.038952 2.232 ° 0.263430 0.033283 1.907 ° n = 3 : 0.472270 0.061512 3.525 ° 0.399089 0.087453 5.012 ° 0.349729 0.084336 4.834 ° 0.314058 0.073291 4.200 ° 0.287129 0.062266 3.569 ° 0.266030 0.053278 3.053 ° n = 4 : 0.484325 0.062078 3.557 ° 0.405105 0.097435 5.585 ° 0.354900 0.109893 6.300 ° 0.318405 0.102186 5.857 ° 0.290666 0.089274 5.117 ° 0.268900 0.077050 4.416 ° n = 5 : 0.504866 0.137602 7.890 ° 0.412380 0.098467 5.644 ° 0.359923 0.127387 7.304 ° 0.322743 0.130108 7.460 ° 0.294333 0.118937 6.819 ° 0.271938 0.104628 5.998 ° .
F ( m ) = n = 0 2 j F n K K n , m j , F n K = m = j j K n , m j F ( m ) ,
K n , m j Φ n j ( m ) ,
F ( m ) = n = 0 2 j F n S S n , m , F n S = m = j j ( S 1 ) n , m F ( m ) ,
S n , m Ψ n ( s ¯ ) ( m ) .
Δ ( n + 1 ) ( m ) = Δ ( n ) ( m ) f n Ψ n ( s ¯ ) ( m ) ,
f n ( Δ ( n ) , Ψ n ( s ¯ ) ) N , Δ ( 0 ) ( m ) F ( m ) .
F ( m ) = f 0 Ψ 0 ( s ¯ ) ( m ) + f 1 Ψ 1 ( s ¯ ) ( m ) + + f N 1 Ψ N 1 ( s ¯ ) ( m ) + Δ ( N ) ( m ) ,
F ¯ ( x ) = n = 0 F ¯ n Ψ n ( x ) , F ¯ n = d x Ψ n ( x ) F ¯ ( x ) , n 0 ,

Metrics