Abstract

The problem of object restoration in the case of spatially incoherent illumination is considered. A regularized solution to the inverse problem is obtained through a probabilistic approach, and a numerical algorithm based on the statistical analysis of the noisy data is presented. Particular emphasis is placed on the question of the positivity constraint, which is incorporated into the probabilistically regularized solution by means of a quadratic programming technique. Numerical examples illustrating the main steps of the algorithm are also given.

© 2000 Optical Society of America

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References

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  1. M. Bertero, P. Boccacci, Introduction to Inverse Problems in Imaging (Institute of Physics, Bristol, UK, 1998).
  2. J. Hadamard, Lectures on the Cauchy Problem in Linear Differential Equations (Yale University, New Haven, Conn., 1923).
  3. C. W. Groetsch, The Theory of Tikhonov Regularization for Fredholm Equations of the First Kind (Pitman, Boston, Mass., 1984).
  4. A. Tikhonov, V. Arsenine, Méthodes de Rèsolution de Problémes Mal Posès (Mir, Moscow, 1976).
  5. E. De Micheli, N. Magnoli, G. A. Viano, “On the regularization of Fredholm integral equations of the first kind,” SIAM (Soc. Ind. Appl. Math.) J. Math. Anal. 29, 855–877 (1998).
    [CrossRef]
  6. K. Miller, “Least square methods for ill-posed problems with a prescribed bound,” SIAM (Soc. Ind. Appl. Math.) J. Math. Anal. 1, 52–74 (1970).
    [CrossRef]
  7. K. Miller, G. A. Viano, “On the necessity of nearly-best-possible methods for analytic continuation of scattering data,” J. Math. Phys. 14, 1037–1048 (1973).
    [CrossRef]
  8. A. N. Kolmogorov, V. M. Tihomirov, “∊-entropy and ∊-capacity of sets in functional spaces,” Uspekhi 14, 3–86 (1959).
  9. E. Scalas, G. A. Viano, “Resolving power and information theory in signal recovery,” J. Opt. Soc. Am. A 10, 991–996 (1993).
    [CrossRef]
  10. A. V. Balakrishnan, Applied Functional Analysis (Springer-Verlag, New York, 1976).
  11. J. N. Franklin, “Well-posed stochastic extensions of ill-posed linear problems,” J. Math. Anal. Appl. 31, 682–716 (1970).
    [CrossRef]
  12. I. M. Gel’fand, A. M. Yaglom, “Calculation of the amount of information about a random function contained in another such function,” Am. Math. Soc. Trans. Ser. 2 12, 199–246 (1959).
  13. J. L. Doob, Stochastic Processes (Wiley, New York, 1953).
  14. D. Middleton, An Introduction to Statistical Communication Theory (McGraw-Hill, New York, 1960).
  15. G. M. Jenkins, D. G. Watts, Spectral Analysis and Its Applications (Holden-Day, San Francisco, Calif., 1968).
  16. M. S. Bartlett, Stochastic Processes—Methods and Applications, 3rd ed. (Cambridge U. Press, Cambridge, UK, 1978).
  17. M. Bertero, V. Dovı́, “Regularized and positive-constrained inverse methods in the problem of object restoration,” Opt. Acta 28, 1635–1649 (1981).
    [CrossRef]
  18. G. de Villiers, B. McNally, E. R. Pike, “Positive solutions to linear inverse problems,” Inverse Probl. 15, 615–635 (1999).
    [CrossRef]
  19. B. McNally, E. R. Pike, “Quadratic programming for positive solutions of linear inverse problems,” in Proceedings of the Workshop on Scientific Computing, F. T. Luk, R. J. Plemmons, eds. (Springer, Berlin, 1997), pp. 101–109.
  20. M. Piana, M. Bertero, “Projected Landweber method and preconditioning,” Inverse Probl. 13, 441–463 (1997).
    [CrossRef]
  21. M. S. Bazaraa, H. D. Sherali, C. M. Shetty, Nonlinear Programming—Theory and Algorithms, 2nd ed. (Wiley, New York, 1993).
  22. W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes (Cambridge U. Press, Cambridge, UK, 1992).
  23. G. E. P. Box, G. M. Jenkins, Time Series Analysis (Holden-Day, San Francisco, Calif., 1976).
  24. F. Gori, “Integral equations for incoherent imagery,” J. Opt. Soc. Am. 64, 1237–1243 (1974).
    [CrossRef]

1999 (1)

G. de Villiers, B. McNally, E. R. Pike, “Positive solutions to linear inverse problems,” Inverse Probl. 15, 615–635 (1999).
[CrossRef]

1998 (1)

E. De Micheli, N. Magnoli, G. A. Viano, “On the regularization of Fredholm integral equations of the first kind,” SIAM (Soc. Ind. Appl. Math.) J. Math. Anal. 29, 855–877 (1998).
[CrossRef]

1997 (1)

M. Piana, M. Bertero, “Projected Landweber method and preconditioning,” Inverse Probl. 13, 441–463 (1997).
[CrossRef]

1993 (1)

1981 (1)

M. Bertero, V. Dovı́, “Regularized and positive-constrained inverse methods in the problem of object restoration,” Opt. Acta 28, 1635–1649 (1981).
[CrossRef]

1974 (1)

1973 (1)

K. Miller, G. A. Viano, “On the necessity of nearly-best-possible methods for analytic continuation of scattering data,” J. Math. Phys. 14, 1037–1048 (1973).
[CrossRef]

1970 (2)

K. Miller, “Least square methods for ill-posed problems with a prescribed bound,” SIAM (Soc. Ind. Appl. Math.) J. Math. Anal. 1, 52–74 (1970).
[CrossRef]

J. N. Franklin, “Well-posed stochastic extensions of ill-posed linear problems,” J. Math. Anal. Appl. 31, 682–716 (1970).
[CrossRef]

1959 (2)

I. M. Gel’fand, A. M. Yaglom, “Calculation of the amount of information about a random function contained in another such function,” Am. Math. Soc. Trans. Ser. 2 12, 199–246 (1959).

A. N. Kolmogorov, V. M. Tihomirov, “∊-entropy and ∊-capacity of sets in functional spaces,” Uspekhi 14, 3–86 (1959).

Arsenine, V.

A. Tikhonov, V. Arsenine, Méthodes de Rèsolution de Problémes Mal Posès (Mir, Moscow, 1976).

Balakrishnan, A. V.

A. V. Balakrishnan, Applied Functional Analysis (Springer-Verlag, New York, 1976).

Bartlett, M. S.

M. S. Bartlett, Stochastic Processes—Methods and Applications, 3rd ed. (Cambridge U. Press, Cambridge, UK, 1978).

Bazaraa, M. S.

M. S. Bazaraa, H. D. Sherali, C. M. Shetty, Nonlinear Programming—Theory and Algorithms, 2nd ed. (Wiley, New York, 1993).

Bertero, M.

M. Piana, M. Bertero, “Projected Landweber method and preconditioning,” Inverse Probl. 13, 441–463 (1997).
[CrossRef]

M. Bertero, V. Dovı́, “Regularized and positive-constrained inverse methods in the problem of object restoration,” Opt. Acta 28, 1635–1649 (1981).
[CrossRef]

M. Bertero, P. Boccacci, Introduction to Inverse Problems in Imaging (Institute of Physics, Bristol, UK, 1998).

Boccacci, P.

M. Bertero, P. Boccacci, Introduction to Inverse Problems in Imaging (Institute of Physics, Bristol, UK, 1998).

Box, G. E. P.

G. E. P. Box, G. M. Jenkins, Time Series Analysis (Holden-Day, San Francisco, Calif., 1976).

De Micheli, E.

E. De Micheli, N. Magnoli, G. A. Viano, “On the regularization of Fredholm integral equations of the first kind,” SIAM (Soc. Ind. Appl. Math.) J. Math. Anal. 29, 855–877 (1998).
[CrossRef]

de Villiers, G.

G. de Villiers, B. McNally, E. R. Pike, “Positive solutions to linear inverse problems,” Inverse Probl. 15, 615–635 (1999).
[CrossRef]

Doob, J. L.

J. L. Doob, Stochastic Processes (Wiley, New York, 1953).

Dovi´, V.

M. Bertero, V. Dovı́, “Regularized and positive-constrained inverse methods in the problem of object restoration,” Opt. Acta 28, 1635–1649 (1981).
[CrossRef]

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes (Cambridge U. Press, Cambridge, UK, 1992).

Franklin, J. N.

J. N. Franklin, “Well-posed stochastic extensions of ill-posed linear problems,” J. Math. Anal. Appl. 31, 682–716 (1970).
[CrossRef]

Gel’fand, I. M.

I. M. Gel’fand, A. M. Yaglom, “Calculation of the amount of information about a random function contained in another such function,” Am. Math. Soc. Trans. Ser. 2 12, 199–246 (1959).

Gori, F.

Groetsch, C. W.

C. W. Groetsch, The Theory of Tikhonov Regularization for Fredholm Equations of the First Kind (Pitman, Boston, Mass., 1984).

Hadamard, J.

J. Hadamard, Lectures on the Cauchy Problem in Linear Differential Equations (Yale University, New Haven, Conn., 1923).

Jenkins, G. M.

G. M. Jenkins, D. G. Watts, Spectral Analysis and Its Applications (Holden-Day, San Francisco, Calif., 1968).

G. E. P. Box, G. M. Jenkins, Time Series Analysis (Holden-Day, San Francisco, Calif., 1976).

Kolmogorov, A. N.

A. N. Kolmogorov, V. M. Tihomirov, “∊-entropy and ∊-capacity of sets in functional spaces,” Uspekhi 14, 3–86 (1959).

Magnoli, N.

E. De Micheli, N. Magnoli, G. A. Viano, “On the regularization of Fredholm integral equations of the first kind,” SIAM (Soc. Ind. Appl. Math.) J. Math. Anal. 29, 855–877 (1998).
[CrossRef]

McNally, B.

G. de Villiers, B. McNally, E. R. Pike, “Positive solutions to linear inverse problems,” Inverse Probl. 15, 615–635 (1999).
[CrossRef]

B. McNally, E. R. Pike, “Quadratic programming for positive solutions of linear inverse problems,” in Proceedings of the Workshop on Scientific Computing, F. T. Luk, R. J. Plemmons, eds. (Springer, Berlin, 1997), pp. 101–109.

Middleton, D.

D. Middleton, An Introduction to Statistical Communication Theory (McGraw-Hill, New York, 1960).

Miller, K.

K. Miller, G. A. Viano, “On the necessity of nearly-best-possible methods for analytic continuation of scattering data,” J. Math. Phys. 14, 1037–1048 (1973).
[CrossRef]

K. Miller, “Least square methods for ill-posed problems with a prescribed bound,” SIAM (Soc. Ind. Appl. Math.) J. Math. Anal. 1, 52–74 (1970).
[CrossRef]

Piana, M.

M. Piana, M. Bertero, “Projected Landweber method and preconditioning,” Inverse Probl. 13, 441–463 (1997).
[CrossRef]

Pike, E. R.

G. de Villiers, B. McNally, E. R. Pike, “Positive solutions to linear inverse problems,” Inverse Probl. 15, 615–635 (1999).
[CrossRef]

B. McNally, E. R. Pike, “Quadratic programming for positive solutions of linear inverse problems,” in Proceedings of the Workshop on Scientific Computing, F. T. Luk, R. J. Plemmons, eds. (Springer, Berlin, 1997), pp. 101–109.

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes (Cambridge U. Press, Cambridge, UK, 1992).

Scalas, E.

Sherali, H. D.

M. S. Bazaraa, H. D. Sherali, C. M. Shetty, Nonlinear Programming—Theory and Algorithms, 2nd ed. (Wiley, New York, 1993).

Shetty, C. M.

M. S. Bazaraa, H. D. Sherali, C. M. Shetty, Nonlinear Programming—Theory and Algorithms, 2nd ed. (Wiley, New York, 1993).

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes (Cambridge U. Press, Cambridge, UK, 1992).

Tihomirov, V. M.

A. N. Kolmogorov, V. M. Tihomirov, “∊-entropy and ∊-capacity of sets in functional spaces,” Uspekhi 14, 3–86 (1959).

Tikhonov, A.

A. Tikhonov, V. Arsenine, Méthodes de Rèsolution de Problémes Mal Posès (Mir, Moscow, 1976).

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes (Cambridge U. Press, Cambridge, UK, 1992).

Viano, G. A.

E. De Micheli, N. Magnoli, G. A. Viano, “On the regularization of Fredholm integral equations of the first kind,” SIAM (Soc. Ind. Appl. Math.) J. Math. Anal. 29, 855–877 (1998).
[CrossRef]

E. Scalas, G. A. Viano, “Resolving power and information theory in signal recovery,” J. Opt. Soc. Am. A 10, 991–996 (1993).
[CrossRef]

K. Miller, G. A. Viano, “On the necessity of nearly-best-possible methods for analytic continuation of scattering data,” J. Math. Phys. 14, 1037–1048 (1973).
[CrossRef]

Watts, D. G.

G. M. Jenkins, D. G. Watts, Spectral Analysis and Its Applications (Holden-Day, San Francisco, Calif., 1968).

Yaglom, A. M.

I. M. Gel’fand, A. M. Yaglom, “Calculation of the amount of information about a random function contained in another such function,” Am. Math. Soc. Trans. Ser. 2 12, 199–246 (1959).

Am. Math. Soc. Trans. Ser. 2 (1)

I. M. Gel’fand, A. M. Yaglom, “Calculation of the amount of information about a random function contained in another such function,” Am. Math. Soc. Trans. Ser. 2 12, 199–246 (1959).

Inverse Probl. (2)

G. de Villiers, B. McNally, E. R. Pike, “Positive solutions to linear inverse problems,” Inverse Probl. 15, 615–635 (1999).
[CrossRef]

M. Piana, M. Bertero, “Projected Landweber method and preconditioning,” Inverse Probl. 13, 441–463 (1997).
[CrossRef]

J. Math. Anal. Appl. (1)

J. N. Franklin, “Well-posed stochastic extensions of ill-posed linear problems,” J. Math. Anal. Appl. 31, 682–716 (1970).
[CrossRef]

J. Math. Phys. (1)

K. Miller, G. A. Viano, “On the necessity of nearly-best-possible methods for analytic continuation of scattering data,” J. Math. Phys. 14, 1037–1048 (1973).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Opt. Acta (1)

M. Bertero, V. Dovı́, “Regularized and positive-constrained inverse methods in the problem of object restoration,” Opt. Acta 28, 1635–1649 (1981).
[CrossRef]

SIAM (Soc. Ind. Appl. Math.) J. Math. Anal. (2)

E. De Micheli, N. Magnoli, G. A. Viano, “On the regularization of Fredholm integral equations of the first kind,” SIAM (Soc. Ind. Appl. Math.) J. Math. Anal. 29, 855–877 (1998).
[CrossRef]

K. Miller, “Least square methods for ill-posed problems with a prescribed bound,” SIAM (Soc. Ind. Appl. Math.) J. Math. Anal. 1, 52–74 (1970).
[CrossRef]

Uspekhi (1)

A. N. Kolmogorov, V. M. Tihomirov, “∊-entropy and ∊-capacity of sets in functional spaces,” Uspekhi 14, 3–86 (1959).

Other (13)

A. V. Balakrishnan, Applied Functional Analysis (Springer-Verlag, New York, 1976).

M. Bertero, P. Boccacci, Introduction to Inverse Problems in Imaging (Institute of Physics, Bristol, UK, 1998).

J. Hadamard, Lectures on the Cauchy Problem in Linear Differential Equations (Yale University, New Haven, Conn., 1923).

C. W. Groetsch, The Theory of Tikhonov Regularization for Fredholm Equations of the First Kind (Pitman, Boston, Mass., 1984).

A. Tikhonov, V. Arsenine, Méthodes de Rèsolution de Problémes Mal Posès (Mir, Moscow, 1976).

B. McNally, E. R. Pike, “Quadratic programming for positive solutions of linear inverse problems,” in Proceedings of the Workshop on Scientific Computing, F. T. Luk, R. J. Plemmons, eds. (Springer, Berlin, 1997), pp. 101–109.

J. L. Doob, Stochastic Processes (Wiley, New York, 1953).

D. Middleton, An Introduction to Statistical Communication Theory (McGraw-Hill, New York, 1960).

G. M. Jenkins, D. G. Watts, Spectral Analysis and Its Applications (Holden-Day, San Francisco, Calif., 1968).

M. S. Bartlett, Stochastic Processes—Methods and Applications, 3rd ed. (Cambridge U. Press, Cambridge, UK, 1978).

M. S. Bazaraa, H. D. Sherali, C. M. Shetty, Nonlinear Programming—Theory and Algorithms, 2nd ed. (Wiley, New York, 1993).

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes (Cambridge U. Press, Cambridge, UK, 1992).

G. E. P. Box, G. M. Jenkins, Time Series Analysis (Holden-Day, San Francisco, Calif., 1976).

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Figures (4)

Fig. 1
Fig. 1

Example 1: f1(x)=(1-x)sin2[4(1+x)], =10-3, c=20. The global SNR, defined as the ratio of the mean power of the noiseless data to the noise variance, was SNR40 dB. A, Noiseless Fourier coefficients gk. B, Modulus of the autocorrelation function. From the analysis of δg¯(n) we have n0=14, Q={1, 2 ,, 13, 14}. I={1, 2, 3, 4, 6, 7 ,, 14, 15}, each element with maximum inner compatibility, i.e., 13. Horizontal straight line, 95% confidence limit 1.96σδ(n, 0) for a purely random sequence. This limit was used to select the elements of Q for nn014 (solid part). Curve, confidence limit 1.96σδ(n, 14) for n>n0 (solid part) that we used for rejecting the autocorrelations that are spuriously inflated by statistical fluctuations, whereas for nn0 (dashed part) it shows only how the final confidence limit 1.96σδ(n, 14) was reached during the maximization procedure for setting n0 [see text and, in particular, Eq. (26)]. C, Regularized solution. Solid curve, actual solution f1(x). Dashed curve, reconstruction fI(x). D, Two examples of the approximation f0(x) [see Eq. (5)]. k0() was chosen as the largest integer such that λk/M, where M=f1(x)L2(-1, 1); that is, in the current case, k0=29. Solid curve, actual solution f1(x). Dashed curve, f0(x) computed with k0=27 (see in particular the rightmost peak); dotted–dashed curve, f0(x) computed with k0=28. The approximation f0(x) computed with k0=29, as prescribed by the above truncation criterion, is not displayed, since it is extremely different from the real solution.

Fig. 2
Fig. 2

Example 2: f2(x)=exp[-(x-x0)2/2σ2]+exp[-(x+x0)2/2σ2] with x0=0.5, σ=0.1, =3×10-2, SNR6.2 dB, c=20. A, Noiseless Fourier coefficients gk. B, Modulus of the autocorrelation function. n0=8, Q={2, 4, 6, 8}, I={1, 3, 5, 7, 9}. C, Unconstrained regularized solution. Solid curve, actual solution f2(x). Dashed curve, reconstruction fI(x). D, Comparison between the actual solution f2(x) and the constrained regularized solution fI(p)(x) [see Eq. (33)]. The number of corrective eigenfunctions used was m¯I-mI=8. The positivity constraint was set over Np=64 points of the interval [-1, 1] [see relation (35)]. After the optimization procedure the coefficients dk were such that maxmImm¯I|dm-λmg¯m| 0.01<.

Fig. 3
Fig. 3

Example 3: f3(x)=sin2[5(1-x)]sin2[5(1+x)][exp(-3x)+exp(-x)]. =5×10-2, SNR16.1 dB, c=20. A, Noiseless Fourier coefficients gk. B, Modulus of the autocorrelation function. n0=12, Q={1, 2, 4, 5, 6, 7, 8, 10, 11, 12}, I={1, 2, 3, 7, 8, 9, 13}. Each element of I has maximum inner compatibility, i.e., 6, with respect to the set Q. C, Unconstrained regularized solution. Solid curve, actual solution f3(x). Dashed curve, reconstruction fI(x). D, Comparison between the actual solution f3(x) and the constrained regularized solution fI(p)(x) [see Eq. (33)]. The number of corrective eigenfunctions used was m¯I-mI=40. The positivity constraint was explicitly imposed on Np=64 points of the interval [-1, 1] [see relation (35)]. After the optimization procedure the coefficients dk were such that maxmImm¯I|dm-λmg¯m| 0.02<.

Fig. 4
Fig. 4

Example 4: f4(x)=1 for -0.6x0.6 and null elsewhere. =8×10-2, SNR3.2 dB, c=20. A, Noiseless Fourier coefficients gk. B, Modulus of the autocorrelation function. n0=17, Q={2, 4, 6, 17}. The set Q led to selection of the Fourier components k=1, 3, 5, 7, 18. However, the component at k=18 has the minimum compatibility index, i.e. 1, which strongly indicates that the correlation at the lag n=17 was spuriously generated by the noise. Then I={1, 3, 5, 7}. C, Unconstrained regularized solution. Solid line, actual solution f4(x). Dashed curve, reconstruction fI(x). D, Comparison between the actual solution f4(x) and the constrained regularized solution fI(p)(x) [see Eq. (33)]. The number of corrective eigenfunctions used was m¯I-mI=70. The positivity constraint was set over Np=64 points of the interval [-1, 1] [see relation (35)]. After the optimization procedure the coefficients dk were such that maxmImm¯I|dm-λmg¯m| 0.01<.

Equations (52)

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

(Af )(y)=-11K(y-x)f(x)dx=g(y),
-1y1,
K(x)=sin2(cx)πcx2,c=πR,
f(x)=k=1gkλk ψk(x),
g¯-gL2(-1, 1)=nL2(-1, 1).
f0(x)=k=1k0()g¯kλk ψk(x),
lim0(f-f0, v)L2(-1, 1)=0,
[vL2(-1, 1),vL2(-1, 1)1].
Lmax()2k0()log2(1/).
Aξ+ζ=η,
λkξk+ζk=ηk,(k=1, 2 ,),
pξk(x)=12πρkexp-x22ρk2, (k=1, 2 ,),
pζk(x)=12πνkexp-x222νk2,(k=1, 2 ,).
pηk(y|x)=12πνkexp-(y-λkx)222νk2=12πνkexp-λk222νk2x-yλk2.
pξk(x|y)=pξk(x)pηk(y|x)pηk(y).
pξk(x|g¯k)=Akexp-x22ρk2exp-λk222νk2x-g¯kλk2.
J(ξk, ηk)=-12ln(1-rk2),
rk2=|E{ξkηk}|2E{|ξk|2}E{|ηk|2}=(λkρk)2(λkρk)2+(νk)2.
J(ξk, ηk)=12ln1+λk2ρk22νk2.
I={k : λkρkνk},N={k : λkρk<νk}.
p1(x)=Ak(1)exp(-x2/2ρk2),
p2(x)=Ak(2)exp{-(λk2/22νk2)[x-(g¯k/λk)]2},
[Ak=Ak(1)Ak(2)],
ξk=g¯k/λkkI0kN.
Bˆg¯=kIg¯kλk ψk.
lim0 E{ξ-Bˆη2}=0.
Δη(k1, k2)
=E{(ηk1-E{ηk1})(ηk2-E{ηk2})}E{(ηk1-E{ηk1})2}1/2E{(ηk2-E{ηk2})2}1/2.
δg¯(n)=k=1N-n(g¯k-g¯k)(g¯k+n-g¯k+n)[k=1N-n(g¯k-g¯k)2k=1N-n(g¯k+n-g¯k+n)2]1/2,
n=0, . . . , N-1,
g¯k=1N-nk=1N-ng¯k,g¯k+n=1N-nk=1N-ng¯k+n.
σδ(n, n0)=1N-n1+2v=1n0δg¯2(v)1/2
forn>n0.
n0=max{n¯0 : n(n¯, N-1],|δg¯(n)|<1.96σδ(n, n¯)}.
Q={0<nn0 : |δg¯(n)|>1.96σδ(n, 0)}.
Fi={(g¯ki,g¯ki+ni)}ki=1(N-ni),
niQ,i=1 ,, NQ,
ki=argmaxk[1, N-ni]{|g¯k g¯k+ni|},i=1 ,, NQ.
I={ki}1NQ{ki+ni}1NQ,
fI(x)=kIg¯kλk ψk(x).
fI(x)=m=1mI()g¯mλm ψm(x),
fI(p)(x)=fI(x)+ m=mI()+1m¯I dmψm(x),
F(d)=fI(p)(x)-fI(x)L2(-1,1)2,
d=(dmI+1 dmI+2 ,, dm¯I),
fI(p)(xi)0,i=1, 2 ,, Np,
|λmdm-g¯m|m,m=mI+1 ,, m¯I,
fn(x)=np+121/p(1-|x|)n,x[-1, 1]
(p1).
supx[-1,1]|fn(x)|=np+121/pn(p1),
fnLp =1(p1),
|Afn| constnp+121/p1n+1 .
AfL1lfL1,l=infx[-1,1]-11K(x, y)dy>0,

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