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The black-body radiation inversion problem, its instability and a new universal function set method
Physics Letters A 348 (2006) 141–146 www.elsevier.com/locate/pla
The black-body radiation inversion problem, its instability and a new universal function set method JiPing Ye a , FengMin Ji a , Tao Wen a , Xian-Xi Dai a,∗ , Ji-Xin Dai b,∗ , William E. Evenson c,1 a Research Group of Quantum Statistics and Methods of Theoretical Physics & Surface Physics Laboratory, Department of Physics,
Fudan University, Shanghai 200433, PR China b New York University, Department of Chemistry, Washington Place, New York, NY 10003, USA c Department of Physics, Utah Valley State College, Orem, UT 84058-5999, USA
Received 29 April 2005; received in revised form 17 August 2005; accepted 22 August 2005 Available online 30 August 2005 Communicated by P.R. Holland
Abstract The black-body radiation inversion (BRI) problem is ill-posed and requires special techniques to achieve stable solutions. In this Letter, the universal function set method (UFS), is developed in BRI. An improved unique existence theorem is proposed. Asymptotic behavior control (ABC) is introduced. A numerical example shows that practical calculations are possible with UFS. 2005 Elsevier B.V. All rights reserved. Keywords: Universal function set method; Instability problem; Asymptotic behavior control; Inverse black-body radiation problem
1. Introduction With advances in techniques to deal with instabilities in ill-posed inverse problems in physics, these problems have generated increasing interest over a wide range of areas, including specific heat-phonon * Corresponding authors.
E-mail addresses: [email protected] (X.-X. Dai), [email protected] (J.-X. Dai). 1 Brigham Young University, emeritus. 0375-9601/$ – see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2005.08.051
spectrum inversion (SPI) in quantum statistical mechanics, geological prospecting inverse problems in geophysics or remote sensing, inverse scattering problems in acoustics or electromagnetism, nondestructive testing problems, and medical imaging. It is well known that there are many practical difficulties in physical inverse problems, including the inherent ill-posed nature of the inversions, data incompleteness, and insufficient accuracy of data. Specific heat-phonon spectrum inversion (SPI) was first realized for a real system (YBCO), using the exact so-
142
J. Ye et al. / Physics Letters A 348 (2006) 141–146
lution formula (with parameter s for canceling divergences) and the universal function set (UFS) method, both developed in our group [1]. In this Letter, in order to overcome a series of numerical and theoretical difficulties, the UFS method is generalized and calculated in high precision to study black-body radiation inversion (BRI), in which the Riemann zeta function is defined out of the Riemann strip [2] by a different set of constraints than for SPI. The black-body radiation inversion problem (BRI) has drawn attention recently due to its potential applications in remote sensing [3]. BRI was first proposed by Bojarski in the early 1980s [4] and stated as follows: For a given or measured power spectrum, W (ν), radiated by a black-body with area-temperature distribution a(T ), ∞ W (ν) = 0
2πhν 3 a(T ) dT . c2 exp( KhνT ) − 1
(1)
B
The black-body radiation inversion problem is to obtain a(T ) by solving this integral equation. Much progress has been made in the past 20 years, and numerous important papers have been published on methods of solution of BRI problems, including iteration and Laplace transform [4–7], Fourier transform [8,9], Rayleigh–Jeans approximation [5], Wien approximation [10], the Möbius inverse method in number theory [11], and the technique for eliminating divergences [12]. The exact solution formula and the uniqueness and existence theorem were proven, and the exact solution of an inverse black-body radiation problem hierarchy with Planck integral spectrum was presented in work from our group [13,14]. An important and difficult topic in the study of BRI is to expose conditions of stability for this ill-posed problem. Regularization of ill-posed problems and stability control have been proposed via sum rules [15] (but with no concrete example) and Tikhonov’s regularization technique [16]. However, in studies of the global structure of Green’s functions, one can prove that even if all the sum rules are satisfied exactly, the differences between solutions can still be arbitrarily large [17]. The Hausdorff moment problem specifies necessary and sufficient conditions under which the result of certain types of inversions can be a function of bounded variation [18].
In this Letter we present a more general method to control the fundamental instabilities of this problem. This method is useful for practical calculations and valid for general cases, not limited to a few individual cases, as is much previous work. The method has been checked for some exact solutions in BRI, and the agreement in these admittedly artificial examples is excellent. The method proposed here is also a generalization of the UFS method developed in SPI, whose success has been demonstrated by comparing phonon spectrum inversion results for real data with those obtained from a neutron inelastic scattering experiment. The Letter is organized as follows: In Section 2, an exact solution formula and an improved uniqueness and existence theorem are presented. Asymptotic behavior control conditions are discussed, and a set of exact solutions is also introduced. In Section 3, the UFS method in BRI is developed. The universal function set is calculated to high precision. Some exactly soluble examples are presented and used as input to calculate numerical solutions. Finally, in Section 4, we discuss the results presented in the Letter and draw conclusions.
2. Improved uniqueness and existence theorem To obtain an exact solution for the inversion of Eq. (1), we introduce new variables x and y, defined by hν = kB T0 ex and T = T0 ey , with temperature scaling factor T0 . We further define the following functions, with which we transform Eq. (1) into an integral with difference kernel:
k B T0 x W e , g0 (x) = h 2πkB3 T04 A0 (y) = e4y a T0 ey , h2 c2
K0 (ξ ) =
e3ξ . exp(eξ ) − 1
(2)
By the Fourier transform convolution theorem, we obtain the exact solution formula [13,14] a(T ) =
1 2π
T0 T
4 ∞ −∞
g˜ 0 (k)( TT0 )ik dk (3 − ik)ζ (3 − ik)
,
(3)
J. Ye et al. / Physics Letters A 348 (2006) 141–146
where g˜ 0 (k) is the Fourier transform of g0 (x), and (· · ·) and ζ (· · ·) are the gamma function and Riemann zeta function, respectively. While this is an exact inversion formula, many difficulties still remain for practical calculations. The first problem is the limited precision and domain of measured power spectra W (ν). These data limitations then eliminate any possibility that the measured power spectrum could control the accuracy of g˜ 0 (k) for large k in Eq. (3). That is why one cannot in general obtain a successful solution with real data. This situation is similar to that of SPI, in which we overcame the problem by analyzing the asymptotic behaviors required by physical considerations and using those behaviors as constraints in the inversion. Therefore, in the present case we need to understand the asymptotic behavior of W (ν) or g˜ 0 (k) and find asymptotic behavior control conditions (ABC). With ABC conditions we can prove an improved unique existence theorem, as follows: When the asymptotic behavior of the radiated power, W (ν), satisfies the following necessary and sufficient condition: 1 k g˜ 0 (k) = o k 2+ 2 e−k arctan( 3 ) , k → ±∞, (4) then the inversion of Eq. (1) is the unique, exact solution (3). The condition (4) will be referred to as the asymptotic behavior control condition (ABC), a necessary and sufficient condition for the existence of a unique solution to the inversion problem. To prove the unique existence theorem, we note that the solution (3) is, in fact, an inverse Fourier transform: ∞ a(T ) = (1/2π )
˜ exp[ik ln T /T0 ] dk, φ(k)
(5)
−∞
where ˜ φ(k) ≡
T0 T
4
g˜ 0 (k) . (3 − ik)ζ (3 − ik)
(6)
The uniqueness and existence condition is the require˜ ment that φ(k) → 0, (k → ±∞). This means that as k → ±∞, g˜ 0 (k) must be an infinitesimal of higher order than (3 − ik)ζ (3 − ik). Thus we have g˜ 0 (k) = o (3 − ik) 1 k = o k 2+ 2 e−k arctan( 3 ) , k → ±∞, (7)
143
where the asymptotic representation of (z) and the bounded variation of ζ (z) have been used. This proves the improved unique existence theorem. The ABC condition is similar to the boundary conditions usually encountered in solving partial differential equations. However, the ABC condition is very special because it does not impose a constraint on W (ν) but on g˜ 0 (k), the Fourier transform of W (in the new variable, x). This may seem confusing at first, but it is, in fact, the necessary and sufficient condition for uniqueness and existence of the solution to our problem. Previous examples of this type in physics include studies of the singular state in quantum mechanics, in which pair-wise boundary conditions are found to guarantee the orthogonality of the eigenstates [19]. The ABC condition is the key point for developing the UFS.
3. The UFS method in BRI As we pointed out, the limited precision and domain of measured W (ν) make it difficult to satisfy the ABC condition directly from real data, in general. But the ABC condition is obtained from the convergence of the integral in (5) (a mathematical requirement) and the basic integral equation is dictated by the physics of the problem, so the only way to obtain satisfactory solutions is to constrain the solutions to satisfy ABC in advance. This approach leads us to the UFS in BRI: One should choose a complete set, satisfying ABC term by term, and carry out the inversion within that complete set. Although the choice of complete set for expansion is far from unique, we require one that consists of elementary functions whose Fourier transforms can be expressed exactly and that satisfy ABC naturally. One set with these properties is the Hermite function set, which we have chosen for the universal basis in the present work: 1/2 1 2 2 α e− 2 α x Hn (αx), un (x) = √ n π 2 n! n = 0, 1, . . . . (8) This set is particularly convenient because its Fourier transform can be obtained exactly and analytically: √ 1/2 2 k − k2 n 2 π 2α e Hn , u˜ n (k) = (−i) (9) α2n n! α
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J. Ye et al. / Physics Letters A 348 (2006) 141–146
where α is a parameter, and Hn (x) are the Hermite polynomials. We can expand any g0 (x) up to n terms in the form g0 (x) =
n−1
cl ul (x),
(10)
l=0
and the Fourier transform of g0 (x) can be obtained analytically, in terms of the same coefficients, cl : n−1
g˜ 0 (k) =
cl u˜ l (k).
(11)
l=0
The expansion coefficients can be obtained by the orthonormality of the Hermite function set: ∞ cl =
ul (x)g0 (x) dx.
Fig. 1. The UFS AL (T ) for L from 0 to 3, with parameters T0 = 4000, α = 1.
(12)
−∞
Hence, according to the exact solution formula, Eq. (3), and using (11), we can obtain an approximation to the area-temperature distribution a(T ) as a(T ) =
n−1
cl Al (T ),
(13)
l=0
where {Al (T )} is a (purely mathematical) universal function set (UFS), which is system independent: 1 T0 4 Al (T ) = 2π T ∞ ik u˜ l (k) dk T × (14) . T0 (3 − ik)ζ (3 − ik)
Fig. 2. The UFS AL (T ) for L from 4 to 7, with parameters T0 = 4000, α = 1.
−∞
Graphs of the first four Al (T ), l from 0 to 3, and the second four, l from 4 to 7, are shown in Figs. 1 and 2, respectively. We have calculated 50 UFS functions (l from 0 to 49) in high precision. The parameters of our UFS are T0 = 4000, α = 1, n = 50. We first test the use of UFS in BRI on some exact solutions. Fortunately, we can find a complete set of exact solutions of the form Wm (ν) =
2πhν 3 ξ µ+m+1 (µ + m + 1) c2 ν µ+m+1 ξ × ζ µ + m + 1; 1 + , 2ν
(15)
where ξ is a parameter and ζ (z, q) is the generalized Riemann zeta function (e.g., see [20]): ζ (z, q) ≡
Fig. 3. The UFS method’s inversion result shown with the analytical exact solution for the single mode Wm (ν), with m = 1, µ = 2, ξ = 6.0 × 1013 .
J. Ye et al. / Physics Letters A 348 (2006) 141–146
∞
+ q)−z , (Re(z) > 1). The exact solution of Eq. (1) for this form of power spectrum is 1 ξ h µ+m+1 ξh exp − . am (T ) = (16) T kB T 2kB T n=0 (n
Now we select a single mode of the form of Eq. (15) as input, and calculate the solution in BRI by the UFS formula Eq. (13), choosing convenient values of the parameters, e.g., µ = 2, m = 1, ξ = 6.0 × 1013 Hz. By the UFS calculated with high precision above, we obtain the coefficients, cl , using the orthonormality relation, Eq. (12). The final result obtained by UFS is shown in Fig. 3. In the primary temperature region from 200 to 1000 K, the results are in excellent agreement with the exact inversion a1 (T ) from Eq. (16).
4. Discussion and conclusions Inversion problems are difficult because they are often ill-posed and unstable, coupled with data incompleteness and limited accuracy. In our study, it is shown that these difficulties can be circumvented to some extent. For example, in previous studies of specific heatphonon spectrum inversion, the Fourier transform of the real data is required by ABC to go to zero sufficiently rapidly for large k if the inverse transform is to converge. Although we have successfully inverted real data several times in SPI, we do not yet have a real example here because of the difficulty of finding data of sufficient precision. Of course, all real data have limited precision, and even if the precision is sufficient as a function of ν, the ABC condition in k space cannot be guaranteed within the data. In particular, g˜ 0 (k) must go to zero more quickly than an exponential. This is very difficult to ensure. In our method, the ABC condition is guaranteed in advance by the expansion set, which satisfies ABC term by term. In summary, we have established the following: (1) In the UFS method, the improved unique existence theorem and asymptotic behavior control (ABC) condition play an essential role. ABC can also help us to check the validity of different calculations. Some noticeable differences are found in the low temperature region (see Fig. 3, for example) due to the limited accuracy of the calculations given the strong dependence of the solutions on 1/T (see Eq. (16), for exam-
145
ple). Exact solutions, like Eq. (16), can provide useful comparisons for evaluating inversions of realistic data. (2) In UFS almost all calculations can be finished exactly. The procedure is quite straightforward. One only needs to expand g0 (x), and the expansion coefficients can be obtained by the orthonormality relation, Eq. (10). (3) The universal function set, {Al (T )}, is system independent. It only needs to be calculated once, and it can be calculated with high precision in advance. (4) Although one cannot expect that the stability and data incompleteness problems in BRI can be solved completely, the difficulties can be reduced, as in the UFS procedure, resulting in improvements and progress for the general problem. The UFS method is a powerful approach to the black-body radiation inverse problem. The existence and uniqueness theorem demonstrates that the UFS method will be superior to any method that does not satisfy ABC. This method is suitable for real systems in BRI because it ensures compliance with the ABC condition, so it should be helpful in real applications in the future.
Acknowledgements The authors would like to acknowledge valuable support in part from the National Science Foundation of China (Project Nos. 10375012, 10174016 and 19975009).
References [1] X. Dai, T. Wen, G. Ma, J. Dai, Phys. Lett. A 264 (1999) 68; T. Wen, G. Ma, X.-X. Dai, W.E. Evenson, J. Phys.: Condens. Matter 15 (2003) 223. [2] G. Riemann, Monatsberichte Berliner Akademie (1859) 671. [3] F.T. Ulaby, R.K. Morse, A.K. Fung, Microwave Remote Sensing, I, Addison–Wesley, Reading, MA, 1981. [4] N.N. Bojarski, IEEE Trans. Antennas Propagation 30 (1982) 778. [5] M. Harmid, H.A. Ragheb, IEEE Trans. Antennas Propagation 31 (1983) 810. [6] Y. Kim, D.L. Jaggard, IEEE Trans. Antennas Propagation 33 (1985) 797. [7] J.D. Hunter, IEEE Trans. Antennas Propagation 34 (1986) 261. [8] N.X. Chen, Chinese Phys. Lett. 4 (1987) 337. [9] N.X. Chen, et al., Acta Electron. Sinica 17 (1987) 59.
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[10] M.N. Lakhtakia, A. Lakhtakia, IEEE. Trans. Antennas Propagation 32 (1984) 972. [11] N.X. Chen, Phys. Rev. Lett. 64 (1990) 1193. [12] X. Dai, X. Xu, J. Dai, Phys. Lett. A 147 (1990) 445. [13] X. Dai, J. Dai, Phys. Lett. A 161 (1991) 45. [14] X. Dai, J. Dai, IEEE Trans. Antennas Propagation 40 (1992) 257. [15] M.N. Lakhtakia, A. Lakhtakia, Appl. Phys. B 39 (1986) 191.
[16] X. Sun, D.L. Jaggard, J. Appl. Phys. 82 (1987) 110, 4382. [17] X. Dai, Commun. Theor. Phys. 3 (1984) 51. [18] T.W. Körner, Fourier Analysis, Cambrige Univ. Press, Cambrige, 1988 (Chaper 6). [19] X. Dai, J. Dai, J. Dai, Phys. Rev. A 55 (1997) 2617. [20] I.S. Gradshtyen, I.M. Ryzhik, Tables of Integrals, Series and Products, Academic Press, New York, 1980.
The black-body radiation inversion problem, its instability and a new universal function set method JiPing Ye a , FengMin Ji a , Tao Wen a , Xian-Xi Dai a,∗ , Ji-Xin Dai b,∗ , William E. Evenson c,1 a Research Group of Quantum Statistics and Methods of Theoretical Physics & Surface Physics Laboratory, Department of Physics,
Fudan University, Shanghai 200433, PR China b New York University, Department of Chemistry, Washington Place, New York, NY 10003, USA c Department of Physics, Utah Valley State College, Orem, UT 84058-5999, USA
Received 29 April 2005; received in revised form 17 August 2005; accepted 22 August 2005 Available online 30 August 2005 Communicated by P.R. Holland
Abstract The black-body radiation inversion (BRI) problem is ill-posed and requires special techniques to achieve stable solutions. In this Letter, the universal function set method (UFS), is developed in BRI. An improved unique existence theorem is proposed. Asymptotic behavior control (ABC) is introduced. A numerical example shows that practical calculations are possible with UFS. 2005 Elsevier B.V. All rights reserved. Keywords: Universal function set method; Instability problem; Asymptotic behavior control; Inverse black-body radiation problem
1. Introduction With advances in techniques to deal with instabilities in ill-posed inverse problems in physics, these problems have generated increasing interest over a wide range of areas, including specific heat-phonon * Corresponding authors.
E-mail addresses: [email protected] (X.-X. Dai), [email protected] (J.-X. Dai). 1 Brigham Young University, emeritus. 0375-9601/$ – see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2005.08.051
spectrum inversion (SPI) in quantum statistical mechanics, geological prospecting inverse problems in geophysics or remote sensing, inverse scattering problems in acoustics or electromagnetism, nondestructive testing problems, and medical imaging. It is well known that there are many practical difficulties in physical inverse problems, including the inherent ill-posed nature of the inversions, data incompleteness, and insufficient accuracy of data. Specific heat-phonon spectrum inversion (SPI) was first realized for a real system (YBCO), using the exact so-
142
J. Ye et al. / Physics Letters A 348 (2006) 141–146
lution formula (with parameter s for canceling divergences) and the universal function set (UFS) method, both developed in our group [1]. In this Letter, in order to overcome a series of numerical and theoretical difficulties, the UFS method is generalized and calculated in high precision to study black-body radiation inversion (BRI), in which the Riemann zeta function is defined out of the Riemann strip [2] by a different set of constraints than for SPI. The black-body radiation inversion problem (BRI) has drawn attention recently due to its potential applications in remote sensing [3]. BRI was first proposed by Bojarski in the early 1980s [4] and stated as follows: For a given or measured power spectrum, W (ν), radiated by a black-body with area-temperature distribution a(T ), ∞ W (ν) = 0
2πhν 3 a(T ) dT . c2 exp( KhνT ) − 1
(1)
B
The black-body radiation inversion problem is to obtain a(T ) by solving this integral equation. Much progress has been made in the past 20 years, and numerous important papers have been published on methods of solution of BRI problems, including iteration and Laplace transform [4–7], Fourier transform [8,9], Rayleigh–Jeans approximation [5], Wien approximation [10], the Möbius inverse method in number theory [11], and the technique for eliminating divergences [12]. The exact solution formula and the uniqueness and existence theorem were proven, and the exact solution of an inverse black-body radiation problem hierarchy with Planck integral spectrum was presented in work from our group [13,14]. An important and difficult topic in the study of BRI is to expose conditions of stability for this ill-posed problem. Regularization of ill-posed problems and stability control have been proposed via sum rules [15] (but with no concrete example) and Tikhonov’s regularization technique [16]. However, in studies of the global structure of Green’s functions, one can prove that even if all the sum rules are satisfied exactly, the differences between solutions can still be arbitrarily large [17]. The Hausdorff moment problem specifies necessary and sufficient conditions under which the result of certain types of inversions can be a function of bounded variation [18].
In this Letter we present a more general method to control the fundamental instabilities of this problem. This method is useful for practical calculations and valid for general cases, not limited to a few individual cases, as is much previous work. The method has been checked for some exact solutions in BRI, and the agreement in these admittedly artificial examples is excellent. The method proposed here is also a generalization of the UFS method developed in SPI, whose success has been demonstrated by comparing phonon spectrum inversion results for real data with those obtained from a neutron inelastic scattering experiment. The Letter is organized as follows: In Section 2, an exact solution formula and an improved uniqueness and existence theorem are presented. Asymptotic behavior control conditions are discussed, and a set of exact solutions is also introduced. In Section 3, the UFS method in BRI is developed. The universal function set is calculated to high precision. Some exactly soluble examples are presented and used as input to calculate numerical solutions. Finally, in Section 4, we discuss the results presented in the Letter and draw conclusions.
2. Improved uniqueness and existence theorem To obtain an exact solution for the inversion of Eq. (1), we introduce new variables x and y, defined by hν = kB T0 ex and T = T0 ey , with temperature scaling factor T0 . We further define the following functions, with which we transform Eq. (1) into an integral with difference kernel:
k B T0 x W e , g0 (x) = h 2πkB3 T04 A0 (y) = e4y a T0 ey , h2 c2
K0 (ξ ) =
e3ξ . exp(eξ ) − 1
(2)
By the Fourier transform convolution theorem, we obtain the exact solution formula [13,14] a(T ) =
1 2π
T0 T
4 ∞ −∞
g˜ 0 (k)( TT0 )ik dk (3 − ik)ζ (3 − ik)
,
(3)
J. Ye et al. / Physics Letters A 348 (2006) 141–146
where g˜ 0 (k) is the Fourier transform of g0 (x), and (· · ·) and ζ (· · ·) are the gamma function and Riemann zeta function, respectively. While this is an exact inversion formula, many difficulties still remain for practical calculations. The first problem is the limited precision and domain of measured power spectra W (ν). These data limitations then eliminate any possibility that the measured power spectrum could control the accuracy of g˜ 0 (k) for large k in Eq. (3). That is why one cannot in general obtain a successful solution with real data. This situation is similar to that of SPI, in which we overcame the problem by analyzing the asymptotic behaviors required by physical considerations and using those behaviors as constraints in the inversion. Therefore, in the present case we need to understand the asymptotic behavior of W (ν) or g˜ 0 (k) and find asymptotic behavior control conditions (ABC). With ABC conditions we can prove an improved unique existence theorem, as follows: When the asymptotic behavior of the radiated power, W (ν), satisfies the following necessary and sufficient condition: 1 k g˜ 0 (k) = o k 2+ 2 e−k arctan( 3 ) , k → ±∞, (4) then the inversion of Eq. (1) is the unique, exact solution (3). The condition (4) will be referred to as the asymptotic behavior control condition (ABC), a necessary and sufficient condition for the existence of a unique solution to the inversion problem. To prove the unique existence theorem, we note that the solution (3) is, in fact, an inverse Fourier transform: ∞ a(T ) = (1/2π )
˜ exp[ik ln T /T0 ] dk, φ(k)
(5)
−∞
where ˜ φ(k) ≡
T0 T
4
g˜ 0 (k) . (3 − ik)ζ (3 − ik)
(6)
The uniqueness and existence condition is the require˜ ment that φ(k) → 0, (k → ±∞). This means that as k → ±∞, g˜ 0 (k) must be an infinitesimal of higher order than (3 − ik)ζ (3 − ik). Thus we have g˜ 0 (k) = o (3 − ik) 1 k = o k 2+ 2 e−k arctan( 3 ) , k → ±∞, (7)
143
where the asymptotic representation of (z) and the bounded variation of ζ (z) have been used. This proves the improved unique existence theorem. The ABC condition is similar to the boundary conditions usually encountered in solving partial differential equations. However, the ABC condition is very special because it does not impose a constraint on W (ν) but on g˜ 0 (k), the Fourier transform of W (in the new variable, x). This may seem confusing at first, but it is, in fact, the necessary and sufficient condition for uniqueness and existence of the solution to our problem. Previous examples of this type in physics include studies of the singular state in quantum mechanics, in which pair-wise boundary conditions are found to guarantee the orthogonality of the eigenstates [19]. The ABC condition is the key point for developing the UFS.
3. The UFS method in BRI As we pointed out, the limited precision and domain of measured W (ν) make it difficult to satisfy the ABC condition directly from real data, in general. But the ABC condition is obtained from the convergence of the integral in (5) (a mathematical requirement) and the basic integral equation is dictated by the physics of the problem, so the only way to obtain satisfactory solutions is to constrain the solutions to satisfy ABC in advance. This approach leads us to the UFS in BRI: One should choose a complete set, satisfying ABC term by term, and carry out the inversion within that complete set. Although the choice of complete set for expansion is far from unique, we require one that consists of elementary functions whose Fourier transforms can be expressed exactly and that satisfy ABC naturally. One set with these properties is the Hermite function set, which we have chosen for the universal basis in the present work: 1/2 1 2 2 α e− 2 α x Hn (αx), un (x) = √ n π 2 n! n = 0, 1, . . . . (8) This set is particularly convenient because its Fourier transform can be obtained exactly and analytically: √ 1/2 2 k − k2 n 2 π 2α e Hn , u˜ n (k) = (−i) (9) α2n n! α
144
J. Ye et al. / Physics Letters A 348 (2006) 141–146
where α is a parameter, and Hn (x) are the Hermite polynomials. We can expand any g0 (x) up to n terms in the form g0 (x) =
n−1
cl ul (x),
(10)
l=0
and the Fourier transform of g0 (x) can be obtained analytically, in terms of the same coefficients, cl : n−1
g˜ 0 (k) =
cl u˜ l (k).
(11)
l=0
The expansion coefficients can be obtained by the orthonormality of the Hermite function set: ∞ cl =
ul (x)g0 (x) dx.
Fig. 1. The UFS AL (T ) for L from 0 to 3, with parameters T0 = 4000, α = 1.
(12)
−∞
Hence, according to the exact solution formula, Eq. (3), and using (11), we can obtain an approximation to the area-temperature distribution a(T ) as a(T ) =
n−1
cl Al (T ),
(13)
l=0
where {Al (T )} is a (purely mathematical) universal function set (UFS), which is system independent: 1 T0 4 Al (T ) = 2π T ∞ ik u˜ l (k) dk T × (14) . T0 (3 − ik)ζ (3 − ik)
Fig. 2. The UFS AL (T ) for L from 4 to 7, with parameters T0 = 4000, α = 1.
−∞
Graphs of the first four Al (T ), l from 0 to 3, and the second four, l from 4 to 7, are shown in Figs. 1 and 2, respectively. We have calculated 50 UFS functions (l from 0 to 49) in high precision. The parameters of our UFS are T0 = 4000, α = 1, n = 50. We first test the use of UFS in BRI on some exact solutions. Fortunately, we can find a complete set of exact solutions of the form Wm (ν) =
2πhν 3 ξ µ+m+1 (µ + m + 1) c2 ν µ+m+1 ξ × ζ µ + m + 1; 1 + , 2ν
(15)
where ξ is a parameter and ζ (z, q) is the generalized Riemann zeta function (e.g., see [20]): ζ (z, q) ≡
Fig. 3. The UFS method’s inversion result shown with the analytical exact solution for the single mode Wm (ν), with m = 1, µ = 2, ξ = 6.0 × 1013 .
J. Ye et al. / Physics Letters A 348 (2006) 141–146
∞
+ q)−z , (Re(z) > 1). The exact solution of Eq. (1) for this form of power spectrum is 1 ξ h µ+m+1 ξh exp − . am (T ) = (16) T kB T 2kB T n=0 (n
Now we select a single mode of the form of Eq. (15) as input, and calculate the solution in BRI by the UFS formula Eq. (13), choosing convenient values of the parameters, e.g., µ = 2, m = 1, ξ = 6.0 × 1013 Hz. By the UFS calculated with high precision above, we obtain the coefficients, cl , using the orthonormality relation, Eq. (12). The final result obtained by UFS is shown in Fig. 3. In the primary temperature region from 200 to 1000 K, the results are in excellent agreement with the exact inversion a1 (T ) from Eq. (16).
4. Discussion and conclusions Inversion problems are difficult because they are often ill-posed and unstable, coupled with data incompleteness and limited accuracy. In our study, it is shown that these difficulties can be circumvented to some extent. For example, in previous studies of specific heatphonon spectrum inversion, the Fourier transform of the real data is required by ABC to go to zero sufficiently rapidly for large k if the inverse transform is to converge. Although we have successfully inverted real data several times in SPI, we do not yet have a real example here because of the difficulty of finding data of sufficient precision. Of course, all real data have limited precision, and even if the precision is sufficient as a function of ν, the ABC condition in k space cannot be guaranteed within the data. In particular, g˜ 0 (k) must go to zero more quickly than an exponential. This is very difficult to ensure. In our method, the ABC condition is guaranteed in advance by the expansion set, which satisfies ABC term by term. In summary, we have established the following: (1) In the UFS method, the improved unique existence theorem and asymptotic behavior control (ABC) condition play an essential role. ABC can also help us to check the validity of different calculations. Some noticeable differences are found in the low temperature region (see Fig. 3, for example) due to the limited accuracy of the calculations given the strong dependence of the solutions on 1/T (see Eq. (16), for exam-
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ple). Exact solutions, like Eq. (16), can provide useful comparisons for evaluating inversions of realistic data. (2) In UFS almost all calculations can be finished exactly. The procedure is quite straightforward. One only needs to expand g0 (x), and the expansion coefficients can be obtained by the orthonormality relation, Eq. (10). (3) The universal function set, {Al (T )}, is system independent. It only needs to be calculated once, and it can be calculated with high precision in advance. (4) Although one cannot expect that the stability and data incompleteness problems in BRI can be solved completely, the difficulties can be reduced, as in the UFS procedure, resulting in improvements and progress for the general problem. The UFS method is a powerful approach to the black-body radiation inverse problem. The existence and uniqueness theorem demonstrates that the UFS method will be superior to any method that does not satisfy ABC. This method is suitable for real systems in BRI because it ensures compliance with the ABC condition, so it should be helpful in real applications in the future.
Acknowledgements The authors would like to acknowledge valuable support in part from the National Science Foundation of China (Project Nos. 10375012, 10174016 and 19975009).
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