Blaschke Decompositions on Weighted Hardy Spaces

Stephen D. Farnham

REVIEW 4 major objections 5 minor 11 references

Blaschke Decompositions on Weighted Hardy Spaces

T0 review · 4 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read With bounded, concave weights whose tail sums converge, every Hardy-space function satisfies a sharper Blaschke-decomposition inequality: the norm of the zero-free factor is at most the norm of the original function minus a convergent sum…

desk verdict A genuine but narrowly scoped extension of Coifman–Steinerberger whose main theorems are plausible, yet the written proof has a sign error in the key coefficient identity and an invalid limit exchange; referee it, but ask for repairs. read the letter →

arxiv 1908.04665 v1 pith:CB7USSU2 submitted 2019-08-13 math.CV

Stephen D. Farnham This is my paper · ORCID

classification math.CV MSC 30B30J30H

keywords BlaschkedecompositionweightedHardyspaceHardy-SobolevDirichletunwindingseriesadaptiveFourierproduct

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper studies what happens to a function's norm when it is factored as a Blaschke product times a zero-free function, $F=B\cdot G$, in weighted Hardy spaces whose Fourier coefficients are weighted by a sequence $\gamma_n$. Its central claim is that, for bounded, increasing, concave weights that approach their limit fast enough, every $F$ in the ordinary Hardy space $H^2$ obeys an explicit norm drop: the weighted norm of $G$ is at most the weighted norm of $F$ minus a convergent sum over the zeros of $F$. The same telescoping machinery gives sharper bounds for two large weight classes when the zero set is finite: increasing differences (convex weights) pull the bound from $G$, while decreasing differences (concave weights) pull it from $F$. This extends earlier bounds that required analyticity in a strictly larger disk, and it supplies a quantitative mechanism behind the rapid convergence seen in iterated Blaschke decompositions.

What carries the argument

The central object is the single-zero reflection operator $\varphi_\alpha$: when $F=(\cdot-\alpha)H_\alpha$, it sends $F$ to $(1-\overline{\alpha}\cdot)H_\alpha$, reflecting the zero across the unit circle. Applied one zero at a time, it produces the partial decompositions $F_k$ and converts the norm drop into an exactly telescoping sum over zeros. The monotonicity of the difference sequence $\Gamma_n=\gamma_{n+1}-\gamma_n$ decides which endpoint of the intermediate $Y_\gamma$ norms is usable, and the tail condition $\sum_n(M-\gamma_n)<\infty$ makes the infinite collection of contributions summable. This reflection mechanism is what carries the finite-zero results all the way to $H^2$.

What would settle it

Take $\gamma_n = 1 - 2^{-n}$ and let $F$ be an infinite Blaschke product with zeros $\alpha_j = 1 - 2^{-j}$, so $G=1$. After reflecting the first $m$ zeros, the residual function is a tail Blaschke product; the theorem requires its $X_\gamma$ norm to tend to $0$ as $m\to\infty$. Numerically computing these tail-product norms for increasing $m$ and finding a nonvanishing limit would refute Theorem 6.

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Extended reading notes

Core claim

For a single zero the paper proves an exact reflection identity. If $F(z)=(z-\alpha)H(z)$, then replacing the factor $(z-\alpha)$ by $(1-\overline{\alpha}z)$ changes the weighted norm by $\|\varphi_\alpha(F)\|^2_{X_\gamma} = \|F\|^2_{X_\gamma} - (1-|\alpha|^2)\|H\|^2_{Y_\gamma}$, where $Y_\gamma$ is the weighted space with weights $\gamma_{n+1}-\gamma_n$. Reflecting one zero at a time telescopes to $\|G\|^2_{X_\gamma} = \|F\|^2_{X_\gamma} - \sum_j (1-|\alpha_j|^2)\|H_{\alpha_j}\|^2_{Y_\gamma}$. When the differences $\Gamma_n=\gamma_{n+1}-\gamma_n$ are monotone increasing or decreasing, each intermediate $H_{\alpha_j}$ term is bounded between the $G$-based and $F$-based expressions, which yields the finite-zero inequalities of Theorems 2 and 5. The final result, Theorem 6, lets the zero set be infinite: for $\gamma_n\uparrow M$ with $\Gamma_n$ decreasing and $\sum_n (M-\gamma_n)<\infty$, every $F\in H^2$ has $\sum_j (1-|\alpha_j|^2)\|F(e^{it})/(e^{it}-\alpha_j)\|^2_{Y_\gamma}<\infty$ and $\|G\|^2_{X_\gamma} \le \|F\|^2_{X_\gamma} - \sum_j (1-|\alpha_j|^2)\|F(e^{it})/(e^{it}-\alpha_j)\|^2_{Y_\gamma}$.

Load-bearing premise

The result needs the norm of the leftover function after many zero-reflections to converge to the norm of the zero-free factor; the printed argument assumes a limit exchange that is not automatic.

Editorial extensions

If this is right

For every function in $H^2$ with bounded, concave, fast-tending weights, the infinite zero-sum in Theorem 6 is finite, so the weighted norm of the zero-free factor is reduced by an explicit contribution from each zero.
In the constant-difference case, the inequality becomes an identity, recovering the exact Dirichlet-space formula for the norm of the zero-free factor.
The Hardy-Sobolev bound follows by decomposing the $W^{1,2}$ norm into $X_\gamma$ and $H^2$ pieces with $\gamma_n=n^2$, giving an explicit correction involving Dirichlet- and $H^2$-type terms.
The finite-zero results cover convex weights such as $\gamma_n=n$ and $\gamma_n=n^2$, hence Dirichlet and Hardy-Sobolev spaces, as well as concave weights such as partial sums of $1/k^\beta$, hence weighted Bergman spaces.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The infinite-zero limit can be justified by dominated convergence on the boundary, where the partial decompositions have the same modulus as the zero-free factor; this supplies a direct route to the norm-convergence step.
The tail condition $\sum_n(M-\gamma_n)<\infty$ is structurally similar to the Blaschke condition on zeros, hinting at a trade-off: slow-tending weights should make the zero-sum diverge for some $H^2$ functions, and locating that threshold would sharpen the theorem.
The identity behind equation (14) suggests a practical convergence-rate estimate: the norm removed at a Blaschke step is computable from derivative data of the current remainder at the zero, which could be used to predict how many unwinding terms a signal needs.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Quick take: this is a real extension of the Coifman–Steinerberger weighted-Hardy estimates, and the coefficient-reflection idea is sensible. But the manuscript as written has proof gaps that are more basic than the reader's report suggests. I would send it to a referee, not desk-reject it, and not accept it as is.

What's new: Theorems 2, 5, 6 and Corollary 4 genuinely go beyond [2] by treating γ with convex or concave increments and, under boundedness plus the summability condition, functions in H² with infinitely many zeros. The framework is inherited, but the results are not restatements; Corollary 3 recovers the Dirichlet identity as a special case. The coefficient-based strategy is the right way to see why the norm drops. There is no circularity and no fitted-parameter inflation. Credit where due.

Soft spots, in increasing order of seriousness.

1. Identity (14) in the introduction is false as stated. For F(z)=z−a and k=1, the claimed correction gives (1−|a|²)(1+|a|²)² instead of 1−|a|². It is advertised as an improvement of Qian's tail inequality but is not used later; it can be removed or repaired, but as printed it is wrong.

2. Lemma 14's proof exchanges sup_{r<1} and lim_m. Uniform convergence on compact subsets gives lim_m∫|Fm−G|²=0 for each fixed r, not lim_m sup_r. The conclusion is probably salvageable because |Fm|=|G| on the boundary and γ is bounded, but the submitted argument is not valid.

3. More serious: Proposition 9, the engine for everything, has a sign error in the finite-difference computation. The algebra gives (1−|α|²)(Σ_{j=0}^{N−1}Γ_j|a_j|² − γ_N|a_N|²), not with a plus sign. The subsequent proof that γ_N|a_N|²→0 assumes coefficients are eventually nonzero and the ratio a_{n−1}/a_n converges; neither is guaranteed for H² functions with zeros. So the zero-counting identity (34) and the finite-zero Theorems 2 and 5 rest on an unproved limit. The gap is likely repairable—this is not a counterexample to the theorems—but it is load-bearing.

The citation pattern is fine; the paper builds openly on [2], [7], and [9], and the debt is acknowledged. No invented entities.

Who should read it: people working on unwinding series, adaptive Fourier decomposition, and weighted Hardy-space inequalities. If the author fixes Proposition 9 and Lemma 14, the paper becomes a solid contribution. As it stands, it is a conditional accept with requested revision, not a reject. I would bring it to a reading group, but I would not cite it until the corrections are in the arXiv version.

Referee Report

4 major / 5 minor

Summary. The paper studies the behavior of the Blaschke decomposition F = B·G under weighted Hardy norms. It defines spaces Xγ and Yγ through coefficient weights γ_n and Γ_n = γ_{n+1} - γ_n, and Proposition 9 asserts that reflecting a single zero α of F across the unit circle changes the Xγ norm by exactly (1 - |α|²)‖Hα‖²_Yγ. Iterating this identity and comparing the intermediate functions Hα_j with either G/(1 - \bar α_j z) or F/(z - α_j) yields Theorems 2 and 5 for functions in Xγ with finitely many zeros, under convex (Γ increasing) and concave (Γ decreasing) weight sequences. Theorem 6 extends the concave-weight inequality to arbitrary F ∈ H² with infinitely many zeros, assuming γ_n is bounded, increasing, concave, and satisfies Σ(M - γ_n) < ∞. The paper also claims an identity (14) that would improve Qian's tail inequality for coefficient magnitudes.

Significance. If the main theorems hold, they extend the Coifman-Steinerberger bounds from functions analytic in a larger disk to the full Hardy space H², with explicit quantitative control for Dirichlet, Hardy-Sobolev, and weighted Bergman spaces, and they provide a sufficient condition for the infinite-zero case. The coefficient-based strategy is transparent, parameter-free, and the overall plan is convincing. However, the current manuscript contains a false displayed identity and several invalid proof steps in the engine of the paper, so the results are not yet established as written. The gaps appear repairable: Proposition 9's statement is true and admits a rigorous coefficient-tail proof, and Lemma 14 can be fixed by a boundary-modulus argument.

major comments (4)

[Section 3.1, Proposition 9] The proof contains a sign error in the finite-difference calculation. Expanding the displayed difference gives (1 - |α|²)(Σ_{j=0}^{N-1} Γ_j |a_j|² - γ_N |a_N|²), with a minus sign before γ_N |a_N|², not the plus sign printed. The subsequent argument that γ_N |a_N|² → 0, based on the claim that |a_{n-1}/a_n - α| cannot tend to 0, is invalid when some coefficients a_n vanish or when the ratio does not converge; both situations are compatible with Hα ∈ H². Since Proposition 9 is used to prove Lemma 12, identity (34), and Theorems 2, 5, and 6, this proof must be repaired. The statement itself is true: one can write a_n as a tail sum of the coefficients of F and use dominated convergence to show γ_n |a_n|² → 0.
[Section 3.3, Lemma 14] The proof exchanges sup_{0<r<1} and lim_m without justification, writing that sup_r lim_m ∫|F_m - G|² = 0 implies lim_m sup_r ∫|F_m - G|² = 0. This exchange is invalid as stated. A correct proof should use that |F_m| = |G| on the boundary (each F_m is G multiplied by a Blaschke product), so the H² norms are equal and the locally uniform convergence gives strong convergence in H², for instance via weak convergence plus norm equality. Lemma 14 is load-bearing because it supplies identity (41) used to pass the limit in the proof of Theorem 6.
[Section 1.2, Eq. (14)] The stated identity is false. For F(z) = z - a with |a| < 1 and k = 1, the left side is |a|², while the right side equals 1 - (1 - |a|²)^3. The correct correction term for the step weight (13) is the Yγ norm of H_{α_j} = F_j/(1 - \bar α_j z), that is, the squared modulus of the coefficient of z^{k-1} in that function, not the derivative of F/∏(1 - \bar α_j z) displayed in (14). This claim should be corrected or removed.
[Section 3.2, Lemma 13, infinite zero case] The passage from monotone sequences of Yγ norms to the claimed inequalities for m = ∞ is not fully justified. In part 1, Fatou's lemma supplies the needed lower semicontinuity, but in part 2 the sequence of norms is increasing and one must additionally show it is bounded by the Yγ norm of the endpoint, for example by weak convergence in the weighted ℓ² space. Without such an argument, the upper bound ‖Hα_j‖²_Yγ ≤ ‖G/(1 - \bar α_j z)‖²_Yγ used in the proof of Theorem 6 is not established for infinite zero sets.

minor comments (5)

[Throughout] There are several citation and numbering slips: in Section 3.2.1 the proof says 'Corollary 2' where Corollary 3 is meant; the proof of Corollary 4 invokes 'Theorem 1' instead of Theorem 2; and the proof of Theorem 6 refers to 'Proposition 12' instead of Lemma 12.
[Lemma 13 proof] The proof begins with the product ∏_{j=0}^m (α_j - z)/(1 - \bar α_j z), but α_0 is not defined in the enumeration; this is presumably an indexing typo.
[Section 3.2] The text says 'we can treat Yγ as XΓ', but Definition 1 requires γ_0 = 0 for Xγ, while Γ_0 = γ_1 is generally positive. The intended statement is that the algebra of Proposition 9 extends to weights with positive initial value, which is true but should be stated explicitly.
[Lemma 15] In the case |α| ≥ 1/2, the pole should be β = 1/\bar α rather than 1/α, and the lower bound 1/3 for the ratio (Σ_{k=0}^n |β|^{2k})/|β|^{2n+2} is not correct as stated; the natural bound is 1/|β|² ≥ 1/4. These issues do not affect the finiteness conclusion.
[Eq. (14)] The notation d/dk appears to be a typo for d^k/dz^k, and the factor 1/k! is dimensionally inconsistent with a coefficient formula; if a corrected identity is provided, the normalization should be 1/((k-1)!) for the coefficient of z^{k-1}.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main estimates are derived from coefficient calculations rather than from their conclusions or from self-citation.

full rationale

The paper's central results, Theorem 2, Theorem 5, and Theorem 6, are obtained from Proposition 9, which computes the difference between the weighted norms of F and phi_alpha(F) in terms of the Fourier coefficients of H_alpha. This is a direct coefficient identity: from F(z)=(z-alpha)H_alpha(z) and phi_alpha(F)(z)=(1-\bar{alpha}z)H_alpha(z), the proof derives ||F||_Xgamma^2 - ||phi_alpha(F)||_Xgamma^2 = (1-|alpha|^2)||H_alpha||_Ygamma^2. No fitted parameter is introduced, and no target inequality is assumed in the calculation. The later identities and inequalities, including equation (34), Lemma 12, Lemma 13, and the final theorems, are obtained by iterating this same coefficient identity and by comparing Ygamma norms under the stated monotonicity conditions; they are not restatements of the conclusions. The paper cites Coifman-Steinerberger and Qian for context, motivation, and some known inequalities, but these citations are not used as the sole justification of the paper's new claims, and the author does not cite himself in a load-bearing way. A reviewer-identified proof gap in Lemma 14, concerning the interchange of a supremum over r and a limit in m, is a correctness concern about the internal argument, not an instance of circular reasoning; it does not make the derivation equivalent to its inputs by definition. Accordingly, no circular step can be exhibited and the circularity score is 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper introduces no fitted parameters and no new entities. The only inputs are the hypothesis on the weight sequence γ_n, which is a theorem hypothesis rather than a fitted value, and standard Hardy-space background. The main gap is the missing justification of H2 convergence in Lemma 14, and the false identity (14).

assumptions (5)

standard math Blaschke factorization theorem: each F in Hp with zeros satisfying the Blaschke condition factors as F = B·G, with B an inner Blaschke product and G zero-free in Hp.
Invoked in Section 2.1 around Equation (19) and used as the basis for all decompositions.
standard math Convergence of finite partial Blaschke decompositions to G uniformly on compact subsets of the disk.
Assumed as background from Ricci [9] in Section 3.3; needed for the infinite-root case.
standard math Qian's tail inequality: for F=B·G, sum_{n≥k}|b_n|^2 ≤ sum_{n≥k}|a_n|^2.
Cited in Section 1 from [7]; used as the comparison benchmark for identity (14).
standard math Boundary dominated convergence for Blaschke products: partial products have unit modulus on ∂D, so |F_m|=|G| and H2 norm convergence follows.
This is the missing justification in Lemma 14; the paper only gives compact convergence and an invalid limit swap.
standard math Shapiro-Shields criterion controlling which zero sets are allowed in a weighted Hardy space Xγ.
Mentioned in Section 1.2 to explain why Theorem 2 is restricted to finitely many zeros; not used in the main proofs.

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Cite this review

Pith. "Pith review of Blaschke Decompositions on Weighted Hardy Spaces." pith.science (2026). https://pith.science/paper/CB7USSU2

@misc{pith2026190804665,
  author       = {Pith},
  title        = {Pith review of: Blaschke Decompositions on Weighted Hardy Spaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CB7USSU2}},
  note         = {Machine review of arXiv:1908.04665}
}

abstract

Recently, several papers have considered a nonlinear analogue of Fourier series in signal analysis, referred to as either nonlinear phase unwinding or adaptive Fourier decomposition. In these processes, a signal is represented as the real component of a complex function $F: \partial\mathbb{D}\to \mathbb{C}$, and by performing an iterative method to obtain a sequence of Blaschke decompositions, the signal can be approximated using only a few terms. To better understand the convergence of these methods, the study of Blaschke decompositions on weighted Hardy spaces was studied by Coifman and Steinerberger, under the assumption that the complex valued function $F$ has an analytic extension to $\mathbb{D}_{1+\epsilon}$ for some $\epsilon>0$. This provided bounds on weighted Hardy norms involving a function and its Blaschke decomposition. That work also noted that in many examples, the nonlinear unwinding series of a function converges at an exponential rate to the original signal, which when coupled with an efficient algorithm to perform a Blaschke decomposition, has lead to a new and efficient way to approximate signals. In this work, we continue the study of Blaschke decompositions on weighted Hardy Spaces for functions in the larger space $\mathcal{H}^2(\mathbb{D})$ under the assumption that the function has finitely many roots in $\mathbb{D}$. By studying the growth rate of the weights, we improve the bounds provided by Coifman and Steinerberger. This provides us with new insights into Blaschke decompositions on classical function spaces including the Hardy-Sobolev spaces and weighted Bergman spaces. Further, we state a sufficient condition on the weights for our improved bounds to hold for any function in the Hardy space, $\mathcal{H}^2(\mathbb{D})$. These results may help to better explain why the exponential convergence of the unwinding series is seen in many numerical examples.

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Reference graph

Works this paper leans on

11 extracted references · 11 canonical work pages

[2]

Coifman, S

R. Coifman, S. Steinerberger , Nonlinear Phase Unwinding of Functions , J. Fourier Anal. Appl, 23, pp. 778–809 (2017)

work page 2017
[7]

T. Qian. Adaptive Fourier decomposition, rational approximation, part 1: theory, International Jour- nal of Wavelets, Multiresolution and Information Processi ng, 12 (05) (2014)

work page 2014
[9]

Ricci , Hardy Spaces in One Complex Variable , Lecture Notes Scuola Normale Superiore di Pisa http://homepage.sns.it/fricci/papers/hardy.pdf (2004-2005)

F. Ricci , Hardy Spaces in One Complex Variable , Lecture Notes Scuola Normale Superiore di Pisa http://homepage.sns.it/fricci/papers/hardy.pdf (2004-2005)

work page 2004
[1]

Coifman, J

R. Coifman, J. Peyriere , Phase Unwinding, or Invariant Subspace Decompositions of H ardy Spaces, J. Fourier Anal. Appl, 25 (3), pp. 684–695 (2019)

work page 2019
[3]

Coifman, S

R. Coifman, S. Steinerberger, H. Wu , Carrier Frequencies, Holomorphy, and Unwinding , SIAM Journal on Mathematical Analysis 49 (6), pp. 4838–4864 (201 8)

work page
[4]

Y. Gao, M. Ku, T. Qian, J. W ang , FFT formulations of adaptive Fourier decomposition , Journal of Computational and Applied Mathematics. 324 , pp. 204–215 (2017)

work page 2017
[5]

Nahon , Phase Evaluation and Segmentation , Ph.D

M. Nahon , Phase Evaluation and Segmentation , Ph.D. Thesis,Yale University (2000)

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[6]

Qian , Intrinsic Mono-component Decomposition of Functions: an a dvance of Fourier theory , Math

T. Qian , Intrinsic Mono-component Decomposition of Functions: an a dvance of Fourier theory , Math. Methods Appl. Sci. 33 (7), pp. 880–891 (2010)

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[8]

W. Qu, P. Dang , Rational Approximation in a Class of Weighted Hardy Spaces Complex Analysis and Operator Theory Volume 13, Issue 4, pp. 1827–1852 (2019)

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Shapiro, A

H. Shapiro, A. Shields , On the Zeros of Functions with Finite Dirichlet Integral and S ome Related Function Spaces, Math. Zeitschr. 80, pp. 217–229 (1962)

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A. Zygmund , Trigonometric Series, Third Edition , Cambridge University Press (2002) 19

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [2]

Coifman, S

R. Coifman, S. Steinerberger , Nonlinear Phase Unwinding of Functions , J. Fourier Anal. Appl, 23, pp. 778–809 (2017)

work page 2017

[2] [7]

T. Qian. Adaptive Fourier decomposition, rational approximation, part 1: theory, International Jour- nal of Wavelets, Multiresolution and Information Processi ng, 12 (05) (2014)

work page 2014

[3] [9]

Ricci , Hardy Spaces in One Complex Variable , Lecture Notes Scuola Normale Superiore di Pisa http://homepage.sns.it/fricci/papers/hardy.pdf (2004-2005)

F. Ricci , Hardy Spaces in One Complex Variable , Lecture Notes Scuola Normale Superiore di Pisa http://homepage.sns.it/fricci/papers/hardy.pdf (2004-2005)

work page 2004

[4] [1]

Coifman, J

R. Coifman, J. Peyriere , Phase Unwinding, or Invariant Subspace Decompositions of H ardy Spaces, J. Fourier Anal. Appl, 25 (3), pp. 684–695 (2019)

work page 2019

[5] [3]

Coifman, S

R. Coifman, S. Steinerberger, H. Wu , Carrier Frequencies, Holomorphy, and Unwinding , SIAM Journal on Mathematical Analysis 49 (6), pp. 4838–4864 (201 8)

work page

[6] [4]

Y. Gao, M. Ku, T. Qian, J. W ang , FFT formulations of adaptive Fourier decomposition , Journal of Computational and Applied Mathematics. 324 , pp. 204–215 (2017)

work page 2017

[7] [5]

Nahon , Phase Evaluation and Segmentation , Ph.D

M. Nahon , Phase Evaluation and Segmentation , Ph.D. Thesis,Yale University (2000)

work page 2000

[8] [6]

Qian , Intrinsic Mono-component Decomposition of Functions: an a dvance of Fourier theory , Math

T. Qian , Intrinsic Mono-component Decomposition of Functions: an a dvance of Fourier theory , Math. Methods Appl. Sci. 33 (7), pp. 880–891 (2010)

work page 2010