On the solutions to linear systems of moment differential equations with variable coefficients

Alberto Lastra

REVIEW 2 major objections 4 minor 29 references

On the solutions to linear systems of moment differential equations with variable coefficients

T0 review · 2 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Every linear system of moment differential equations with analytic coefficients has a unique local analytic solution, with the radius controlled by the growth of the moment sequence.

desk verdict Plausible and new variable-coefficient existence result, but Lemma 2's majorant comparison is wrong as printed; the paper deserves revision and then a serious referee. read the letter →

arxiv 2501.04396 v1 pith:AKYTDDCG submitted 2025-01-08 math.CA

Alberto Lastra This is my paper · ORCID

classification math.CA MSC 30D1534M0334A08

keywords momentderivativedifferentialequationsstronglyregularsequencesgeneralizedsummabilityCaputofractionallocalanalyticsolutionlinearsystems

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 first-order linear systems of moment differential equations $\partial_m y = A(z)y + b(z)$ whose coefficients are analytic near the origin. Its main claim is that, whenever the moment sequence is strongly regular and arises from a pair of kernel functions for $M$-summability, the Cauchy problem has a unique solution whose components are analytic on a disc around the origin. The proof separates two growth regimes: sequences that grow at least like a factorial power preserve the radius of convergence of the data, while sequences growing like a fractional power of the factorial (the Caputo-type case) give a smaller radius determined by the data and the parameter $\alpha$. This unifies existence results for ordinary, $q$-difference, and fractional Caputo differentiation, which are all cases of moment differentiation.

What carries the argument

The central object is the moment derivative $\partial_m$, defined by $\partial_m\bigl(\sum_{p\ge 0} a_p z^p/m_p\bigr) = \sum_{p\ge 0} a_{p+1} z^p/m_p$. The argument writes the unknown as a formal series, derives the coefficient recursion, and bounds the coefficients by a scalar majorant series. For factorial-type sequences the majorant solves an ordinary differential equation; for fractional-type sequences it solves the Caputo fractional integral equation on the real axis, using the relation between moment differentiation and Caputo differentiation for $m = (\Gamma(1+\alpha p))_{p\ge0}$. The strongly regular condition, together with kernel functions for $M$-summability, guarantees that the moment sequence is comparable to a factorial power and thus falls into one of the two growth regimes.

What would settle it

For the scalar example (17) with $0<\alpha<1$, compute the formal coefficients and the two bounding sequences $c_p$ and $\tilde c_p$ used in Lemma 2: if $c_p \le \tilde c_p$ fails while the solution still has the claimed radius, the theorem survives but the printed proof does not; if the radius differs from the claimed value $r_0$, the theorem's bound is wrong.

Watch

Extended reading notes

Core claim

The central claim is Theorem 1: if $M$ is a strongly regular sequence admitting a pair of kernel functions for $M$-summability, and $m$ is the associated moment sequence, then the Cauchy problem $\partial_m y = A(z)y$, $y(0) = y_0$, with $A$ analytic near $0$, has a unique solution whose components are analytic on some disc $D(0,R)$. The proof splits into two regimes: sequences whose successive quotients grow at least linearly (Assumption (A)) give $R$ equal to the radius of $A$, while sequences of fractional-factorial growth (Assumption (B), $0 < \alpha < 1$) give a possibly smaller radius computed from $\alpha$ and the data. The same conclusion extends to nonhomogeneous systems. The paper also shows that, unless the moment sequence is essentially factorial, the moment derivative does not commute with translation, so the local result at $0$ cannot be transported to other base points by the classical shift.

Load-bearing premise

The printed proof of the fractional case relies on comparing the coefficient bound to a classical-derivative recursion, a comparison that is not justified as written; the existence claim may still hold via the Caputo majorant equation, but that repair is not the printed argument.

Editorial extensions

If this is right

Every strongly regular moment sequence falls into Assumption (A) or (B), so the local existence theorem covers all such sequences and not only the explicit Gevrey-type examples.
Under Assumption (A) the solution's disc of convergence is at least the disc of analyticity of $A$ and $b$; no radius loss occurs.
Under Assumption (B) the radius shrinks to an explicit value determined by $\alpha$ and the data, matching the picture for Caputo fractional systems.
For constant-coefficient homogeneous equations the solution space has an explicit basis of entire functions built from the kernel function $E$, with growth in generalized order and type governed by the roots of the characteristic polynomial.
Translation-commutation fails except for the factorial moment sequence, so the origin-centered theory cannot be moved to arbitrary points by the classical shift trick.

Reading between the lines

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

The printed comparison of bounding sequences in Lemma 2 has a gap; the intended route through the Caputo integral equation would repair it, but an extra argument is required.
The absence of a Leibniz rule for moment derivatives (noted in the paper) makes a closed-form variation-of-constants formula unlikely; the nonhomogeneous result is obtained by majorants, not by such a formula.
Theorem 2 suggests a quantitative measure of 'momentness': the defect in the translation identity for a non-factorial sequence could be compared with the radius loss in Assumption (B), giving a testable relation between algebraic commutativity and analytic radius.
The cyclic-vector reduction in Theorem 3 is likely sharp: if the matrix $A(0)$ has no cyclic vector, no companion-form reduction to one scalar moment equation should exist, by the standard linear-algebra obstruction.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

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

Candid take: the paper extends your constant-coefficient work to analytic variable coefficients and gives two genuinely new pieces — Theorem 2 (translation invariance forces the factorial sequence) and Theorem 3 (transforming a system into a higher-order moment equation under a cyclic-vector hypothesis). The central existence claim, Theorem 1, is plausible and fits the expected shape, but the proof of Lemma 2 has a real gap. The majorant sequence c̃_p is defined by the ordinary-derivative recursion (8), then claimed to solve the moment problem (11), whose coefficient recursion requires the factor Γ(1+αp)/Γ(1+α(p+1)). Those are different for α<1, and the comparison c_p ≤ c̃_p fails — for m_p=Γ(1+p/2), A=1/(1-z), the intended c_1 exceeds c̃_1. The stress-test note is on target. This is fixable: define the majorant as the formal solution of (11) and then apply the Caputo integral-equation argument to that sequence. As printed, the proof does not establish the radius bound.

There is also a missing invertibility hypothesis in Lemma 5: A(z)=(∂_m Y)Y^{-1} requires det Y(0)≠0, and without it the lemma is false (Y(z)=zI is a counterexample). The recursion for c_p also leaves c_1 undefined since it is stated for p≥1. These are smaller, easy repairs.

What the paper does well: the setup is careful, the relation to Caputo fractional systems and q-difference moments is explicit, Theorem 2 is a neat observation, and the worked examples help. The constant-coefficient adaptation in Section 4 is a reasonable echo of [15].

Who is this for: specialists in moment summability and fractional differential equations. For that audience it deserves referee time — the main theorem is likely true and the gaps are specific, not fatal. I would send it out, with a request that the author fix the Lemma 2 comparison before acceptance. I would not cite it in my own work until the correction appears.

Referee Report

2 major / 4 minor

Summary. This paper studies first-order linear systems of moment differential equations with variable analytic coefficients, ∂_m y = A(z)y + b(z), on a disc centered at the origin. The main result, Theorem 1, asserts that for a strongly regular sequence M admitting a pair of kernel functions for M-summability, and for its associated moment sequence m, the homogeneous Cauchy problem has a unique solution with components holomorphic in a neighborhood of the origin. The proof splits into Assumption (A), where the moment ratios grow at least linearly in p and the full disc D(0,r) is preserved, and Assumption (B), where the ratios behave like Γ(1+αp)/Γ(1+α(p−1)) with 0<α<1 and only a possibly smaller disc is obtained. The paper also relates moment differentiation to Caputo fractional derivatives, shows that translation equivariance forces m=(C p!), converts systems to higher-order equations under a cyclic-vector condition, and extends constant-coefficient results from [15].

Significance. If correct, the paper gives a clean structural statement: local analytic solvability for variable-coefficient moment systems holds for all strongly regular moment sequences, and the radius is controlled by whether the moment growth is at least factorial or only fractional-factorial. The comparison with Caputo fractional equations is explicit and the two regimes are illustrated by Example 1. The proofs are based on explicit majorant sequences rather than fitted parameters, and the paper draws on published external theory (strongly regular sequences, generalized summability, [15]) rather than on assumptions invented for the paper. The main obstacle is the majorant comparison in Lemma 2: the written proof does not justify the estimate that produces the convergence radius in the α<1 case. The theorem may still be true, but the proof needs repair.

major comments (2)

[Section 3, Lemma 2 (p. 9)] The comparison c_p ≤ c̃_p is not established and is in fact false as printed. The sequence c_p is defined with the factor Γ(1+αp)/Γ(1+α(p+1)), while c̃_p is defined by (8) with the classical factor 1/(p+1); consequently ĉ(z)=∑ c̃_p z^p is not a formal solution of the moment-differential problem (11), whose recursion requires the Γ-ratio. For m_p=Γ(1+p/2) (α=1/2, C=1), A(z)=1/(1−z), K=2, c=c0=1, the intended definition gives c_1=Γ(1)/Γ(3/2)=1.128..., whereas (8) with C̃=1 gives c̃_1=1, so c_1>c̃_1. Since the final display of Lemma 2 uses c_p≤c̃_p to bound ‖y_p‖ and obtain a positive radius, the printed proof does not establish the conclusion in the Assumption (B) case. A repaired proof should take the majorant to be the formal solution of (11) and show directly that its coefficients dominate c_p and have positive radius of convergence.
[Section 3, Lemma 2 (definition of c_p)] The recursion c_{p+1}=... is written 'for all p≥1', so c_1 is never defined, although the induction claim for p=1 uses c_1. The intended inequality Y_1 Γ(1)/Γ(1+α)≤c_1 requires a separate definition of c_1 from the p=0 contribution. This is a small but real gap in the written induction.

minor comments (4)

[Example 1, p. 11] For α=1, the displayed solution y(z)=y0/(r−z) does not satisfy (17) with A(z)=r/(r−z) unless r=1; the correct solution is y(z)=y0(1−z/r)^{−r}. The conclusion that the radius remains r is still correct, but the explicit formula should be fixed.
[Section 3.1, pp. 14-15] The sentence 'Under Assumption (B), one achieves the recursion formula (2 1)' is misleading because (21) is the classical-derivative majorant problem; the fractional majorant is (22)–(23), and the Γ-ratio recursion should be stated explicitly to avoid conflating the two cases.
[Theorem 4, proof of linear independence, p. 20] The intermediate Laurent-type functions Φ_{k-1}(z)=z^{-k}∑ C_{jk}∆^{k-1}E(λ_j,z) are not holomorphic at the origin, and the step 'applying classical derivation recurrently' is not written precisely. The subsequent Vandermonde argument on the first r Taylor coefficients gives the independence directly and could replace this part.
[Abstract and Section 2.1] The abstract contains the typo 'stablished'; also the notation M_α=(α!^α) in the paragraph after Definition 1 should presumably be M_α=(p!^α). These should be corrected in the final version.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Theorem 1 is derived against external strongly regular sequence theory and a Caputo integral-equation majorant; the flagged Lemma 2 issue is a proof gap, not a circular reduction.

full rationale

Circularity pass found no circular step. The main result, Theorem 1, is not defined in terms of its own conclusion: it takes as input a strongly regular sequence M admitting kernel functions and an associated moment sequence m, then derives existence and analyticity from external benchmark results (Sanz [25], Petzsche [24], Stirling's formula) and applies Lemmas 1 and 2. Lemma 1 uses the classical majorant h' = c Ctilde (1 - Kz)^{-1} h, an external first-order ODE with explicit solution, while Lemma 2 compares against the Caputo integral equation (13) and solves it by Picard iteration on [0, r1]. There are no fitted parameters renamed as predictions, and no ansatz is smuggled in through self-citation: the kernel functions, moment sequences, and strongly regular sequence facts are imported from published external theories [12, 24, 25], not from the present paper's conclusions. Self-citations to [15] and the summability program [16, 17, 18, 19] are used as contextual ingredients or for the later constant-coefficient rewriting in Section 4; they do not carry the proof of Theorem 1. One genuine issue is flagged, but it is not circularity: in Lemma 2 the text states 'The recursion formula for the coefficients of the formal Puiseux series solution of (12) coincides with that of (8) and also with the initial data.' For alpha < 1 the coefficient recursion for (12) involves Gamma(1 + alpha p)/Gamma(1 + alpha(p+1)) factors, whereas (8) uses 1/(p+1), so the printed identification is not correct and the written proof has a gap in the Assumption (B) case. This is a mathematical error or omitted estimate, not a self-referential reduction, because the intended comparison is still against an external Caputo majorant problem rather than against the theorem being proved. Accordingly, the circularity score is 0.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No free parameters are fitted and no new entities are postulated. The paper's claim rests on background theorems on strongly regular sequences and generalized summability, the growth dichotomy whose proof is sketched, and the correct moment-recursion for the Caputo majorant equation.

assumptions (4)

domain assumption The sequence M is strongly regular (log-convex, moderate growth, non-quasianalytic) and the moment sequence m is equivalent to M as a strongly regular sequence.
Invoked in Definition 1 and Theorem 1 via Proposition 5.8 and Remark 6.6 of [25]; it identifies the class of admissible moment sequences.
domain assumption Every strongly regular sequence grows at least like p!^β for some β>0 (Petzsche's Corollary 1.3(a) [24]).
Used in the proof of Theorem 1 to split the problem into Assumption (A) when β≥1 or Assumption (B) when 0<β<1; the proof in this paper is only sketched.
standard math The Caputo fractional derivative is the left inverse of the Riemann-Liouville integral on the required domains.
Used in Section 2.2 and in the integral-equation step of Lemma 2 to replace the formal moment problem by an integral equation.
domain assumption The real Caputo problem (12)-(13) has a solution obtained from the bounded iterates ω_p on [0,r1).
The induction (14)-(16) in Lemma 2 claims this; the link to the formal Puiseux coefficients is where the recursion mismatch occurs.

how reviews work

0 comments

Cite this review

Pith. "Pith review of On the solutions to linear systems of moment differential equations with variable coefficients." pith.science (2026). https://pith.science/paper/AKYTDDCG

@misc{pith2026250104396,
  author       = {Pith},
  title        = {Pith review of: On the solutions to linear systems of moment differential equations with variable coefficients},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AKYTDDCG}},
  note         = {Machine review of arXiv:2501.04396}
}

read the original abstract

The existence and analyticity of solutions to linear systems of moment differential equations with analytic coefficients is studied. The relation of solutions of such systems with respect to linear moment differential equations is stablished, comparing classical results with the general situation of moment differentiation.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

29 extracted references · 29 canonical work pages

[15]

Lastra, Entire solutions of linear systems of moment diﬀerential eq uations and related asymptotic growth at inﬁnity, Diﬀer

A. Lastra, Entire solutions of linear systems of moment diﬀerential eq uations and related asymptotic growth at inﬁnity, Diﬀer. Equ. Dyn. Syst. (2022)

work page 2022
[1]

Agarwal, S

R. Agarwal, S. Hristova, D O’Regan, Mittag-Leﬄer stability for impulsive Caputo fractional diﬀerential equations , Diﬀer. Equ. Dyn. Syst. 29, No. 3 (2021) 689–705

work page 2021
[2]

Balser, Formal power series and linear systems of meromorphic ordin ary diﬀerential equations

W. Balser, Formal power series and linear systems of meromorphic ordin ary diﬀerential equations. Universitext. Springer-Verlag, New York, 2000

work page 2000
[3]

Balser, M

W. Balser, M. Yoshino, Gevrey order of formal power series solutions of inhomogene ous partial diﬀerential equations with constant coeﬃcients, Funkcial. Ekvac. 53 (2010) 411–434

work page 2010
[4]

Bonilla, M

B. Bonilla, M. Rivero, J. J. Trujillo, Linear diﬀerential equations of fractional order , J. Sabatier (ed.) et al., Advances in fractional calculus. The oretical developments and appli- cations in physics and engineering. Dordrecht: Springer (2 007) 77–91

work page
[5]

Bonilla, M

B. Bonilla, M. Rivero, J. J. Trujillo, On systems of linear fractional diﬀerential equations with constant coeﬃcients, Appl. Math. Comput. 187 (2007) No. 1, 68–78

work page 2007
[6]

Caputo, Lineal model of dissipation whose Q is almost frequency indep endent II , Geo- phys

M. Caputo, Lineal model of dissipation whose Q is almost frequency indep endent II , Geo- phys. J. R. Astronom. Soc. 13 (1967) 529–539

work page 1967
[7]

I. M. Gelfand, G. E. Shilov, Generalized functions, Vol. 2. Academic New York, 1965

work page 1965

Show all 29 references

[8]

Hayek, J

N. Hayek, J. Trujillo, M. Rivero, B. Bonilla, J. C. Moreno , An extension of Picard-Lindel¨ oﬀ theorem to fractional diﬀerential equations , Appl. Anal. 70, No. 3-4 (1999) 347–361

1999
[9]

Immink, Accelero-summation of the formal solutions of nonlinear di ﬀerence equations, Ann

G. Immink, Accelero-summation of the formal solutions of nonlinear di ﬀerence equations, Ann. Inst. Fourier (Grenoble) 61, no. 1 (2011) 1–51

2011
[10]

Immink, Exact asymptotics of nonlinear diﬀerence equations with le vels 1 and 1+ , Ann

G. Immink, Exact asymptotics of nonlinear diﬀerence equations with le vels 1 and 1+ , Ann. Fac. Sci. Toulouse Math. (6) 17, no. 2, (2008) 309–356

2008
[11]

Jim´ enez-Garrido, S

J. Jim´ enez-Garrido, S. Kamimoto, A. Lastra, J. Sanz, Multisummability in Carleman ul- traholomorphic classes by means of nonzero proximate order s, J. Math. Anal. Appl. 472 (1) (2019) 627–686

2019
[12]

Jim´ enez-Garrido, J

J. Jim´ enez-Garrido, J. Sanz, G. Schindl, Log-convex sequences and nonzero proximate or- ders, J. Math. Anal. Appl. 448(2) (2017) 1572–1599

2017
[13]

A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory an d applications of fractional diﬀer- ential equations. North-Holland Math. Stud. 204, Elsevier , Amsterdam, 2006

2006
[14]

Komatsu, Ultradistributions

H. Komatsu, Ultradistributions. I: Structure theorems and a characteri zation, J. Fac. Sci. Univ. Tokyo, Sect. I A 20 (1973) 25–105

1973
[16]

Lastra, S

A. Lastra, S. Malek, J. Sanz, Summability in general Carleman ultraholomorphic classes , J. Math. Anal. Appl. 430 (2015) 1175–1206

2015
[17]

Lastra, S

A. Lastra, S. Michalik, M. Suwi´ nska, Summability of formal solutions for a family of gen- eralized moment integro-diﬀerential equations, Fract. Calc. Appl. Anal. 24 (2021) No. 5, 1445–1476. 23

2021
[18]

Lastra, S

A. Lastra, S. Michalik, M. Suwi´ nska, Multisummability of formal solutions for a family of generalized singularly perturbed moment diﬀerential equa tions, Results Math. 78, 49 (2023)

2023
[19]

Lastra, S

A. Lastra, S. Michalik, M. Suwi´ nska, Estimates of formal solutions for some generalized moment partial diﬀerential equations, J. Math. Anal. Appl. 500(1), 18 (2021)

2021
[20]

M. M. Matar, E. S. A. Skhail, On stability analysis of semi-linear fractional diﬀerenti al systems, Math. Methods Appl. Sci. 43, No. 5 (2020) 2528–2537

2020
[21]

Michalik, Multisummability of formal solutions of inhomogeneous lin ear partial diﬀeren- tial equations with constant coeﬃcients , J

S. Michalik, Multisummability of formal solutions of inhomogeneous lin ear partial diﬀeren- tial equations with constant coeﬃcients , J. Dyn. Control Syst. 18 (2012) 103–133

2012
[22]

Michalik, Analytic solutions of moment partial diﬀerential equation s with constant coef- ﬁcients, Funkcial

S. Michalik, Analytic solutions of moment partial diﬀerential equation s with constant coef- ﬁcients, Funkcial. Ekvac. 56 (1) (2013) 19–50

2013
[23]

Michalik, B

S. Michalik, B. Tkacz, The Stokes phenomenon for some moment partial diﬀerential eu qa- tions, J. Dyn. Control Syst. 25 (4) (2019) 573–598

2019
[24]

Petzsche, On E

H.-J. Petzsche, On E. Borel’s theorem , Math. Ann. 282 (1988) 299–313

1988
[25]

Sanz, Flat functions in Carleman ultraholomorphic classes via pro ximate orders, J

J. Sanz, Flat functions in Carleman ultraholomorphic classes via pro ximate orders, J. Math. Anal. Appl. 415(2) (2014) 623–643

2014
[26]

Suwi´ nska,Gevrey estimates of formal solutions for certain moment par tial diﬀerential equations with variable coeﬃcients, J

M. Suwi´ nska,Gevrey estimates of formal solutions for certain moment par tial diﬀerential equations with variable coeﬃcients, J. Dyn. Control Syst. 27(2) (2021) 355–370

2021
[27]

Thilliez, Division by ﬂat ultradiﬀerentiable functions and sectoria l extensions , Result

V. Thilliez, Division by ﬂat ultradiﬀerentiable functions and sectoria l extensions , Result. Math. 44 (2003) 169–188

2003
[28]

Thilliez, Smooth solutions of quasianalytic or ultraholomorphic equ ations, Monatsh

V. Thilliez, Smooth solutions of quasianalytic or ultraholomorphic equ ations, Monatsh. Math. 160 (2010) 443–453

2010
[29]

U¸ car, N

E. U¸ car, N. ¨Ozdemir, A fractional model of cancer-immune system with Caputo and Ca- puto–Fabrizio derivatives, Eur. Phys. J. Plus 136 (2021) 43

2021

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [15]

Lastra, Entire solutions of linear systems of moment diﬀerential eq uations and related asymptotic growth at inﬁnity, Diﬀer

A. Lastra, Entire solutions of linear systems of moment diﬀerential eq uations and related asymptotic growth at inﬁnity, Diﬀer. Equ. Dyn. Syst. (2022)

work page 2022

[2] [1]

Agarwal, S

R. Agarwal, S. Hristova, D O’Regan, Mittag-Leﬄer stability for impulsive Caputo fractional diﬀerential equations , Diﬀer. Equ. Dyn. Syst. 29, No. 3 (2021) 689–705

work page 2021

[3] [2]

Balser, Formal power series and linear systems of meromorphic ordin ary diﬀerential equations

W. Balser, Formal power series and linear systems of meromorphic ordin ary diﬀerential equations. Universitext. Springer-Verlag, New York, 2000

work page 2000

[4] [3]

Balser, M

W. Balser, M. Yoshino, Gevrey order of formal power series solutions of inhomogene ous partial diﬀerential equations with constant coeﬃcients, Funkcial. Ekvac. 53 (2010) 411–434

work page 2010

[5] [4]

Bonilla, M

B. Bonilla, M. Rivero, J. J. Trujillo, Linear diﬀerential equations of fractional order , J. Sabatier (ed.) et al., Advances in fractional calculus. The oretical developments and appli- cations in physics and engineering. Dordrecht: Springer (2 007) 77–91

work page

[6] [5]

Bonilla, M

B. Bonilla, M. Rivero, J. J. Trujillo, On systems of linear fractional diﬀerential equations with constant coeﬃcients, Appl. Math. Comput. 187 (2007) No. 1, 68–78

work page 2007

[7] [6]

Caputo, Lineal model of dissipation whose Q is almost frequency indep endent II , Geo- phys

M. Caputo, Lineal model of dissipation whose Q is almost frequency indep endent II , Geo- phys. J. R. Astronom. Soc. 13 (1967) 529–539

work page 1967

[8] [7]

I. M. Gelfand, G. E. Shilov, Generalized functions, Vol. 2. Academic New York, 1965

work page 1965