Analog for the Wiener Lemma for Wolff-Denjoy Series

A. A. Atvinovskii; A. R. Mirotin

Analog for the Wiener Lemma for Wolff-Denjoy Series

T0 review · 0 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The reciprocal of a positive Wolff–Denjoy series with real poles is again a Wolff–Denjoy series up to an explicit affine term.

desk verdict A sound Nevanlinna-class reciprocal formula with explicit coefficients; the proof checks out, and the operator corollary is a legitimate application. read the letter →

arxiv 1908.08029 v2 pith:KCUZQ6Z6 submitted 2019-08-21 math.FA

A. R. Mirotin , A. A. Atvinovskii This is my paper · ORCID

classification math.FA MSC 47A6030E2047A05

keywords Wolff-DenjoyseriesWienerlemmamultiplicativeinverseNevanlinnaclassclosedoperatorleftresolventfunctionalcalculus

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 note establishes a multiplicative analogue of the classical Wiener lemma for a class of meromorphic functions given by Wolff–Denjoy series. The central claim is that if $f(z)=\sum_{k=1}^\infty c_k/(\lambda_k-z)$ with positive summable coefficients $c_k$ and a bounded, monotonically increasing real pole sequence $\{\lambda_k\}$, then the reciprocal satisfies $\frac{1}{f(z)}=\alpha+\beta z-\sum_{n=1}^\infty b_n/(t_n-z)$, where the $t_n$ are the zeros of $f$, $b_n=1/f'(t_n)>0$, and $\sum b_n<\infty$. The coefficients are explicit: $\alpha=(\sum c_k\lambda_k)/(\sum c_k)^2$ and $\beta=-1/\sum c_k$. This is the analogue of saying that a function with an absolutely convergent Fourier series has a reciprocal of the same type, with a linear correction term. The result matters because it turns inversion into the same structural class used to build functions of closed operators, giving an explicit left inverse of $f(A)$ from resolvents of $A$.

What carries the argument

The engine is the imaginary-part identity $\operatorname{Im} f(z)=y\sum_{k=1}^\infty c_k/|\lambda_k-z|^2$, which shows that $f$ has the sign of its imaginary part controlled by $y$. This places $-1/f$ in the Nevanlinna class, the class of functions holomorphic in the upper half-plane with nonnegative imaginary part, whose integral representation is given by the Nevanlinna representation theorem. Applying the Stieltjes–Perron inversion formula then locates the atoms of the representing measure exactly at the zeros of $f$, producing the partial-fraction expansion with positive residues $b_n=1/f'(t_n)$. The positivity of the coefficients is what makes $f$ strictly increasing between consecutive poles, so zeros are simple and the inverse develops only simple poles.

What would settle it

Compute a concrete example, say $c_k=2^{-k}$ and $\lambda_k=1-1/k$, and locate the zeros of $f$ numerically on the real interval $[0,1]$; the theorem would be refuted by any non-real zero, a repeated zero, or a sequence of residues $1/f'(t_n)$ whose sum fails to converge. A finite two-pole example can be checked symbolically by verifying that the partial-fraction decomposition of $1/f$ exactly matches the claimed formula with $b=1/f'$ at the unique zero.

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Extended reading notes

Core claim

Theorem 1 asserts that if $f(z)=\sum_{k=1}^\infty c_k/(\lambda_k-z)$ with $c_k>0$, $\sum c_k<\infty$, and $\{\lambda_k\}$ a monotone increasing bounded sequence of real numbers, then all zeros $t_n$ of $f$ are real, simple, and lie in the pole interval; moreover $\frac{1}{f(z)}=\alpha+\beta z-\sum_{n=1}^\infty b_n/(t_n-z)$ with $b_n=1/f'(t_n)>0$, $\sum b_n<\infty$, $\alpha=(\sum c_k\lambda_k)/(\sum c_k)^2$, and $\beta=-1/\sum c_k$. Thus the reciprocal is, up to one affine term, again a Wolff–Denjoy series, this time with negative coefficients. The paper also proves a finite-analogue for complex poles: when the pole set is finite and the spectrum of the operator avoids its convex hull, the left inverse of $f(A)$ is $\alpha I+\beta A+\sum_{j,k} c_{jk}R(t_j,A)^k$, where higher powers of resolvents appear according to the multiplicity of the zero.

Load-bearing premise

The proof needs the pole sequence to be real and the coefficients to be strictly positive, because together they force $f$ to be real on the real axis, strictly increasing between poles, and $-1/f$ to lie in the Nevanlinna class, which yields only simple real zeros for the reciprocal.

Editorial extensions

If this is right

The inverse of such an $f$ can have no poles outside the interval spanned by the original poles, so the original pole interval controls where inverse singularities can appear.
The coefficients of the inverse series are computable once the zeros are known: $b_n=1/f'(t_n)$, with no further integral or limit needed.
Because $\sum b_n<\infty$, the reciprocal expansion converges absolutely in the same sense as the original Wolff–Denjoy series, preserving the summability structure used in applications.
For a closed operator $A$ whose spectrum avoids the pole interval, $f(A)^{-1}=\alpha I+\beta A-\sum_{n=1}^\infty b_nR(t_n,A)$, giving an explicit left inverse built from resolvents of $A$.
In the finite complex-pole case, the same conclusion holds with resolvent powers up to the multiplicity of each zero, provided the spectrum avoids the convex hull of the poles.

Reading between the lines

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

The explicit identity for $\beta=-1/\sum c_k$ suggests that the unbounded linear term in the inverse is unavoidable whenever $f$ vanishes at infinity; the fractional part alone can never represent the inverse as a bounded resolvent series.
Because the inverse's fractional part carries negative coefficients, iterating this inversion theorem is blocked by the positivity hypothesis; one could look for a signed-cone version in which positivity is replaced by a total-variation condition, and each inversion doubles the pole structure.
From the regularization remark in the paper, the same formula gives a concrete way to regularize $f(A)x=y$ whenever the underlying equation $Ax=y$ is regularizable; this route is stated implicitly rather than developed as a numerical method.
The convex-hull argument for complex poles suggests a natural testable extension to infinite series with complex poles: the inverse should remain of the same type when the zeros stay simple and the spectrum avoids the convex hull, but multiple zeros are then expected to appear for some coefficient choices.

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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

Short version: the paper proves a clean quantitative reciprocal formula for a natural class of Nevanlinna functions, and the proof is sound. It deserves a serious referee.

The genuinely new thing is Theorem 1: if f(z)=Σ c_k/(λ_k−z) with c_k>0, λ_k real, increasing, bounded, then 1/f(z)=α+βz − Σ b_n/(t_n−z), with explicit α, β and positive b_n=1/f′(t_n). This extends the finite-sum result [11] and complements [15]. The proof is standard Nevanlinna/Stieltjes–Perron business, and I checked the steps. Positivity gives Im f(z)=yΣ c_k/|λ_k−z|², so −1/f is in the Nevanlinna class R; f′>0 between poles makes zeros simple, one per interval. Stieltjes–Perron inversion shows the representing measure is flat away from the zeros, with jumps 1/f′(t_n); the endpoint-atom concern at b is a genuine subtlety, but the sign of Re f(b+iy) rules it out, as the stress-test note says. Asymptotics at infinity give α and β. The proof is complete modulo routine justification of limit interchange, which the authors gesture at.

The soft spots are modest. The operator-theoretic corollary (explicit left inverse for f(A)) leans on the authors' functional calculus [12] to identify the left inverse with φ(A); that is a black box, but it is prior published work, independent of the scalar theorem, and the conclusion is plausible. Theorem 2, for complex poles in a finite sum, is a nice add-on, but the proof that all zeros of f lie in the convex hull of the λ_j is compressed, and the hypothesis conv(λ_j)∩σ(A)=∅ is considerably stronger than the necessary condition discussed in Remark 3; it is fine as a sufficient condition. The Wiener-lemma title is a bit rhetorical, but the actual statement is clean.

Citation pattern is fine: the self-citations are to the earlier papers that posed the problem and to the functional calculus being used, which is legitimate. Remarks 1 and 2 honestly identify where positivity and realness are needed. In short: a well-executed short paper with no load-bearing flaw. I would take it for a reading group and cite it in connection with reciprocal Nevanlinna series. For peer review: yes, send it out; a referee should push on the measure-theoretic details and on whether Theorem 2's convexity condition is optimal, but the central result stands.

Referee Report

0 major / 5 minor

Summary. The paper studies functions f(z) = Σ_{k=1}^∞ c_k/(λ_k − z) with positive summable coefficients c_k and a monotone bounded real pole sequence {λ_k}. Theorem 1 states that 1/f(z) = α + βz − Σ_{n=1}^∞ b_n/(t_n − z), where t_n are the zeros of f, b_n = 1/f'(t_n) > 0 are summable, and α, β are given explicitly in terms of c_k and λ_k. The proof uses the Nevanlinna representation for −1/f and Stieltjes-Perron inversion to identify the representing measure as atomic at the zeros. A corollary gives a left inverse for f(A) for a closed operator A with spectrum avoiding [a,b], and a second theorem treats finite rational functions with positive coefficients and complex poles, providing a left inverse expressed through powers of resolvents and requiring the spectrum to avoid the convex hull of the poles. Remarks show that the positivity and reality assumptions are essential.

Significance. The result is a clean and explicit analogue of Wiener's lemma for Wolff-Denjoy series: under natural positivity and reality conditions, the reciprocal is again a Wolff-Denjoy series up to a linear term, with completely explicit coefficients. The proofs are transparent and mostly self-contained, relying on classical Nevanlinna theory and Stieltjes-Perron inversion, and the final formulas are directly checkable. The operator-theoretic applications, including the left-inverse formulas for f(A) and the regularization remark, are natural and potentially useful. The paper also provides concrete counterexamples showing the hypotheses cannot simply be dropped. Apart from a small proof-completeness point concerning endpoint atoms, the mathematics is sound and the presentation is adequate.

minor comments (5)

[Theorem 1 proof, around Eq. (2)] The passage from the Stieltjes-Perron inversion to the series (2) implicitly assumes that the representing measure τ has no atoms at the endpoints a and b. The text shows τ is constant on the open intervals outside [a,b] and between zeros, but this does not by itself exclude jumps at a and b. This omission is harmless: for z = b + iy, Re f(b+iy) = −Σ c_k(b−λ_k)/((b−λ_k)^2 + y^2) < 0 and does not tend to 0 as y↓0, so 1/f cannot have a pole at b; the argument at a is analogous. Please add a sentence to make this explicit.
[Theorem 2 proof] The phrase 'прямая Ref = a' appears to be a typo; it should refer to the real part of the rotated variable w (or ζ), not to the function f. Please correct the notation.
[Theorem 1 statement] In the statement of Theorem 1, the summation condition '∑_{k=i}^∞ c_k < ∞' should read '∑_{k=1}^∞ c_k < ∞'; the index i is undefined.
[Theorem 1, assumptions on {λ_k}] The phrase 'монотонно возрастающая' for the sequence {λ_k} is ambiguous if equal values are allowed. If the sequence is only non-decreasing, the interval argument requires merging repeated poles; the authors should either state that the sequence is strictly increasing or note that the repeated-pole case reduces to the strictly increasing one.
[Title and metadata] The title on the arXiv abstract page ('Analog for the Wiener Lemma for Wolff-Denjoy Series') differs from the title in the full text ('Multiplicative inverse for Wolff-Denjoy Series'). Please align them, since the mismatch may confuse readers.

Circularity Check

0 steps flagged · score 0.0 of 10

No material circularity: Theorem 1 is derived from the Nevanlinna representation and Stieltjes-Perron inversion, not from its conclusion; self-citations appear only in non-load-bearing or independently published applications.

full rationale

No step in the derivation reduces to its inputs. Theorem 1 begins with the identity Im f(z) = y Σ c_k / |λ_k − z|^2, which is used only to place −1/f in the Nevanlinna class; the Nevanlinna representation theorem and the Stieltjes-Perron inversion formula are standard external results ([5], [7]). The zero set {t_n} and residues b_n = 1/f'(t_n) > 0 are not assumed: they are produced by the monotonicity of f on R \ ({λ_k} ∪ {b}) and by the constancy of the representing measure between consecutive zeros, with the point mass at t_n equal to the residue. The affine coefficients α and β are computed from asymptotic limits of 1/f, not fitted to the target expansion. Theorem 2 uses a geometric separating-line argument for the location of zeros and the Riesz-Dunford calculus; it does not import the conclusion. The only self-citations ([10], [12], [13], [15]) are not load-bearing for the scalar theorem: [10] is a parenthetical comparison, and [12] is an independent published functional-calculus theorem used to move the already-proved identity for φ to the operator level in Corollary 1. Under the rule that independent published results are real evidence, these citations do not raise the circularity score. A proof-completeness nuance about a possible endpoint atom at the accumulation point b is not circularity, and the sign of f excludes it.

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

The main theorem is built on two classical external tools (Nevanlinna representation and Stieltjes-Perron inversion) plus elementary real analysis. The operator-theoretic corollary additionally invokes the authors' own earlier functional calculus theorem ([12]); this is a support citation for the application, not for the scalar theorem. No free parameters are fitted and no new entities are postulated.

assumptions (4)

standard math Nevanlinna representation theorem for functions of class R: every function holomorphic in the upper half-plane with positive imaginary part admits the integral representation used in the proof.
Invoked at the start of the proof of Theorem 1 to write -1/f(z) = α1 + β1 z + ∫ (1/(t-z) - t/(1+t^2)) dτ(t); cited to [5], [6].
standard math Stieltjes-Perron inversion formula recovers the measure τ from boundary values of Im(-1/f).
Used to show τ is constant outside [a,b] and between the zeros t_n, yielding the atomic series in Theorem 1; cited to [7].
standard math Riesz-Dunford holomorphic functional calculus for closed operators in Banach space, including polynomial identities.
Used in Section 3 to pass from the scalar identity 1 = g(z)f(z) to the operator identity I = g(A)f(A); cited to [14].
domain assumption From [12]: for f in the class R[a,b], the left inverse of f(A) exists and equals (1/f)(A).
Used in Corollary 1 to assert f(A)^{-1} = φ(A) with φ = 1/f; this is a prior result by the same authors, not re-proved here. It is independent of Theorem 1.

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Cite this review

Pith. "Pith review of Analog for the Wiener Lemma for Wolff-Denjoy Series." pith.science (2026). https://pith.science/paper/KCUZQ6Z6

@misc{pith2026190808029,
  author       = {Pith},
  title        = {Pith review of: Analog for the Wiener Lemma for Wolff-Denjoy Series},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KCUZQ6Z6}},
  note         = {Machine review of arXiv:1908.08029}
}

read the original abstract

Let a function f with real poles be expanded in a Wolff-Denjoy series with positive coefficients. The main result of the note states that if we subtract its linear part from the function 1/f, then the remaining fractional part of this function will also expand into Wolff-Denjoy series (its poles are also real, and the coefficients of the series are negative). In other words, for Wolff-Denjoy series of the indicated form, an analogue of the well-known Wiener lemma in the theory of Fourier series is true up to a linear term. Applications of the result to operator theory are given.

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Reference graph

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Pith tools

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

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Р., Атвиновский А

Миротин А. Р., Атвиновский А. А. Обращение линейной ком бинации значений резольвенты замкнутого оператора. Проблемы физики, матем атики и техники. 2014. № 3(20). С. 77–79

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Миротин А. Р. Обращение операторно-монотонных функци й негативных операторов в банаховом пространстве. Труды Института математики. Мин ск. 2004. Т. 12, № 1. С. 104 – 108

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А., Миротин А

Атвиновский А. А., Миротин А. Р. Об одном функционально м исчислении замкнутых операторов в банаховом пространстве. Изв. вузов. Матем. 20 13, Т. 10. С. 3 – 15; английский перевод: A. A. Atvinovskii, A. R. Mirotin, On som e functional calcu- lus of closed operators in a Banach space, Russian Math. (Iz. VUZ), 57: 10 (2013), 1 – 12

work page 2013

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Сибилев, Р. В. Теорема единственности для рядов Вольфа– Данжуа, Алгебра и анализ, 1995, Т. 7, выпуск 1, 170 – 199

work page 1995

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В. Б. Шерстюков, Разложение обратной величины целой фун кции с нулями в полосе в ряд Крейна, Матем. сб., 2011, т. 202, № 12, 137 – 156

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Р., Атвиновский А

Миротин А. Р., Атвиновский А. А. О мультипликативном об- ращении рядов Вольфа- Данжуа, Труды Института математики и механики УрО РАН, - 201 9 - Т. 25, № 4, С. 147 - 154. DOI: 10.21538/0134-4889-2019-25-4-147-154

work page doi:10.21538/0134-4889-2019-25-4-147-154 2019

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Nevanlinna R., Asymptotische Entwicklungen Beschr¨ ankter Funktionen und das Stieltjess- che Momentenproblem, Ann. Acad. Sci. Fenn., Ser. A, vol. 18 ( 1922), 1–53

work page 1922