{"id":"a1d9afcc-c640-4830-a62e-1f416a2ca5e8","arxiv_id":"1908.08029","paper_version":2,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"For a positive-coefficient Wolff-Denjoy series with real poles, the reciprocal equals a linear term plus another Wolff-Denjoy series whose coefficients are the reciprocals of the derivative at the zeros.","lead":"This paper proves that the reciprocal of a Wolff-Denjoy series with positive coefficients and real poles is again a Wolff-Denjoy series up to a linear term. The result gives explicit left inverses for certain resolvent-built operators in Banach spaces, together with a regularization strategy.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: the Nevanlinna/Stieltjes-Perron proof of Theorem 1 is sound, and the endpoint-atom subtlety is excluded by the sign of f.","rationale":"The reader's ACCEPT verdict is justified. The central theorem is a clean consequence of Nevanlinna's representation and Stieltjes-Perron inversion; the sign identity is the key mechanism and is used correctly. I independently checked the location and simplicity of zeros, atomicity of the measure, the residue formula b_n=1/f'(t_n), and the asymptotic computation of α and β. The reader's weakest_assumption correctly identifies positivity and realness of poles as the structural crux, but those are hypotheses of the theorem rather than an unexamined weak point. The only issue I found is that the written proof does not explicitly rule out an atom of the representing measure at the accumulation point b (or at a); the intervalwise inversion only sees interior mass. This is easily excluded from the negative real part of f on the vertical ray, so correctness is unaffected. No change to the reader's verdict is needed.","tokens_in":7415,"tokens_out":33075,"duration_ms":366590,"concrete_test":"Verify the endpoint-atom exclusion analytically: use Re f(b+iy)<0 and monotone convergence to show that the representing measure for -1/f has no atom at b, and similarly at a. A numerical cross-check: take c_k=2^{-k}, λ_k=1-1/k, truncate at 20 terms, and compare the right-hand side of the theorem with direct partial fractions; the residue at b=1 should be exactly zero, and the computed coefficients should approach α, β, and b_n.","verdict_should_be":"UNCHANGED","load_bearing_attack":"I read Theorem 1's proof line by line. The positivity and realness assumptions are not a hidden weakness: they give Im f(z)=yΣc_k/|λ_k-z|^2>0, so -1/f is in the Nevanlinna class, and f'(x)=Σc_k/(λ_k-x)^2>0 makes every zero simple and forces exactly one zero in each interval between consecutive poles. Stieltjes-Perron inversion is applied on intervals where 1/f is real-analytic, so the representing measure has no mass away from the zeros and the atoms are the residues b_n=1/f'(t_n). The asymptotics of 1/f give the stated α and β. One subtle point deserves attention: the inversion is performed on open intervals, so a possible atom at the accumulation point b (and similarly at a) is not directly seen. Such an atom would force -1/f to have angular limit +i∞ at b, equivalently f to have angular limit 0 along the vertical ray. But for z=b+iy, Re f(b+iy)=-Σc_k(b-λ_k)/((b-λ_k)^2+y^2)<0, and as y↓0 this is monotone decreasing to a strictly negative limit or -∞; it cannot tend to 0. Hence no endpoint atom exists and (2) is complete. This is a proof-completeness subtlety, not a real gap in the central claim.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":7667,"tokens_out":20895,"duration_ms":183581,"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.","major_comments":[],"minor_comments":[{"comment":"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.","section":"Theorem 1 proof, around Eq. (2)"},{"comment":"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.","section":"Theorem 2 proof"},{"comment":"In the statement of Theorem 1, the summation condition '∑_{k=i}^∞ c_k < ∞' should read '∑_{k=1}^∞ c_k < ∞'; the index i is undefined.","section":"Theorem 1 statement"},{"comment":"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.","section":"Theorem 1, assumptions on {λ_k}"},{"comment":"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.","section":"Title and metadata"}],"recommendation":"minor_revision","confidential_remarks":"The paper is a short note whose central result is correct. The only substantive point is the implicit exclusion of endpoint atoms in the Stieltjes measure in the proof of Theorem 1, which should be made explicit. Once that sentence is added and the minor typographical issues are fixed, the paper is suitable for publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nThe 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.\n\nThe 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.\n\nCitation 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.","headline":"A sound Nevanlinna-class reciprocal formula with explicit coefficients; the proof checks out, and the operator corollary is a legitimate application.","tokens_in":8225,"tokens_out":2214,"would_cite":true,"duration_ms":22742,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["47A60","30E20","47A05"],"pacs":[],"model":"deepseek-v4-flash","headline":"The reciprocal of a positive Wolff–Denjoy series with real poles is again a Wolff–Denjoy series up to an explicit affine term.","keywords":["Wolff-Denjoy series","Wiener lemma","multiplicative inverse","Nevanlinna class","closed operator","left inverse","resolvent","functional calculus"],"falsifier":"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.","tokens_in":7188,"feed_emoji":"🔄","tokens_out":7686,"duration_ms":158829,"temperature":0.7,"pith_summary":"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 $A$.","feed_headline":"Reciprocal of a pole series is a pole series plus a line","feed_subtitle":"Positive Wolff–Denjoy series invert into the same class, with explicit coefficients and resolvent formulas.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the Nevanlinna representation theorem that converts the sign of the imaginary part into the integral representation used for $-1/f$.","marker":"[5]"},{"why":"Supplies the Stieltjes–Perron inversion formula used to locate the representing measure's atoms at the zeros of $f$ and to compute the jumps $b_n$.","marker":"[7]"},{"why":"Provides the integral representation for the class of analytic functions to which the reciprocal is compared.","marker":"[10]"},{"why":"Builds the functional calculus for closed operators used to pass from the function identity to the operator identity for $f(A)^{-1}$.","marker":"[12]"},{"why":"Gives the Riesz–Dunford functional calculus used to justify composing the rational identity with a closed operator.","marker":"[14]"},{"why":"The finite-sum special case that Corollary 1 generalizes, and the source of the question for complex poles.","marker":"[11]"},{"why":"The continuous analogue whose inversion result is extended by the series version.","marker":"[15]"}],"fun_headline_variants":["Wolff–Denjoy reciprocal: affine term plus pole series","Pole series inverse: same class up to an affine term","Reciprocal series stays Wolff–Denjoy modulo a line","Inversion in Wolff–Denjoy series: affine plus pole part"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Wolff–Denjoy reciprocal: affine term plus pole series","Pole series inverse: same class up to an affine term","Reciprocal series stays Wolff–Denjoy modulo a line","Inversion in Wolff–Denjoy series: affine plus pole part"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000634,"raw_usage":{"total_tokens":2897,"prompt_tokens":887,"completion_tokens":2010,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":503,"completion_tokens_details":{"reasoning_tokens":1937}},"tokens_in":503,"tokens_out":2010,"duration_ms":426376,"temperature":1.0,"reasoning_tokens":1937,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T11:52:28.781794+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the Nevanlinna representation theorem that converts the sign of the imaginary part into the integral representation used for $-1/f$."},{"cited_title":"Г., Нудельман, А","cited_arxiv_id":null,"evidence_quote":"Supplies the Stieltjes–Perron inversion formula used to locate the representing measure's atoms at the zeros of $f$ and to compute the jumps $b_n$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the integral representation for the class of analytic functions to which the reciprocal is compared."},{"cited_title":"А., Миротин А","cited_arxiv_id":null,"evidence_quote":"Builds the functional calculus for closed operators used to pass from the function identity to the operator identity for $f(A)^{-1}$."},{"cited_title":"Данфорд, Дж","cited_arxiv_id":null,"evidence_quote":"Gives the Riesz–Dunford functional calculus used to justify composing the rational identity with a closed operator."},{"cited_title":"Р., Атвиновский А","cited_arxiv_id":null,"evidence_quote":"The finite-sum special case that Corollary 1 generalizes, and the source of the question for complex poles."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The continuous analogue whose inversion result is extended by the series version."}],"review_version":1}