{"id":"365d16a5-620f-4621-9453-53d4422cb8e9","arxiv_id":"1908.03604","paper_version":1,"verdict":"REJECT","confidence":"HIGH","novelty_score":4.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper defines fractional powers of bounded operators via Newton and Shannon interpolation of integer powers, but the sinc formula fails on the eigenspace λ=-1 and the zeta interpolation lacks a convergence proof.","lead":"A math preprint proposes building fractional powers of operators by interpolating integer powers with Newton-series and sinc formulas. It also applies the interpolation to the Riemann zeta function and the Fourier transform; the sinc branch fails for operators with eigenvalue -1.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 1's sinc identity is false at x = -1: the series evaluates to cos(πα), not (-1)^α for noninteger α. Since -1 lies in the spectrum of unitary operators including the Fourier transform, Theorem 3 fails and the proposed fractional powers lose the semigroup property.","rationale":"The reader and I locate the same load-bearing weak point. I independently re-derived the partial-fraction evaluation: for α∉Z, Σ_{n∈Z} sinc(α-n)(-1)^n = cos πα, using Σ_{n∈Z} 1/(α-n)=π cot πα. This proves the scalar identity behind Theorem 3 is false at x=-1, and since -1 is an eigenvalue of the Fourier transform used in Section 4.3, the paper's own application collapses: the alternate fractional Fourier transform is a linear combination of f, f-hat, and their reflections, but it is not a semigroup of powers. The counterexample T=-I is the cleanest witness: Theorem 3 would assign (-I)^{1/2}=0, which is not a square root of -I. This is a decisive correctness failure. I also note the paper explicitly admits in Remark 2 that the Shannon interpolation fails for translation operators, which is consistent with the scalar identity failing at spectral points. Additional defects (Theorem 4's hypothesis, the zeta interpolation at k=1) strengthen the rejection but are secondary. The Newton-series branch is not implicated in this concern; it follows the standard functional calculus and is reasonably argued. Because the central claim of the paper is falsified, the REJECT verdict stands.","tokens_in":10504,"tokens_out":6247,"duration_ms":66045,"concrete_test":"Evaluate Proposition 1 at x=-1, α=1/3. Compute the symmetric partial sums S_M = Σ_{n=-M}^M sinc(1/3-n)(-1)^n; analytically they tend to cos(π/3)=1/2, whereas (-1)^{1/3}=e^{iπ/3}≈0.5+0.8660i. Equivalently, set T=-I on C in Theorem 3 and verify T^{1/2}=0, so T^{1/2}T^{1/2}=0≠T^1. Either computation settles that Eq. (4), and hence Eq. (8), is not an identity for unitaries with -1 in their spectrum.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central load-bearing step is Proposition 1 (Eq. 4): x^α = Σ_{n∈Z} sinc(α-n)x^n claimed for |x|=1. The proof for x=-1 only shows convergence and then asserts equality. Direct partial fractions give, for α∉Z, Σ_{n∈Z} sinc(α-n)(-1)^n = (sin πα/π) Σ_{n∈Z} 1/(α-n) = cos πα, using Σ_{n∈Z} 1/(α-n)=π cot πα. The principal value of (-1)^α is e^{iπα}, so equality holds only at integer α. Hence Proposition 1 is false at a point of the unit circle. The operator-level consequence is immediate: Theorem 3 (Eq. 8) asserts T^α = Σ sinc(α-n)T^n for every unitary T. For T=-I on C, the right side is cos(πα) I, so for α=1/2 it gives the zero operator; then T^{1/2}T^{1/2}=0 ≠ T^1=-I, destroying the semigroup property. The paper's own application, the Fourier transform, has -1 as an eigenvalue (F^4=I, spectrum {1,-1,i,-i}), so the alternate fractional Fourier transform in Proposition 9 is not a fractional power and does not satisfy additivity. This is a falsifying instance of the paper's central claim, not merely a convergence gap. The Newton-series branch (Theorems 1 and 2) is largely standard and is not the problem; the Shannon branch is the load-bearing failure. A secondary defect supports the same conclusion: Theorem 4 states T^N=T but the proofs use T^N=I, so the periodic-interpolation formula is mis-stated.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes to construct fractional powers T^α of bounded linear operators by analytically interpolating the integer powers of T. Two series are used: a Newton-series expansion (Theorems 1 and 2, Eqs. (6)–(7)) for spectra contained in a disk, and a Shannon sampling series (Theorem 3, Eq. (8)) for unitary operators. The manuscript further derives a periodic sampling formula (Theorem 4) and applies the constructions to fractional integration and differentiation, to Dirichlet series and the Riemann zeta function (Theorem 5), and to an 'alternate fractional Fourier transform' (Proposition 9). The central claimed novelty is that fractional powers, including semigroup-property-preserving powers of unitary operators, can be built from integer powers alone.","tokens_in":10913,"tokens_out":5838,"duration_ms":58043,"significance":"The Newton-series branch of the paper is a legitimate binomial-series functional calculus under the stated spectral restrictions, and the exposition of that part is a useful point of reference. The paper contains no fitted parameters and no empirical predictions; its claims are formal identities. If the Shannon-sampling branch were correct, it would give an extremely simple formula for fractional powers of unitaries and a fractional Fourier transform as a four-term linear combination of the integer Fourier powers. However, the key scalar identity (Proposition 1) is false at x = -1, which invalidates Theorem 3 and the alternate fractional Fourier transform. The zeta-function application is a restatement of the standard identity η(s) = (1 - 2^{1-s})ζ(s) evaluated at integers rather than an independent construction. The paper would need a correct replacement for Proposition 1 to support its main claims; no such replacement is present.","major_comments":[{"comment":"Proposition 1 is false. For x = -1 and noninteger α, the series evaluates to Σ_{n∈Z} sinc(α-n)(-1)^n = (sin πα / π) Σ_{n∈Z} 1/(α-n) = cos πα, whereas the principal value of (-1)^α is e^{iπα}. The proof's Case 2 establishes convergence of the series at x = -1 but does not establish equality with (-1)^α; at α = 1/2 the series gives 0 instead of i. Since -1 lies on the unit circle, the claimed identity fails on the asserted domain |x| = 1.","section":"§2.2, Proposition 1 (Eq. 4)"},{"comment":"Because Proposition 1 fails at x = -1, Theorem 3 fails for unitary operators with -1 in the spectrum. For T = -I on C, Eq. (8) gives T^{1/2} = cos(π/2) I = 0, so (T^{1/2})^2 = 0 ≠ T = -I; the semigroup property, which the paper lists as a required property, is destroyed. The Fourier transform F satisfies F^4 = I and has -1 as an eigenvalue, so the 'alternate fractional Fourier transform' in Proposition 9 is not a fractional power of F and does not satisfy additivity.","section":"§3.2, Theorem 3 (Eq. 8) and §4.3, Proposition 9"},{"comment":"The statement 'If T^N = T' is inconsistent with its proof and with its applications. Proposition 2 requires the sequence c_n to be N-periodic, which for c_n = T^n means T^N = I; the applications in Lemma 1 and Proposition 9 use D^4 = I and F^4 = I. As stated, Theorem 4 is not proved, and the applications rely on a different hypothesis than the one announced.","section":"§3.2, Theorem 4"},{"comment":"The zeta-function interpolation is not an independent derivation: it substitutes the known identity η(s) = (1 - 2^{1-s})ζ(s) and the integer values ζ(k) into the Newton-series formula. No convergence domain for the resulting Newton series is established, and the formula does not determine ζ beyond its integer values unless convergence is proved. The claim that this provides an interpolation procedure for the Riemann zeta function is therefore overstated.","section":"§4.2, Lemma 3 and Theorem 5"}],"minor_comments":[{"comment":"The expression for P_n(α) is difficult to verify as printed; the falling-factorial form, the Γ(n+1) factor, and the final gamma ratio should be reconciled with the standard definition of the binomial coefficient.","section":"§2.1, Definition 1"},{"comment":"The fractional-derivative formulas do not specify the domain of f or the branch of λ^α used in the Fourier-series and Fourier-transform representations, even though the introduction states that the principal branch is used.","section":"§4.1.2, Propositions 6 and 7"},{"comment":"The figures lack axis labels, legends, and the precise parameter values used for the comparison, so the plots are not reproducible from the information given.","section":"§4.3, Figures 1-3"},{"comment":"Several classical sources are cited without page or theorem numbers (e.g., [11], [12], [14]), which makes it difficult to verify the convergence criteria invoked in the paper.","section":"References"}],"recommendation":"reject","confidential_remarks":"To the editor: The manuscript's main novelty is invalidated by the failure of Proposition 1 at x = -1, and the remaining Newton-series material is standard functional calculus. The zeta-function application is a restatement of known identities rather than a new interpolation. I recommend rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Let me get straight to it: the paper's central Shannon interpolation claim is false, and the failure sits at a spectral point that matters. Proposition 1 asserts x^α = Σ sinc(α-n)x^n for |x|=1, but the proof only shows convergence at x=-1 and then asserts equality. Direct partial fractions give Σ sinc(α-n)(-1)^n = cos(πα), not (-1)^α, so the identity fails. For T=-I, Theorem 3 then gives T^{1/2}=0, which destroys the semigroup property. The Fourier transform, which the paper's own application relies on, has -1 in its spectrum, so the alternate fractional Fourier transform in Proposition 9 is not a fractional power and does not satisfy additivity as claimed.\n\nThis is not a small gap; it is the load-bearing part of the advertised new theory. That said, I don't want to dismiss the whole paper. The Newton-series branch (Theorems 1 and 2) is legitimate. It is the binomial series applied through the holomorphic functional calculus, with clear conditions on the spectrum, and it gives a real way to approximate fractional powers on the disk. That part works. The zeta and Mellin sections are essentially restatements of known identities through Newton interpolation rather than new derivations, and Theorem 5 is ill-defined because the inner sum hits ζ(1). Theorem 4 is also mis-stated: the hypothesis is T^N = T, but the proof uses T^N = I.\n\nCredit where it is due: the paper is honest enough to include Remark 2, showing that the sinc series does not literally capture the translation semigroup, and it tries to repair this with limits. That suggests the author noticed some tension. But the repair does not fix the Fourier case, which is the one the paper presents as an application.\n\nBottom line: the Newton fragment is a sound exercise but not a new theory, and the Shannon fragment—the advertised alternate fractional Fourier transform and the interpolation of unitaries—collapses. I would not cite it, but I would send it to a referee rather than desk-reject, because the error is instructive and a referee can cleanly separate the salvageable Newton part from the false unitary claim. It could also serve as a useful reading-group example of why convergence of a sampling series is not enough to determine its sum.","headline":"The paper's central Shannon interpolation claim is false at x = -1, which breaks Theorem 3 and the alternate fractional Fourier transform; only the Newton-series part is sound.","tokens_in":11375,"tokens_out":2388,"would_cite":false,"duration_ms":24621,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["47A60","26A33","65D05","11M06","42A38","42A15"],"pacs":[],"model":"deepseek-v4-flash","headline":"Fractional powers of operators can be built by interpolating integer powers.","keywords":["fractional powers of operators","functional calculus","fractional calculus","Newton series","Shannon sampling","Dirichlet series","fractional Fourier transform","Riemann zeta function"],"falsifier":"Take $\\alpha=1/3$ and compute $\\sum_{n\\in\\mathbb{Z}}\\mathrm{sinc}(1/3-n)(-1)^n$. The partial-fraction evaluation gives $\\cos(\\pi/3)=1/2$, while the principal branch of $(-1)^{1/3}$ is $e^{i\\pi/3}$; a discrepancy here would show that the unitary formula fails for operators with $-1$ in the spectrum, such as the Fourier transform.","tokens_in":10303,"feed_emoji":"🧮","tokens_out":9910,"duration_ms":86764,"temperature":0.7,"pith_summary":"The paper sets out to construct fractional powers of bounded linear operators not through the abstract functional calculus, but by analytically interpolating the known sequence of integer powers $T, T^2, \\dots$ (and $T^n$ for $n\\in\\mathbb{Z}$ when $T$ is unitary). It claims that for operators whose spectrum lies strictly inside a disk not containing the origin, $T^{\\alpha}$ equals a Newton-series expression built from the integer powers, and for unitary operators it equals a sinc (Shannon) interpolation. If correct, this gives a more intuitive route to fractional calculus, a new family of fractional Fourier transforms, and a way to interpolate the Riemann zeta function from its values at integers. The paper also derives closed forms for operators of finite order and applies them to fractional derivatives of trigonometric and $L^2$ functions.","feed_headline":"Fractional powers of operators from interpolated integer powers","feed_subtitle":"Newton and sinc series lift operator powers to fractional exponents, touching calculus, Fourier analysis, and zeta.","key_machinery":"The load-bearing machinery is the conversion of a scalar interpolation formula into an operator formula. The scalar Newton identity $$$x^{{\\alpha}}$ = \\sum_{n\\ge 0}\\left[\\sum_{k=0}^{n}P_k(n)$x^{{k}}$\\right]P_n(\\$\\alpha$),\\qquad |x-1|<1,$$ with $P_n(\\alpha)=(-1)^n\\binom{\\alpha}{n}$, and the scalar Shannon identity $$$x^{{\\alpha}}$ = \\sum_{n\\in\\mathbb{Z}}\\mathrm{sinc}(\\$\\alpha$-n)$x^{{n}}$,\\qquad |x|=1,$$ are each fed into a functional calculus: the holomorphic calculus for spectra inside $B(1,1)$ and the continuous functional calculus for unitaries. A periodic version of the Shannon identity turns the unitary case into finite trigonometric sums when $T^N=T$, and that is what produces the paper's fractional Fourier transform and fractional derivatives of sine and cosine.","core_discovery":"The central claim is that nothing more than the integer powers of an operator is needed to define its fractional powers. For a bounded operator $T$ with spectrum strictly inside $B(z,|z|)$, the paper defines $$$T^{{\\alpha}}$ = \\sum_{n\\ge 0}\\left[\\sum_{k=0}^{n}P_k(n)\\$rho^{{\\alpha-k}}$$T^{{k}}$\\right]P_n(\\$\\alpha$),$$ where $P_n$ are the Pochhammer-Newton polynomials, and for unitary $T$ it defines $$$T^{{\\alpha}}$ = \\sum_{n\\in\\mathbb{Z}}\\mathrm{sinc}(\\$\\alpha$-n)$T^{{n}}$.$$ These series are claimed to reproduce integer powers, to satisfy the semigroup property, and to give inverses when $T$ is unitary. The paper proves the operator identities by pushing the scalar interpolation identities through the holomorphic functional calculus (for the disk case) and the continuous functional calculus (for unitaries), then uses the results to identify fractional integrals with Riemann-Liouville integrals, to produce fractional derivatives on $L^2$, to derive an alternate fractional Fourier transform, and to interpolate Dirichlet series such as $\\zeta(s)$.","pith_inferences":["A necessary refinement: the scalar sinc identity at $x=-1$ is asserted on the strength of convergence, but direct evaluation gives $\\sum_{n\\in\\mathbb{Z}}\\mathrm{sinc}(\\alpha-n)(-1)^n=\\cos(\\pi\\alpha)$, which differs from the principal branch $(-1)^\\alpha$ for noninteger $\\alpha$; the unitary formula should therefore be read with a caveat for spectra containing $-1$.","The same interpolation template could be applied to other analytic functions $f$ in place of $x^\\alpha$, producing $f(T)$ from integer iterates whenever the scalar Newton or sinc series converges; testing this on exponentials would link the construction to semigroup generation.","A truncated Newton series gives a parameter-free family of operators that approximate fractional powers; comparing its convergence rate on fractional diffusion against standard spectral methods would show where the interpolation is numerically competitive."],"forward_implications":["For the integration operator $J$, the interpolation series gives $J^{\\alpha}=\\sum_{n\\ge 0}\\left[\\sum_{k=0}^nP_k(n)J^{k}\\right]P_n(\\alpha)$, and the paper shows this coincides with the Riemann-Liouville fractional integral.","A fractional Fourier transform can be expressed as a linear combination of $f$, $\\hat f$, $f(-x)$, and $\\hat f(-x)$ with trigonometric coefficients in $\\alpha$, so it is computable directly from a function and its Fourier transform.","The Riemann zeta function is interpolated from its integer values by $\\zeta(s)=\\sum_{n\\ge 0}\\left[\\sum_{k=0}^nP_k(n)\\frac{1-2^{1-k}}{1-2^{1-s}}\\zeta(k)\\right]P_n(s)$, with a similar formula for $1/\\zeta(s)$ via the Möbius function.","Because fractional powers are approximated by finite sums of integer powers, the construction provides a numerical route to fractional evolution equations without solving for the semigroup generator."],"supporting_citations":[{"why":"supplies the holomorphic functional calculus for fractional powers whose integral representation is used to identify $J^\\alpha$ with the Riemann-Liouville integral.","marker":"[12]"},{"why":"states the Shannon sampling theorem that is the source of the sinc interpolation identity.","marker":"[16]"},{"why":"provides the convergence theory for Newton polynomial expansions used in the series over nonnegative integers.","marker":"[14]"},{"why":"gives the semigroup-based fractional power construction against which the interpolation results are aligned.","marker":"[11]"},{"why":"supports the Newton-series convergence framework through factorial-series Hilbert spaces.","marker":"[15]"},{"why":"supplies the semigroup theory behind the translation-operator and $e^{At}$ applications.","marker":"[10]"}],"fun_headline_variants":["Interpolate integer powers to get fractional operators","Fractional operator powers via analytic interpolation","Integer powers interpolate into fractional ones","New fractional operators from old integer powers","Interpolation turns integer powers fractional"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"Both constructions rest on scalar identities that express fractional powers as infinite sums of integer powers; the unitary case requires the sinc identity to hold at every spectral value of unit modulus, including $x=-1$, where the paper's proof establishes convergence but not equality.","fun_headline_variants_meta":{"raw":{"variants":["Interpolate integer powers to get fractional operators","Fractional operator powers via analytic interpolation","Integer powers interpolate into fractional ones","New fractional operators from old integer powers","Interpolation turns integer powers fractional"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000371,"raw_usage":{"total_tokens":1950,"prompt_tokens":876,"completion_tokens":1074,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":492,"completion_tokens_details":{"reasoning_tokens":1012}},"tokens_in":492,"tokens_out":1074,"duration_ms":9258,"temperature":1.0,"reasoning_tokens":1012,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T14:10:45.380555+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take $\\alpha=1/3$ and compute $\\sum_{n\\in\\mathbb{Z}}\\mathrm{sinc}(1/3-n)(-1)^n$. The partial-fraction evaluation gives $\\cos(\\pi/3)=1/2$, while the principal branch of $(-1)^{1/3}$ is $e^{i\\pi/3}$; a discrepancy here would show that the unitary formula fails for operators with $-1$ in the spectrum, such as the Fourier transform.","supporting_citations":[{"cited_title":"Komatsu, Fractional powers of operators","cited_arxiv_id":null,"evidence_quote":"supplies the holomorphic functional calculus for fractional powers whose integral representation is used to identify $J^\\alpha$ with the Riemann-Liouville integral."},{"cited_title":"Whittaker, On the functions which are represented by the expansions of the interpolation theory","cited_arxiv_id":null,"evidence_quote":"states the Shannon sampling theorem that is the source of the sinc interpolation identity."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"provides the convergence theory for Newton polynomial expansions used in the series over nonnegative integers."},{"cited_title":"Balakrishnan, Fractional powers of closed operators and the semigroups generated by them","cited_arxiv_id":null,"evidence_quote":"gives the semigroup-based fractional power construction against which the interpolation results are aligned."},{"cited_title":"Klopfenstein, A note on Hilbert spaces of factorial series","cited_arxiv_id":null,"evidence_quote":"supports the Newton-series convergence framework through factorial-series Hilbert spaces."},{"cited_title":"Hille, R.S","cited_arxiv_id":null,"evidence_quote":"supplies the semigroup theory behind the translation-operator and $e^{At}$ applications."}],"review_version":1}