{"id":"822c87a7-a276-4a6b-87c1-3743a6b237bf","arxiv_id":"1908.07351","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For bandlimited functions of several variables, exact reconstruction is possible from samples of the function and all 2^n mixed first-order partial derivatives on the lattice (2π/σ)Z^n, using the squared sinc kernel.","lead":"This paper proves a new multivariate sampling theorem: a bandlimited function on R^n is exactly recovered from its samples and its 2^n mixed first-order derivatives at a half-Nyquist-rate lattice. It also gives a counterexample showing that an earlier two-variable sampling formula in the literature is incorrect and must include mixed partial derivatives.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 2.3's induction gives f=0 on the hyperplanes, but not the stronger claim (2.13) that every ∂^s f vanishes on H^{m+1}; the later division by sic^2 and Liouville step is unsupported as written.","rationale":"The reader identified the same load-bearing gap: the transition from F≡0 on C^m to the full claim (2.13) is unjustified. My reading confirms that the proof of Proposition 2.3 only establishes vanishing of f on each hyperplane H_m, while the later division by sic^2 requires vanishing of all first-order derivatives on H^{m+1}. The theorem may still be true—the support constraints for bandlimited functions plausibly force the missing transverse derivatives to vanish—but the written proof does not establish this. This is precisely the kind of repairable-but-unsupported step that warrants a conditional verdict rather than acceptance as fully verified. I found no independent reason to raise the confidence level or to reject the paper outright, since the earlier counterexample and the convergence estimates in the proof of Theorem 1.1 are sound, and the main result is plausible. The concrete Fourier-based test would either close the gap by providing an alternative proof or reveal a genuine counterexample.","tokens_in":9293,"tokens_out":26308,"duration_ms":245181,"concrete_test":"Check the n=2 case by Fourier analysis: write f(x,y)=∫_{Q_π^2} e^{i(xξ+yη)} dμ(ξ,η) for a measure μ, and translate the four vanishing conditions f(2a,2b)=f_x(2a,2b)=f_y(2a,2b)=f_{xy}(2a,2b)=0 into the statement that μ, ξμ, ημ, and ξημ have zero Fourier coefficients at all even lattice frequencies. Determine analytically, or by a finite-dimensional SVD of the truncated moment system, whether any nonzero μ supported in Q_π^2 satisfies these equations. A nonzero solution would falsify Proposition 2.3 and Theorem 1.1; if no solution exists, the missing step in (2.13) must be supplied by a separate argument, ideally by handling ∂_{z_1}f on the hyperplane directly.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Proposition 2.3 claims (2.13): ∂^s_z f(z)=0 for all s∈E_{m+1} and all z∈H^{m+1}. The derivation fixes ~z=(2u,~z2,…,~zn), defines F(z1,…,zm)=f(2u,z1,…,zm), and applies the induction hypothesis to conclude F≡0 on C^m. That conclusion yields only f(2u,z′)=0 for all z′∈C^m, i.e. f vanishes on the hyperplane H_1. It does not control ∂_{z1}f(2u,z′) when z′ is not itself an even lattice point, nor does it control mixed derivatives involving ∂_{z1} at such points. The later step 'Using (2.13), we get h(z)=f(z)/sic^2_{m+1}(πz)' requires f to have zeros of order at least two on every hyperplane z_j∈2Z, not merely to vanish there. Without (2.13), h may have poles and the boundedness/Liouville argument collapses. This is load-bearing because Proposition 2.3 is the uniqueness lemma that identifies the sampling series with f in the proof of Theorem 1.1. The gap may well be repairable: Fourier-support considerations suggest functions like sin(πz_1/2)φ(z_2,…) with φ∈B^p_{Q_{π/2}^{m}} are excluded by critical-density sampling for 1≤p<∞. But the written induction does not supply the missing transverse-derivative argument, so the theorem is not fully verified from the text.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves an n-dimensional sampling theorem with derivatives for Bernstein spaces B^p_{Q^n_σ}, 1 ≤ p < ∞. Theorem 1.1 asserts that every such f can be reconstructed from its values and all mixed partial derivatives of order at most one, taken on the lattice (2π/σ)Z^n, through a series whose terms are local Hermite-type polynomials multiplied by squared sinc factors. The proof is reduced to the case σ_j = π by an isometric isomorphism, and the main technical ingredient is Proposition 2.3, a uniqueness statement for B_{Q^n_π} functions whose values and first mixed derivatives vanish on 2Z^n. The paper also gives a counterexample showing that a previously published two-dimensional derivative sampling formula without mixed derivatives fails in general.","tokens_in":9601,"tokens_out":20183,"duration_ms":210028,"significance":"If Theorem 1.1 is correct, it is a clean and natural multidimensional analogue of the one-dimensional derivative sampling expansion, and it correctly identifies the role of mixed derivatives in overcoming the failure of naïve product-type formulas. The counterexample to formula (1.4) is convincing and is a useful contribution in itself. The convergence proof in (2.22)–(2.25) is careful and gives uniform convergence on R^n and on compact subsets of C^n, and the paper is self-contained apart from standard Bernstein-space facts. However, the uniqueness proof in Proposition 2.3 contains a load-bearing gap: the induction step does not establish claim (2.13) as written. The theorem may well be true, and the gap appears repairable, but the manuscript in its current form does not fully verify the central statement.","major_comments":[{"comment":"The proof of (2.13) is not valid. For fixed ~z=(2u,~z2,...,~zn), the function F is defined by F(z1,...,zm)=f(2u,z1,...,zm), so z1 is not a free variable. The induction hypothesis applied to F yields only that f and its derivatives with respect to the last m coordinates vanish at (2u,z′) for all z′∈C^m; it gives no information about ∂_{z1}f(2u,z′) or about mixed derivatives that contain ∂_{z1}. Consequently the displayed argument does not prove the strong claim (2.13) for all s∈E_{m+1} and all z∈H^{m+1}. This matters because the subsequent line 'Using (2.13), we get h(z)=f(z)/sic^2_{m+1}(πz)' requires f to have zeros of order at least two on the hyperplanes H^{m+1}; Lemma 2.1 alone supplies only one factor of sic_{m+1}. The later portions of the induction may be sufficient to obtain the needed factorization directly, but the written proof must be revised to remove the unsupported use of (2.13).","section":"§2, Proposition 2.3, derivation of (2.13)"},{"comment":"The proof of the second claim asserts that F~z∈BQπ^{m+1-r} after Lemma 2.2. Lemma 2.2 shows only that derivatives of f belong to BQπ^{m+1}; it does not show that restricting such a derivative to an affine coordinate subspace produces a function in BQπ^{m+1-r}. This restriction property is true and can be justified by noting that the Fourier transform of the restriction is supported in the projection of Qπ onto the remaining coordinates, but the argument is omitted. Without this justification, the induction hypothesis cannot be applied to F~z.","section":"§2, Proposition 2.3, application of induction to F~z"},{"comment":"The proof of (2.15) does not cover the case r=m+1, i.e., the case where ~z is itself a lattice point. In that case F~z is a constant and the induction hypothesis for dimension m is not available. This case can be handled directly: from the factorization f=sic_{m+1}g, the only non-automatic condition among (2.8) at a lattice point is the full mixed derivative with k=(1,...,1), which forces g(2u)=0. The authors should add this argument or otherwise treat the lattice-point case explicitly.","section":"§2, Proposition 2.3, second claim in the case r=m+1"}],"minor_comments":[{"comment":"The title contains a typo ('functi ons'); the abstract is otherwise clear.","section":"Title and abstract"},{"comment":"There is an extraneous absolute-value sign in the displayed estimate for |sin(πz/2)|; the intended inequality is |sin(πz/2)| ≥ (1/2)e^{π|Im z|/2} on the indicated region.","section":"§2, equation (2.12)"},{"comment":"The sentence 'This proves (2.14)' should read 'This proves (2.15)'.","section":"§2, line after (2.15)"},{"comment":"There are several typos and duplicated words, including 'operator operator', 'modules p rincipe', 'this this', and 'lema'; a careful proofreading pass is needed.","section":"Throughout"},{"comment":"Reference [8] gives the journal name as 'Inform. Theorem'; this should be 'Information Theory', and the spelling 'alliasung' in reference [2] should be corrected.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The central theorem is plausible and the convergence part is solid, but the uniqueness lemma is not fully proved as written. The identified gaps appear repairable within the manuscript's scope, so I would not recommend rejection; a major revision that rewrites the induction step of Proposition 2.3 is appropriate."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a real result and worth a referee, but the written proof has a gap that needs cleaning up before I would trust the details as they stand.\n\nThe new content is solid. Theorem 1.1 gives a multivariate sampling expansion for functions in B^p_{Q^n_σ} using all 2^n mixed first derivatives on a doubled lattice, and the counterexample (1.5) shows that the earlier formula (1.4) from Fang–Li is simply wrong as stated. Both are genuinely new. The counterexample is convincing and precisely targets the missing mixed derivatives. The positive theorem fills a natural hole in the derivative sampling literature, and the author makes the plausibility argument that you really do need every mixed derivative polynomial. The convergence proof is careful and uses standard tools (Hölder, Nikolskii, Bernstein, Splettstösser's estimate).\n\nThe soft spot is Proposition 2.3, the uniqueness lemma. The proof claims (2.13): all first-order derivatives of f vanish on H^{m+1}. The induction step as written only establishes f=0 on the hyperplane, not the transverse derivatives. The later line 'Using (2.13), we get h(z)=f(z)/sic^2' therefore rests on an assertion that is not proven. This is a genuine gap in the exposition.\n\nHowever, I think the stress-test note overstates the damage. The subsequent part of the proof actually proves g=0 on H^{m+1}, and once you have f=sic·g and g=sic·h, the identity h=f/sic^2 follows from the factorizations alone. You do not need the full derivative vanishing (2.13) for that step. So the gap is repairable, and likely not load-bearing for the theorem's truth. What is missing is a clean rewrite: the author should either prove (2.13) properly or remove the claim and restructure the argument around the two factorizations.\n\nCitation pattern is normal; the author engages the relevant sampling and entire-function literature and corrects a specific published formula. The mathematics is self-contained modulo standard results.\n\nWho this is for: researchers in sampling theory, multivariate approximation, or complex analytic factorization. It is a niche but genuine contribution. My recommendation: send to a serious referee in sampling theory or complex analysis. The counterexample alone justifies publication; the uniqueness proof needs revision before I would certify the main theorem from the text as written.","headline":"Genuinely new derivative sampling theorem with a convincing counterexample; the proof has a repairable gap in Proposition 2.3 that should be fixed before publication but does not block peer review.","tokens_in":10086,"tokens_out":9210,"would_cite":true,"duration_ms":86405,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["41A05","41A63","32A15"],"pacs":[],"model":"deepseek-v4-flash","headline":"A bandlimited function of several variables is exactly determined by its values and all mixed first partial derivatives on a lattice twice as coarse as the usual sampling grid.","keywords":["multidimensional sampling with derivatives","Bernstein spaces","entire functions","sampling series","bandlimited functions","mixed partial derivatives","sinc kernel","Nyquist lattice"],"falsifier":"The decisive check is to look in dimension two for a nonzero $f\\in B^p_{Q^2_\\pi}$ with $f(u)=\\partial_{z_1}f(u)=\\partial_{z_2}f(u)=\\partial_{z_1}\\partial_{z_2}f(u)=0$ for every $u\\in 2\\mathbb{Z}^2$; such a function would refute the theorem. A lighter numerical check is to evaluate the right-hand side of (1.7) for a closed-form bandlimited function such as $f(z)=\\mathrm{sinc}_2(z)$ and compare it with $f$ at points away from the lattice, and separately to verify whether the missing derivative step in Proposition 2.3 holds.","tokens_in":9081,"feed_emoji":"📐","tokens_out":16246,"duration_ms":141874,"temperature":0.7,"pith_summary":"The paper claims a multidimensional sampling theorem: every function whose Fourier transform is supported in a rectangular box and which lies in $L^p$ can be reconstructed exactly from its values and all of its mixed first-order partial derivatives at lattice points spaced by twice the usual sampling step. The reconstruction is an explicit series in which a first-degree Taylor polynomial at each sample point is multiplied by the square of the multidimensional sinc kernel. If the theorem is right, halving the sampling density in any number of variables costs exactly the $2^n$ mixed-derivative samples at each retained point, and no smaller set of derivative data works in general. This gives a parameter-free exact reconstruction rule for multidimensional bandlimited signals, the natural $n$-dimensional analogue of the classical one-variable derivative sampling formula.","feed_headline":"Mixed derivatives rebuild functions from half the samples","feed_subtitle":"Values plus all first mixed partials at each lattice point recover the whole function in any dimension","key_machinery":"The central object is the sampling identity itself, with the polynomial factor $P_{f,k,u}(z-u)=(\\partial^k_z f)(u)\\prod_{j=1}^n(z_j-u_j)^{k_j}$ and the squared multidimensional sinc kernel $\\mathrm{sinc}_n^2(\\sigma(z-u)/(2\\pi))$. The load-bearing mechanism is Proposition 2.3, an induction on dimension: if a bandlimited function and all its mixed first partial derivatives vanish on the lattice $2\\mathbb{Z}^n$, the function is identically zero. The induction factors out the product-of-sine factors $\\mathrm{sic}_n(\\pi z)$ using a divisibility criterion for entire functions, controls the quotient by the exponential-type estimate (2.7), and applies Liouville's theorem to force the quotient to be constant. The convergence proof then splits the series by Hölder's inequality, using the $\\ell^p$ summability of derivative samples from Bernstein- and Nikol'skii-type inequalities and the sinc summability estimate (2.19).","core_discovery":"The paper's central claim is Theorem 1.1: for $f\\in B^p_{Q^n_\\sigma}$ with $1\\le p<\\infty$, the series $$f(z)=\\sum_{u\\in(2\\pi/\\$\\sigma$)\\mathbb{Z}^n}\\left(\\sum_{k\\in E_n}P_{f,k,u}(z-u)\\right)\\mathrm{sinc}$_n^{2}$\\left(\\frac{\\$\\sigma$(z-u)}{2\\pi}\\right)$$ converges absolutely and uniformly on $\\mathbb{R}^n$ and on compact subsets of $\\mathbb{C}^n$, and its sum equals $f$. The sample data are the $2^n$ numbers $\\partial^k_z f(u)$ for every multi-index $k\\in\\{0,1\\}^n$ at every lattice point $u\\in(2\\pi/\\sigma)\\mathbb{Z}^n$. The paper also shows that the set of derivative data is minimal in a strong sense: if any one of these mixed terms is dropped from the series, there are nonzero bandlimited functions that vanish at every remaining sample and are therefore sent to zero by the truncated formula.","pith_inferences":["If the theorem is right, the proof's reliance on zero-set divisibility suggests the same sampling scheme should transfer to other lattices or spectral shapes whose kernels have the same divisibility property, so the rectangular box is probably not essential.","The tail bounds already visible in the convergence argument could be converted into explicit truncation-error estimates for the series, which would make the formula usable in numerical reconstruction with controlled error.","Stopping at $p<\\infty$ is not an oversight: the paper's own counterexample lives in $B^\\infty$, so an $L^\\infty$ version of the theorem would need additional hypotheses."],"forward_implications":["For any $f\\in B^p_{Q^n_\\sigma}$, the $2^n$ values $\\partial^k_z f(u)$ with $k\\in\\{0,1\\}^n$ on the lattice $(2\\pi/\\sigma)\\mathbb{Z}^n$ form a complete encoding: they determine $f$ everywhere on $\\mathbb{C}^n$.","Because the series converges uniformly on compact subsets of $\\mathbb{C}^n$, termwise differentiation of the sampling expansion is justified, so derivatives of $f$ can also be recovered from the same samples.","The minimality argument implies that universal exact reconstruction at this halved sampling density is impossible unless every mixed first derivative is included in the data.","In one dimension the theorem reduces to the known formula $f(z)=\\sum_u(f(u)+f'(u)(z-u))\\mathrm{sinc}_1^2(\\sigma(z-u)/(2\\pi))$, showing the multidimensional statement is an extension rather than a new one-dimensional result."],"supporting_citations":[{"why":"Supplies the Paley-Wiener-Schwartz theorem identifying bandlimited $L^p$ functions with entire functions.","marker":"[7, p. 181]"},{"why":"Supplies the Bernstein inequality and related estimates used to place derivative samples in $\\ell^p$.","marker":"[10, p. 116]"},{"why":"States the one-dimensional derivative sampling formula that the theorem generalizes.","marker":"[8, p. 145]"},{"why":"Gives the earlier two-dimensional Hermite-type sampling expression that the paper shows is generally false without mixed derivatives.","marker":"[2]"},{"why":"Provides the entire-function divisibility criterion used to factor out $\\mathrm{sic}$ factors in the uniqueness lemma.","marker":"[1, p. 12]"},{"why":"Provides the multidimensional sinc-summability estimate used in the uniform convergence proof.","marker":"[11, p. 811]"},{"why":"Supplies the convolution-operator facts used to show partial derivatives preserve the space $B_{Q^n_\\pi}$.","marker":"[3, p. 646]"}],"fun_headline_variants":["Partial derivatives double the gap in sampling theory","Multivariate sampling: add derivatives, halve the points","Derivatives at lattice points rebuild bandlimited functions","Fewer samples, more derivatives: new multivariate sampling","Sampling series with partial derivatives for multi-dim signals"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof of the uniqueness lemma depends on an unstated step: that vanishing of all sampled mixed derivatives on the lattice forces the derivatives to vanish on the whole complex hyperplane through those lattice points, not just along the lattice directions; if that step fails, the division used to prove uniqueness cannot be justified and Theorem 1.1 is left unsupported.","fun_headline_variants_meta":{"raw":{"variants":["Partial derivatives double the gap in sampling theory","Multivariate sampling: add derivatives, halve the points","Derivatives at lattice points rebuild bandlimited functions","Fewer samples, more derivatives: new multivariate sampling","Sampling series with partial derivatives for multi-dim signals"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000598,"raw_usage":{"total_tokens":2790,"prompt_tokens":931,"completion_tokens":1859,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":547,"completion_tokens_details":{"reasoning_tokens":1784}},"tokens_in":547,"tokens_out":1859,"duration_ms":14470,"temperature":1.0,"reasoning_tokens":1784,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T12:58:22.355184+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"The decisive check is to look in dimension two for a nonzero $f\\in B^p_{Q^2_\\pi}$ with $f(u)=\\partial_{z_1}f(u)=\\partial_{z_2}f(u)=\\partial_{z_1}\\partial_{z_2}f(u)=0$ for every $u\\in 2\\mathbb{Z}^2$; such a function would refute the theorem. A lighter numerical check is to evaluate the right-hand side of (1.7) for a closed-form bandlimited function such as $f(z)=\\mathrm{sinc}_2(z)$ and compare it with $f$ at points away from the lattice, and separately to verify whether the missing derivative step in Proposition 2.3 holds.","supporting_citations":[{"cited_title":"Chinese Ann","cited_arxiv_id":null,"evidence_quote":"Gives the earlier two-dimensional Hermite-type sampling expression that the paper shows is generally false without mixed derivatives."}],"review_version":1}