On a sampling expansion with partial derivatives for functions of several variables

Saulius Norvidas

REVIEW 3 major objections 5 minor 11 references

On a sampling expansion with partial derivatives for functions of several variables

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 1908.07351 v1 pith:BSAGXJC7 submitted 2019-08-16 math.CA math.CV

Saulius Norvidas This is my paper · ORCID

classification math.CAmath.CV MSC 41A0541A6332A15

keywords multidimensionalsamplingwithderivativesBernsteinspacesentirefunctionsseriesbandlimitedmixedpartialsinckernelNyquistlattice

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

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.

What carries the argument

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).

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

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

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.

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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: 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.

The 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).

The 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.

However, 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.

Citation 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.

Who 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.

Referee Report

3 major / 5 minor

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.

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 (3)

[§2, Proposition 2.3, derivation of (2.13)] 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).
[§2, Proposition 2.3, application of induction to F~z] 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.
[§2, Proposition 2.3, second claim in the case r=m+1] 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.

minor comments (5)

[Title and abstract] The title contains a typo ('functi ons'); the abstract is otherwise clear.
[§2, equation (2.12)] 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.
[§2, line after (2.15)] The sentence 'This proves (2.14)' should read 'This proves (2.15)'.
[Throughout] There are several typos and duplicated words, including 'operator operator', 'modules p rincipe', 'this this', and 'lema'; a careful proofreading pass is needed.
[References] Reference [8] gives the journal name as 'Inform. Theorem'; this should be 'Information Theory', and the spelling 'alliasung' in reference [2] should be corrected.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the paper is a self-contained mathematical proof relying on external analytic facts.

full rationale

The paper derives a multidimensional sampling expansion for bandlimited functions from classical external tools (Paley-Wiener, Bernstein's inequality, Nikol'skii's inequality, Liouville's theorem, and a divisibility lemma for entire functions). The main theorem is proved by showing that the proposed series converges absolutely and uniformly and that the residual satisfies the uniqueness conditions of Proposition 2.3. No fitted parameter is renamed as a prediction, no conclusion is assumed through its own definition, and no load-bearing step is justified by a self-citation: the references are standard external sources, not prior work of the author. The only substantive concern in the proof is the derivation of claim (2.13) in Proposition 2.3, which may be a gap in the induction argument, but a proof gap is a correctness issue, not circularity. There is no reduction of a claimed result to its own inputs.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters are fitted. The central claim rests on standard theorems in Fourier analysis, complex analysis, and approximation theory, all explicitly cited. No new entities are postulated.

assumptions (5)

standard math Paley-Wiener-Schwartz theorem: functions in B^p_{Q^n_σ} extend to entire functions of exponential type
Invoked in Section 1 to identify bandlimited L^p functions with entire functions.
standard math Chirka's zero-division theorem: if entire F and H satisfy ord_z(H) ≤ ord_z(F) for all z, then F = G H for entire G
Used in Lemma 2.1 to factor f = sic_n g; this is the key factorization step in the induction.
standard math Bernstein's inequality for bandlimited functions (derivatives are bounded operators on B^p_{Q^n_π})
Used in Section 2 to ensure derivatives of f are again in the same Bernstein space and satisfy sampling bounds.
standard math Nikol'skii inequality: sampling values at lattice points belong to l^p
Used in (2.18) to justify the convergence estimates for the sampling series.
standard math Splettstösser's summability bound (2.19): Σ |sinc_n(a x - m)|^r is bounded uniformly in x
Used to prove absolute and uniform convergence of the series (1.7).

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Cite this review

Pith. "Pith review of On a sampling expansion with partial derivatives for functions of several variables." pith.science (2026). https://pith.science/paper/BSAGXJC7

@misc{pith2026190807351,
  author       = {Pith},
  title        = {Pith review of: On a sampling expansion with partial derivatives for functions of several variables},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BSAGXJC7}},
  note         = {Machine review of arXiv:1908.07351}
}

abstract

Let $B^p_{\sigma}$, $1\le p<\infty$, $\sigma>0$, denote the space of all $f\in L^p(\mathbb{R})$ such that the Fourier transform of $f$ (in the sense of distributions) vanishes outside $[-\sigma,\sigma]$. The classical sampling theorem states that each $f\in B^p_{\sigma}$ may be reconstructed exactly from its sample values at equispaced sampling points $\{\pi m/\sigma\}_{m\in\mathbb{Z}} $ spaced by $\pi /\sigma$. Reconstruction is also possible from sample values at sampling points $\{\pi \theta m/\sigma\}_m $ with certain $1< \theta\le 2$ if we know $f(\theta\pi m/\sigma) $ and $f'(\theta\pi m/\sigma)$, $m\in\mathbb{Z}$. In this paper we present sampling series for functions of several variables. These series involves samples of functions and their partial derivatives.

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Reference graph

Works this paper leans on

11 extracted references · 11 canonical work pages

[1]

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[3]

textit In: Linear and complex analysis problem book; 199 research problems

Gorin, E.A., Norms and extremals of convolution operato rs on spaces of entire func- tions. textit In: Linear and complex analysis problem book; 199 research problems. Lect. Notes Math., 1043 646-651, Springer-V erlag, Berlin (1984)

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Gosselin, R. P ., On Fourier transforms with small suppor t. J. Math. Anal. Appl., bf 13 166178 (1966)

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Higgins, J.R.: Sampling Theory in Fourier and Signal Ana lysis. Foundations. Claren- don Press, Oxford (1996)

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Pith tools

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

[1] [1]

M., Complex analytic sets

Chirka, E. M., Complex analytic sets. Kluwer Academic Pu blishers Group, Dordrecht (1989)

work page 1989

[2] [2]

Chinese Ann

Fang, G., Li, Y ., Multidimensional sampling theorem of H ermite type and estimates for alliasung error on Sobolev classes, J. Chinese Ann. Math. Se wr., A, 27 217-230, 2006 (in Chinese)

work page 2006

[3] [3]

textit In: Linear and complex analysis problem book; 199 research problems

Gorin, E.A., Norms and extremals of convolution operato rs on spaces of entire func- tions. textit In: Linear and complex analysis problem book; 199 research problems. Lect. Notes Math., 1043 646-651, Springer-V erlag, Berlin (1984)

work page 1984

[4] [4]

A., Bernstein inequalities from the perspecti ve of operator theory

Gorin, E. A., Bernstein inequalities from the perspecti ve of operator theory. (Russian) V estnik Kharkov. Gos. Univ. No. 205 (1980), 77105. A also in: Selecta Math. Soviet, 7 191209, (1988)

work page 1980

[5] [5]

P ., On Fourier transforms with small suppor t

Gosselin, R. P ., On Fourier transforms with small suppor t. J. Math. Anal. Appl., bf 13 166178 (1966)

work page 1966

[6] [6]

Foundations

Higgins, J.R.: Sampling Theory in Fourier and Signal Ana lysis. Foundations. Claren- don Press, Oxford (1996)

work page 1996

[7] [7]

H¨ ormander, L.: The Analysis of Linear Partial differen tial Operators. I. Distribution Theory and Fourier Analysis, 2nd edn. Springer-V erlag, Berlin, Heidelberg, New Y ork (1990)

work page 1990

[8] [8]

Jagerman, D., Fogel, L.J., Some general aspects of the sa mpling theorem, IEEE Trans. Inform. Theorem, 2 139-156 (1956)

work page 1956