Pith. sign in

REVIEW 3 major objections 4 minor 18 references

On extremal problems of Delsarte type for positive definite functions on LCA groups

T0 review · 3 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Boundary-coherent sign sets always admit extremal functions for Delsarte- and Turán-type problems.

desk verdict Solid existence theorem for a new sign-set Delsarte/Turán variant on LCA groups, but the printed boundary-coherence definition is self-contradictory; the fix is one line and the proofs already use the right condition. read the letter →

arxiv 2412.00482 v1 pith:KSI2YU4G submitted 2024-11-30 math.CA

Elena E. Berdysheva , Mita D. Ramabulana , Szilárd Gy. Révész This is my paper · ORCID
classification math.CA MSC 43A35
keywords locallycompactAbeliangroupspositivedefinitefunctionsDelsarteextremalproblemTuránexistenceofboundary-coherentsetssign-setformulation
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 sets up a common umbrella for Delsarte- and Turán-type extremal problems on locally compact Abelian groups, using the weaker condition that the positivity and negativity sets of a function are contained in prescribed sets, instead of requiring containment of supports. Its main theorem states that if those prescribed sets are boundary-coherent, symmetric, and the positive set is a neighbourhood of zero with finite Haar measure, then the extremal value is actually attained by an admissible function. This is an existence result, not just an estimate: it turns the supremum defining the problem into a maximum. The authors also show that for boundary-coherent sets this new formulation is equivalent to the original support-based formulation, so the existence conclusion carries over to the classical Delsarte and Turán constants.

What carries the argument

The central condition is boundary-coherence: a set $\Omega$ is boundary-coherent when every boundary point is also a boundary point of its exterior $\operatorname{ext}\Omega = X\setminus\overline{\Omega}$, so boundary points can be approached from outside the closure. As printed, Definition 1 states $\partial\Omega\subset\operatorname{ext}\Omega$, but the intended and used condition is $\partial\Omega\subset\partial(\operatorname{ext}\Omega)$. The proof machinery is a weak-compactness argument: extremal sequences are bounded in $L^2(G)$ by the finite measure of $\Omega_+$; the convex-combination lemma turns the weak limit into a strong $L^2$ and almost-everywhere limit of convex combinations; the limit is integrally positive definite and hence, by a standard theorem on such functions, agrees almost everywhere with a continuous positive definite function; boundary-coherence then forces the sign preimages of the corrected limit into $\Omega_+$ and $\Omega_-$. This last step is where the topological hypothesis does its work.

What would settle it

Find a locally compact Abelian group $G$ and symmetric boundary-coherent sets $\Omega_+$, $\Omega_-$, with $\Omega_+$ a neighbourhood of 0 having finite Haar measure, for which the supremum defining $C_G(\Omega_+,\Omega_-)$ is not attained by any function in $F_G(\Omega_+,\Omega_-)$. Theorem 17 asserts no such pair exists, so one concrete pair would settle the question.

Watch

Extended reading notes

Core claim

On any locally compact Abelian group $G$ with Haar measure $\lambda_G$, the paper proves that the extremal problem $C_G(\Omega_+,\Omega_-)=\sup\{\int_G f\,d\lambda_G : f\in F_G(\Omega_+,\Omega_-)\}$, where $F_G(\Omega_+,\Omega_-)$ consists of continuous positive definite functions with $f(0)=1$, integrable, and satisfying $f^{-1}(0,\infty)\subset\Omega_+$ and $f^{-1}(-\infty,0)\subset\Omega_-$, has a solution whenever $\Omega_+$ and $\Omega_-$ are symmetric, boundary-coherent, and $\Omega_+$ is a neighbourhood of 0 of finite Haar measure. The proof first reduces the general group case to a $\sigma$-compact open subgroup, constructs an extremal function there by weak compactness in $L^2$ combined with a convex-combination lemma and a standard lifting of integrally positive definite functions to continuous ones, and then extends it by trivial extension. Boundary-coherence is used precisely once, to force the sign sets of the limiting function to lie in the prescribed sets. Corollaries assert the same existence for the Delsarte constant $D_G(\Omega)$ and the Turán constant $T_G(\Omega)$ when the defining set is boundary-coherent.

Load-bearing premise

The load-bearing premise is that both $\Omega_+$ and $\Omega_-$ satisfy the intended boundary-coherence condition — every boundary point of the set is also a boundary point of its exterior, so boundary points can be approached from outside the closure — and the paper notes this condition is used exactly once, in the final step, and could not be removed.

Editorial extensions

If this is right

  • The Delsarte and Turán constants are attained for every boundary-coherent symmetric neighbourhood of 0 with finite Haar measure, on any LCA group (Corollaries 19 and 20).
  • For boundary-coherent sets, the new sign-set class equals the original support-based class, so the two formulations of the extremal problem have the same admissible functions and the same constants (Proposition 24 and Corollary 27).
  • In topological vector spaces over $\mathbb{R}$, for bounded 0-symmetric sets satisfying a radial-containment condition, all six extremal constants in the comparison chain coincide (Corollary 34).
  • Every extremal function can be approximated by compactly supported admissible functions in $L^1$ and uniformly on compact sets, so compactly supported test functions are dense enough to compute the constant (Proposition 18).

Reading between the lines

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

  • The boundary-coherence assumption is probably not necessary for existence: Example 21 gives a non-coherent set with an extremal function, so the exact class of sets guaranteeing existence is still open.
  • Because the main theorem reduces to $\sigma$-compact subgroups, the result applies uniformly to finite-dimensional, infinite-dimensional, and totally disconnected LCA groups; the hypothesis is purely topological and independent of the group's lattice structure.
  • The equivalence results suggest that in future applications of Delsarte-type bounds one may freely switch between support and preimage formulations whenever the defining sets are boundary-coherent, which could simplify the search for extremal functions in concrete packing and energy problems.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper studies a general extremal problem for continuous positive definite functions on locally compact Abelian groups, where the usual support conditions are replaced by the weaker preimage conditions f^{-1}(0,∞) ⊂ Ω_+ and f^{-1}(-∞,0) ⊂ Ω_-. It introduces the corresponding Delsarte and Turán variants, proves reductions to σ-compact open subgroups, and establishes an existence theorem for extremal functions under a new topological condition called "boundary-coherence" (Theorem 17). The paper also compares the new formulation with the original support-based one, proving equivalence of the extremal constants for boundary-coherent sets and for star-shaped sets in topological vector spaces, with several examples illustrating sharpness.

Significance. If the main existence theorem is correct, it provides a broad existence result for Delsarte- and Turán-type extremal problems on general LCA groups under an explicit topological hypothesis, substantially extending previous work restricted to specific groups or sets. The paper is careful in reducing to σ-compact groups, uses standard tools (weak compactness, Mazur's lemma, approximation of unity) in a transparent way, and includes instructive examples showing that boundary-coherence is not necessary for existence. The equivalence section also clarifies the relationship between the two formulations. However, as printed, the central definition is misstated, and the proof of the main theorem relies on a different, intended definition; this must be repaired before the results can be considered reliable.

major comments (3)
  1. [Definition 1; proof of Theorem 14; Theorem 17] Definition 1 defines boundary-coherence as ∂Ω ⊂ ext Ω, where ext Ω = X \ cl(Ω). Since ∂Ω and ext Ω are disjoint by construction, this condition forces ∂Ω = ∅, i.e., Ω is clopen. The paragraph after Definition 1 claims equivalence with ∂Ω ⊂ ∂(ext Ω), but this equivalence is false: for Ω = (-1,1) ⊂ R, ∂Ω = {-1,1} satisfies ∂Ω ⊂ ∂(ext Ω) but does not satisfy ∂Ω ⊂ ext Ω. The proof of Theorem 14 uses the intended condition: for arbitrary x ∉ Ω_+ it invokes "by boundary-coherence there are points z ∈ V ∩ ext Ω_+", which requires x ∈ cl(ext Ω_+), i.e., ∂Ω_+ ⊂ ∂(ext Ω_+). Consequently, as printed, Theorem 17 covers only clopen sets and does not cover the motivating open convex sets, intervals, or Examples 21/28. The definition and the paragraph after it must be corrected to ∂Ω ⊂ ∂(ext Ω), and all statements and proofs (Theorems 14 and 17, Corollaries 19 and 20, Proposition 24, Lemma 16) should be read and restated with this intended notion.
  2. [Section 4, proof of Theorem 14] The proof asserts "The fact that ∫_G f dλ_G ≥ C_G(Ω_+,Ω_-) > 0" without justification. Strict positivity of C_G(Ω_+,Ω_-) is not automatic from the definition of the extremal constant and is used to deduce f(0) = 1. The inequality is true and can be proved from Lemma 12 by choosing a compact symmetric neighbourhood K of 0 with K+K ⊂ Ω_+ and considering a normalized convolution f = (1_K * 1_K)/λ(K), which lies in F_G(Ω_+,Ω_-) and has positive integral λ(K); the argument should be supplied in the text rather than left implicit.
  3. [Section 5, Eq. (18) and preceding paragraph] The sentence "Firstly, the equality F (Ω_+, Ω_-) = F*(Ω_+, Ω_-) is easily seen to be true, always" is false, and the displayed chain (18) ends with the same equality. Example 22 gives a direct counterexample: for Ω = (-1,1) ⊂ R, the triangular extremal function lies in F_R(Ω,Ω) but not in F*_R(Ω,Ω). This is a substantive error in the equivalence discussion, although it appears to be a local slip; the chain and the surrounding claims need to be corrected, for instance by removing the final equality and stating it only under the additional assumptions of Proposition 24 or Corollary 26 where it actually holds.
minor comments (4)
  1. [Examples 21 and 28] Example 21 and Example 28 are identical (same set Ω, same extremal constant, same figures); one of them should be removed or the two occurrences should be cross-referenced instead of repeated.
  2. [Proposition 15, display after (14)] The line "∫_G f_− λ_G dλ_G ≤ lim inf ..." contains a misplaced λ_G; it should read ∫_G f_− dλ_G ≤ lim inf ... . The same typo appears in the inequality two lines above (15).
  3. [Display (18) and (19)] The notation in the chains (18) and (19) is ambiguous because closure bars appear to be missing (e.g., terms like F (int Ω_+, int Ω_-) are not clearly distinguished from F (overline{int Ω_+}, overline{int Ω_-})). The typesetting should be cleaned so that each equality is unambiguously stated.
  4. [Lemma 16] The statement of Lemma 16 says "i.e., ∂_H(Ω ∩ H) ⊂ ext_H(Ω ∩ H)^H", which is the printed (literal) form of boundary-coherence; once Definition 1 is corrected, the statement should be updated to the intended form ∂_H(Ω ∩ H) ⊂ ∂_H(ext_H(Ω ∩ H)) to match the proof, which uses the corrected notion.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; main existence theorem is proved from standard tools, with only minor non-load-bearing self-citations.

full rationale

The paper's central result, Theorem 17 (existence of an extremal function for C_G(Ω+,Ω−)), is proved by a self-contained argument: weak/strong compactness in L2, Mazur's lemma, integral positive definiteness, the Sasvári theorem, and a final boundary-coherence argument. The constant C_G and the class F_G are not defined in terms of any optimizer; existence is an independent conclusion, not an input. The auxiliary facts cited from the authors' own prior work (Proposition 29 from [2], Lemma 31 from [2, Theorem 2.1]) are used only in the equivalence section to extend equalities of extremal constants under convex/star-shaped conditions, not to prove the main existence theorem; each is stated as an established published result, and Theorem 32 is additionally proved independently in the paper, with credit to [14] noted after the proof. The proof of Theorem 14 also explicitly uses boundary-coherence only in the final sign-inclusion step, and the paper candidly states that the condition may be unnecessary. There is no fitted parameter renamed as a prediction, no uniqueness theorem imported from the authors' prior work, and no ansatz smuggled in via citation. Citation to [15] (by co-author Ramabulana) and [13]/[16] (by co-author Révész) is used for background lemmas (Lemma 2, Lemma 3) and for the known Turán constant in Example 21, but these references do not carry the existence proof. A separate concern, flagged in the skeptic headline, is that Definition 1 as printed ('∂Ω ⊂ ext Ω') contradicts the intended condition ('∂Ω ⊂ ∂(ext Ω)'), since ∂Ω and ext Ω are disjoint; as literally stated, Theorem 17 would only cover clopen sets. However, that is a correctness/typo issue in the hypothesis, not circularity: the proof does not assume its conclusion. The asserted unproved step C_G(Ω+,Ω−) > 0 in Theorem 14 is also a gap in exposition rather than circularity, since positivity follows from Lemma 12. On the circularity scale, the paper is a standard derivation from established harmonic analysis with minor self-citations that are not load-bearing for the main theorem, so the appropriate score is 0–1; I set 0 because there is no step that reduces to its own input by construction.

Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

The main theorem relies on standard harmonic analysis and functional analysis theorems, listed above, plus the specific boundary-coherence assumption on the sets. No numerical parameters are fitted, and no new entities are postulated.

assumptions (8)
  • standard math Haar measure exists and is unique up to a positive constant on locally compact abelian groups.
    Used throughout to define integrals, Lp spaces, and the Fourier transform on LCA groups.
  • standard math Closed bounded convex sets in L2 are weakly sequentially compact (Eberlein-Smulian).
    Invoked in Theorem 14 via Brezis [3] to extract a weakly convergent subsequence of an extremal sequence.
  • standard math Mazur's lemma: convex combinations of a weakly convergent sequence converge strongly to the weak limit.
    Used in Theorem 14 to pass from weak to strong convergence in L2 while staying in the convex admissible class.
  • standard math An integrally positive definite function on a sigma-compact LCA group agrees almost everywhere with a continuous positive definite function.
    Used to correct the weak limit to a continuous positive definite representative; cited from Sasvari [17].
  • standard math The support of any L1 function on an LCA group is contained in an open sigma-compact subgroup.
    Used in Theorem 11 and Proposition 18; cited from Deitmar-Echterhoff [6].
  • standard math Approximation of unity: for a compact set C and eta>0 there exists a continuous compactly supported positive definite k with 0<=k<=1, k(0)=1, and k|_C > 1-eta.
    Lemma 12 cited from Kolountzakis-Revesz [13]; used in Proposition 15 and 18 to construct compactly supported approximants.
  • standard math The product of two positive definite functions is positive definite.
    Used in Proposition 15 and 18 to ensure fn k remains in the admissible class.
  • domain assumption Omega+ and Omega- are symmetric, boundary-coherent, with Omega+ a neighbourhood of 0 of finite Haar measure.
    Main hypothesis of Theorems 14 and 17; boundary-coherence is the new topological condition introduced in this paper.

how reviews work

0 comments
Cite this review

Pith. "Pith review of On extremal problems of Delsarte type for positive definite functions on LCA groups." pith.science (2026). https://pith.science/paper/KSI2YU4G

@misc{pith2026241200482,
  author       = {Pith},
  title        = {Pith review of: On extremal problems of Delsarte type for positive definite functions on LCA groups},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KSI2YU4G}},
  note         = {Machine review of arXiv:2412.00482}
}
read the original abstract

A unifying framework for some extremal problems on locally compact Abelian groups is considered, special cases of which include the Delsarte and Tur\'an extremal problems. A slight variation of the extremal problem is introduced and the different formulations are studied for equivalence. Extending previous work, a general result on existence of extremal functions for the new variant is proved under a certain general topological condition.

Figures

Figures reproduced from arXiv: 2412.00482 by the authors.

Figure 1
Figure 1. Extremal functions for our class and if we allow fun [PITH_FULL_IMAGE:figures/full_fig_p011_1.png] view at source ↗
Figure 2
Figure 2. Extremal functions for Ω and for Ω. Notice that in Corollary 26 the set F ∗ (Ω+, Ω−) is missing. Indeed, Example 22 demonstrates that this should be the case and that we can not do better under these conditions. On the other hand, this does not exclude, as demonstrated by Example 22, the possibility that under the same conditions the term C ∗ (Ω+, Ω−) should feature in the equalities in Corollary 27. We consider thi… view at source ↗
Figure 3
Figure 3. Two open, 0-symmetric, star-shaped sets. The left set satisfies the condition rX ⊂ intX, 0 ≤ r < 1; the right set does not satisfy this condition — it is even true that rX 6⊂ X. Theorem 33 and its proof are essentially the same as the argument used in Theorem 32, so we give credit to [14] while observing that our work naturally leads us to consider a result along these lines. Now we have the following general form o… view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

18 extracted references · 17 canonical work pages

  1. [1]

    V. V. Arestov and E. E. Berdysheva, Turán’s problem for a class of polytopes , East J. Approx. 8 (2002) 3, 381–388

    work page 2002

  2. [2]

    E. E. Berdysheva, Sz. Gy. Révész, Delsarte’s Extremal Problem and Packing on Locally Com- pact Abelian Groups , Annali della Scuola Normale di Pisa – Classe di Scienze (5) XXIV (2023), 1007–1052

    work page 2023

  3. [3]

    Functional Analysis, Sobolev Spaces, and Pa rtial Differential Equations

    H. Brezis, “Functional Analysis, Sobolev Spaces, and Pa rtial Differential Equations”, Springer- Verlag, New York (2010)

    work page 2010

  4. [4]

    Carathéodory, Über den Variabilitätsbereich der Fourier’schen Konstanten v on positiven harmonischen Funktionen , Rendiconti del Circolo Matematico di Palermo, 32 (1911), 193– 217

    C. Carathéodory, Über den Variabilitätsbereich der Fourier’schen Konstanten v on positiven harmonischen Funktionen , Rendiconti del Circolo Matematico di Palermo, 32 (1911), 193– 217

    work page 1911

  5. [5]

    H. Cohn, D. De Laat, and A. Salmon, Three-point bounds for sphere packing, arXiv:2206.15373

    arXiv

  6. [6]

    Principles of Harmonic Analy sis

    A. Deitmar, S. Echterhoff, “Principles of Harmonic Analy sis”, Second Edition, Springer (2014)

    work page 2014

  7. [7]

    Domar, An extremal problem for positive definite functions , J

    Y. Domar, An extremal problem for positive definite functions , J. Math. Anal. Appl. 52 (1975), 56–63. 15

    work page 1975

  8. [8]

    A Course in Abstract Harmonic Analysis

    G. Folland, “A Course in Abstract Harmonic Analysis”, Se cond Edition, Taylor & Francis Group (2016)

    work page 2016

Show all 18 references
  1. [9]

    Fejér, Über trigonometrische Polynome , J

    L. Fejér, Über trigonometrische Polynome , J. Reine Angew. Math. 146 (1916), 53-82

    1916

  2. [10]

    Gaál and Zs

    M. Gaál and Zs. Nagy-Csiha, On the Existence of an Extremal Function in the Delsarte Extremal Problem, Mediterr. J. Math, 17:190 (2020)

    2020

  3. [11]

    D. V. Gorbachev, An extremal problem for periodic functions with supports in th e ball , Math. Notes (3) 69 (2001), 313–319

    2001

  4. [12]

    M. N. Kolountzakis and Sz. Gy. Révész, On a problem of Turán about positive definite func- tions, Proc. Amer. Math. Soc. 131 (2003), 3423–3430

    2003

  5. [13]

    M. N. Kolountzakis and Sz. Gy. Révész, Turán’s extremal problem for positive definite functions on groups , J. London Math. Soc. 74 (2006), 475–496

    2006

  6. [14]

    P. G. Mavroudis, On the approximation of positive definite functions by like f unctions of smaller support , Bull. Math. Sci. 3 (2013), 287–298

    2013

  7. [15]

    M. D. Ramabulana, On the existence of an extremal function for the Delsarte extr emal problem, Analysis Mathematica, to appear. See also as arxiv: 2407.04 410

  8. [16]

    Sz. Gy. Révész, Turán’s extremal problem on locally compact Abelian groups , Anal. Math. 37 (2011), 15–50

    2011

  9. [17]

    Positive definite and definitizable functi ons

    Z. Sasvári, “Positive definite and definitizable functi ons”, Vol. 2, Akademie Verlag, Berlin (1994)

    1994

  10. [18]

    C. L. Siegel, Über Gitterpunkte in konvexen Körpern und ein damit zusammenhä n¨ gendes Extremalproblem, Acta Math. 65 (1935), 307–323. 16

    1935

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.