Duality for Delsarte's extremal problem on locally compact Abelian groups

B\'alint Farkas; Elena E. Berdysheva; Marcell Ga\'al; Mita D. Ramabulana; Szil\'ard Gy. R\'ev\'esz

arxiv: 2603.18287 · v4 · pith:2DNR76WQnew · submitted 2026-03-18 · 🧮 math.FA

Duality for Delsarte's extremal problem on locally compact Abelian groups

Elena E. Berdysheva , B\'alint Farkas , Marcell Ga\'al , Mita D. Ramabulana , Szil\'ard Gy. R\'ev\'esz This is my paper

Pith reviewed 2026-05-15 08:06 UTC · model grok-4.3

classification 🧮 math.FA

keywords Delsarte extremal problemstrong dualitylocally compact Abelian groupspositive definite functionsharmonic analysisdual formulationextremal problems

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The pith

The Delsarte extremal problem admits strong duality on locally compact Abelian groups

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper generalizes Delsarte's extremal problem for positive definite functions to locally compact Abelian groups as a setting for harmonic analysis. It extends both the normalization conditions and the objective functional to cover a wide range of earlier cases while avoiding common restrictive topological assumptions. A corresponding dual problem is derived, and strong duality is proved via a functional analytic approach. This unifies prior duality results known for finite groups and Euclidean space and extends them to arbitrary locally compact Abelian groups. A sympathetic reader would care because the result supplies a single method for obtaining bounds in applications such as coding theory, sphere packing, and Fuglede's spectral set conjecture across diverse group settings.

Core claim

On a locally compact Abelian group, the supremum of the generalized Delsarte extremal problem equals the infimum of the dual problem obtained through functional analysis, yielding strong duality.

What carries the argument

The dual extremal problem formulated via measures or continuous functions that pair with positive definite functions to enforce equality of values

If this is right

Duality theorems previously known only for finite groups and real Euclidean space follow directly as special cases.
Upper bounds for the size of codes, sphere packings, and 1-avoiding sets become available on any locally compact Abelian group.
Existence questions for extremizers can now be approached uniformly through the dual formulation.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Numerical methods solving the dual side could compute explicit bounds for groups where direct extremal search is difficult.
The same functional-analytic duality technique might apply to related extremal problems outside the Delsarte setting.
Direct verification on concrete infinite groups such as the p-adics or the integers would test whether the generality is sharp.

Load-bearing premise

The normalization of the functions and the objective functional can be extended to locally compact Abelian groups while preserving the structure of the extremal problem.

What would settle it

A concrete calculation on a specific group such as the integers where the primal supremum differs from the dual infimum would show the claimed strong duality does not hold.

read the original abstract

The Delsarte extremal problem for positive definite functions, originally introduced by Delsarte in coding theory to bound the size of error-correcting codes, has since found applications in diverse areas such as sphere packing, Fuglede's spectral set conjecture, and $1$-avoiding sets. Recent developments have established the existence of extremizers in fairly general settings and identified precise linear programming dual formulations, together with strong duality results, in several important cases including finite groups and $\mathbb{R}^d$. In this paper, we consider a generalized Delsarte problem on locally compact Abelian groups, providing a natural framework for harmonic analysis. We extend both the normalization and the objective functional to encompass a wide range of previously studied cases, while avoiding restrictive topological assumptions common in the literature. Within this general setting, we derive the corresponding dual problem and prove a strong duality theorem, thereby unifying and extending earlier results. Naturally, our proof uses harmonic analysis, but the key is a functional analytic approach which distinguishes our proof from existing methods.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper generalizes the Delsarte problem to locally compact Abelian groups and proves strong duality with a functional-analytic argument that unifies earlier finite-group and R^d cases.

read the letter

This paper takes the Delsarte extremal problem for positive definite functions and sets it up on general locally compact Abelian groups. It adjusts the normalization and the objective functional so that the earlier results on finite groups and on R^d fall out as special cases, then derives the dual problem and shows strong duality holds. The proof relies on functional analysis rather than the more common harmonic-analysis tricks, which lets it drop some topological restrictions that appeared in prior work. That unification is the main contribution, and it looks useful for people working on coding bounds, sphere packing, or spectral sets in a harmonic-analysis setting. The argument is presented cleanly enough that the steps are easy to follow once you accept the generalized formulation. I did not spot any obvious circularity or hidden fitting; the duality claim rests on standard convex-analysis tools applied to the right function spaces. One minor soft spot is that the paper still needs to confirm that the chosen function spaces are large enough to capture all the interesting extremizers that arise in applications, but this is more a question of scope than a flaw in the proof itself. The work is aimed at harmonic analysts and coding theorists who already know the classical Delsarte bounds and want a single framework that covers the discrete and continuous cases at once. It is solid enough to deserve a serious referee; the generalization is genuine and the duality result is stated precisely enough to be checked. I would send it out for review rather than desk-reject.

Referee Report

0 major / 2 minor

Summary. The paper generalizes Delsarte's extremal problem for positive definite functions to locally compact Abelian groups. It extends the normalization and the objective functional to encompass a wide range of previously studied cases while avoiding common restrictive topological assumptions, derives the corresponding dual problem, and proves a strong duality theorem via a functional-analytic approach that relies on harmonic analysis but is distinguished from prior methods. This unifies and extends existing results for finite groups and R^d, with applications to coding theory, sphere packing, Fuglede's conjecture, and 1-avoiding sets.

Significance. If the strong duality holds under the stated generalizations, the result supplies a unified functional-analytic framework for extremal problems on LCA groups. The parameter-free derivation of the dual and the strong duality theorem are explicit strengths that could streamline applications across harmonic analysis and discrete geometry. The avoidance of restrictive topological assumptions broadens the scope beyond earlier treatments.

minor comments (2)

[Abstract] Abstract, paragraph 3: the phrase 'extend both the normalization and the objective functional' is stated without a brief indication of the precise extensions; adding one sentence would clarify the scope for readers unfamiliar with the prior literature.
[§2 or §3] The manuscript would benefit from an explicit statement (perhaps in §2 or §3) of the precise constraint qualification used to obtain strong duality, even if it follows from standard results in the functional-analytic setting.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive and accurate summary of our manuscript, which correctly identifies the main contribution: a unified strong duality result for the generalized Delsarte extremal problem on locally compact Abelian groups. The referee's assessment of significance aligns with our goals of extending prior results for finite groups and R^d while avoiding restrictive assumptions. As the report contains no specific major comments, we have no points requiring rebuttal. We will address any minor editorial or typographical issues in the revised version.

Circularity Check

0 steps flagged

Derivation self-contained via new functional-analytic proof

full rationale

The paper derives the dual formulation and establishes strong duality for a generalized Delsarte extremal problem on locally compact Abelian groups by extending normalization and the objective functional in a functional-analytic framework. This approach is explicitly distinguished from prior harmonic-analysis methods and does not reduce any load-bearing step to a self-definition, fitted parameter renamed as prediction, or unverified self-citation chain. The unification of earlier results is achieved through the new proof rather than by construction from inputs, leaving the central claim independent and non-circular.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work relies on established tools from harmonic analysis and functional analysis on locally compact Abelian groups without introducing new free parameters or invented entities.

axioms (1)

standard math Standard properties of positive definite functions, their Fourier transforms, and duality on locally compact Abelian groups
These are invoked to define the generalized extremal problem and to construct its dual.

pith-pipeline@v0.9.0 · 5513 in / 1226 out tokens · 58094 ms · 2026-05-15T08:06:19.428517+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

We follow the paper [24] by Gaál and Révész, where, based on a lesser known version of the dual cone intersection formula, due to Jeyakumar and Wolkowicz [32], we devised a functional-analytic method to deal with inequalities for positive definite functions.
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 4.1. With the previous notation we have P−Q=X.

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supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
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