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arxiv: 2603.11792 · v3 · submitted 2026-03-12 · 🧮 math.CA

Duality for Delsarte's extremal problem on compact Gelfand pairs

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 11:58 UTC · model grok-4.3

classification 🧮 math.CA
keywords Delsarte problempositive definite functionsGelfand pairslinear programmingstrong dualityTurán problemharmonic analysisextremal problems
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The pith

Delsarte extremal problems on compact Gelfand pairs admit strong duality when cast as infinite-dimensional linear programs.

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

The paper formulates Delsarte-type extremal problems for positive definite functions on compact Gelfand pairs as infinite-dimensional linear programs. It derives the corresponding dual problems explicitly and proves that the optimal values of the primal and dual coincide. The setup covers compact Abelian groups as a special case and distinguishes Turán and Delsarte problems according to sign restrictions on the functions. These duality results supply a common framework for bounding problems that arise in number theory, sphere packing, and statistics.

Core claim

We study Delsarte-type problems for positive definite functions on compact Gelfand pairs as infinite-dimensional linear programming problems. This setup includes, as a particular case, the case of compact Abelian groups. Depending on the restriction on the signs of the functions, we obtain two important particular cases, the Turán and Delsarte problems. In this paper, we describe their duals and prove a strong duality statement.

What carries the argument

Infinite-dimensional linear programming formulation of extremal problems for positive definite functions on compact Gelfand pairs, together with the explicit dual program and the strong duality theorem that equates their optimal values.

If this is right

  • The common optimal value can be computed or bounded by solving either the primal or the dual linear program.
  • Sphere-packing and Turán-type bounds on Gelfand pairs become accessible through the dual formulation.
  • The same duality recovers classical results for compact Abelian groups as a special case.
  • Sign-restricted versions of the problem (Turán versus Delsarte) inherit the same dual structure.

Where Pith is reading between the lines

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

  • The duality statement may serve as a template for extending the linear-programming approach to other classes of symmetric spaces where positive-definite functions are well understood.
  • Numerical solution of the dual programs could produce new explicit bounds for packing densities on specific Gelfand pairs.
  • The framework suggests examining whether weak duality alone suffices for certain applications when strong duality fails.

Load-bearing premise

The standard strong duality theorem for infinite-dimensional linear programs applies directly to the space of positive definite functions on compact Gelfand pairs.

What would settle it

A concrete compact Gelfand pair together with a feasible positive definite function whose objective value strictly exceeds the optimal value of the explicitly described dual program.

read the original abstract

We study Delsarte-type problems for positive definite functions on compact Gelfand pairs as infinite-dimensional linear programming problems. This setup includes, as a particular case, the case of compact Abelian groups. Depending on the restriction on the signs of the functions, we obtain two important particular cases, the Tur\'an and Delsarte problems. These problems have been studied in relation to number theory, sphere packing, and statistics. In this paper, we describe their duals and prove a strong duality statement.

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.

Referee Report

1 major / 1 minor

Summary. The paper formulates Delsarte-type and Turán-type extremal problems for positive definite functions on compact Gelfand pairs (including compact Abelian groups) as infinite-dimensional linear programs, explicitly describes the corresponding dual programs, and proves a strong duality theorem relating the primal and dual values.

Significance. If the strong duality holds under the stated conditions, the work supplies a clean dual description that could streamline the derivation of bounds in sphere packing, coding theory, and statistical applications on Gelfand pairs. The explicit duals also open the possibility of new analytic or numerical attacks on these classical problems.

major comments (1)
  1. [Section 3 (Duality theorem and proof)] The proof of strong duality (the central claim) invokes a standard infinite-dimensional LP duality theorem but does not explicitly verify the required constraint qualification—e.g., existence of a strictly feasible point in the interior of the positive-definite cone or weak*-compactness of the feasible set—in the uniform topology on C(G). Without this check, a duality gap cannot be ruled out, and the dual description loses its guaranteed optimality property.
minor comments (1)
  1. [Section 2] Notation for the cone of positive-definite functions and the dual pairing should be introduced once and used consistently; the current alternation between P and the space of measures is occasionally ambiguous.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting the need for explicit verification of the constraint qualification in the duality theorem. We will revise the manuscript accordingly to strengthen the presentation.

read point-by-point responses

  1. Referee: [Section 3 (Duality theorem and proof)] The proof of strong duality (the central claim) invokes a standard infinite-dimensional LP duality theorem but does not explicitly verify the required constraint qualification—e.g., existence of a strictly feasible point in the interior of the positive-definite cone or weak*-compactness of the feasible set—in the uniform topology on C(G). Without this check, a duality gap cannot be ruled out, and the dual description loses its guaranteed optimality property.

    Authors: We agree that an explicit check strengthens the argument. In the revised manuscript we will add a dedicated paragraph (or short subsection) verifying the hypotheses of the invoked infinite-dimensional LP duality theorem. Weak*-compactness of the feasible set follows from Alaoglu’s theorem: the unit ball of C(G)* is weak*-compact, and the positive-definite cone intersected with the linear constraints is a weak*-closed subset because G is compact and the positive-definiteness condition is weak*-closed. Strict feasibility holds by taking the constant function f ≡ 1, whose Fourier transform is a positive multiple of the delta at the trivial representation; this function lies in the interior of the positive-definite cone in the uniform norm and satisfies all inequality constraints strictly. Consequently the duality gap is zero and the dual attains its optimum. revision: yes

Circularity Check

0 steps flagged

No circularity detected in duality derivation

full rationale

The paper formulates Delsarte and Turán problems as infinite-dimensional linear programs over positive definite functions on compact Gelfand pairs and invokes standard strong duality results from functional analysis to obtain the dual problems. This relies on external theorems for infinite-dimensional LPs rather than any self-definitional reduction, fitted parameters renamed as predictions, or load-bearing self-citations. The derivation chain is self-contained against external benchmarks in convex analysis and harmonic analysis on groups, with no steps that reduce by construction to the paper's own inputs or prior author work.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on the domain assumption that positive definite functions on compact Gelfand pairs admit an infinite-dimensional linear programming formulation whose dual satisfies strong duality.

axioms (1)
  • domain assumption Positive definite functions on compact Gelfand pairs can be treated as variables in an infinite-dimensional linear program with sign restrictions yielding the Turán and Delsarte cases.
    This is the foundational setup stated in the abstract for obtaining the dual problems.

pith-pipeline@v0.9.0 · 5407 in / 1171 out tokens · 55041 ms · 2026-05-15T11:58:17.403770+00:00 · methodology

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

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

    math.FA 2026-03 unverdicted novelty 6.0

    Strong duality is proved for a generalized Delsarte extremal problem on locally compact Abelian groups, unifying prior results for finite groups and R^d.

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