Berdysheva 2026 duality delsarte s extremal problem locally

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Elena E. Berdysheva, Bálint Farkas, Marcell Gaál, Mita D. Ramabulana, Szilárd Gy. Révész, Duality for Delsarte's Extremal Problem on Locally Compact Abelian Groups. arXiv preprint (2026). arXiv:2603.18287.

The authors set up Delsarte's extremal problem for positive definite functions on a general locally compact abelian group G: for a symmetric set Omega with 0 in its interior and compact closure, the Delsarte constant D_G(F,Omega) is the supremum of the Haar integral of f over continuous positive definite f in a function class F with f(0) = 1 and f <= 0 off Omega (Definition 1.1). They generalize further by replacing both the objective (the Haar integral) and the normalization (point evaluation at 0) with almost arbitrary linear functionals, while avoiding the restrictive topological assumptions on Omega common in the literature; since the function classes used in R^d (band-limited by Gorbachev, fast-decaying by Viazovska, Schwartz by Cohn-Elkies and others) are not available on a general group, the problem is posed on the amalgam space C^{infty,1}(G;R), the Wiener algebra. In this generality they derive the dual infinite-dimensional linear program and prove a strong duality theorem, unifying and extending known finite-group and R^d results; the proof uses harmonic analysis, but its key ingredient is functional-analytic, a dual cone intersection formula of Jeyakumar and Wolkowicz (Lemma 4.4, applied in Proposition 4.5), which distinguishes it from prior methods. The main result is Theorem 5.3, stated below; Theorem 5.1 is its compactly generated case, Theorems 1.2 and 5.6 are the discrete and compact cases, and Remark 5.4 recovers the Delsarte constant of Definition 1.1, with the amalgam space as function class, by taking minus the Haar integral as objective and evaluation at 0 as normalization. The abstract lists sphere packing, Fuglede's spectral set conjecture, and 1-avoiding sets among the uses of Delsarte's problem, the last connecting to the Erdos-Moser distance-avoiding set conjecture, which the introduction records as proved by Ambrus, Csiszárik, Matolcsi, Varga and Zsámboki. For problem 1070 it is a method paper only: it names 1-avoiding sets and the Erdos-Moser conjecture without treating them, and gives no planar density or finite ratio.

Source: https://arxiv.org/abs/2603.18287. The held PDF is arXiv:2603.18287v4 (28 May 2026), and page and label citations refer to it. The arXiv record (https://arxiv.org/abs/2603.18287, read 2026-10-02) names the Creative Commons Attribution 4.0 license.

Bears on. #1070

Results to transcribe.

• Definition 1.1 (p. 2): For an LCA group G with Haar measure lambda, a function class F forming a subspace of the real-valued continuous functions on G, and symmetric Omega with 0 in int Omega and compact closure, the Delsarte constant is D_G(F,Omega) = sup of the integral of f over all continuous positive definite f in F with f(0) = 1 and f <= 0 on G minus Omega.
• Theorem 5.3 (p. 18): Let G be an LCA group, Omega a symmetric subset of G with 0 in int Omega (no closure condition), and rho, sigma translation-bounded real Radon measures on G, the dual of the amalgam space X = C^{infty,1}(G;R), with sigma strictly positive definite (Wiener's condition, that its Fourier transform is positive) and rho even. Then the infimum of <f,rho> over positive definite f in X with f <= 0 on G minus Omega and <f,sigma> = 1 equals the supremum of the s in R with rho - s sigma = nu - kappa, where nu is real and positive definite and kappa >= 0 is supported in the closure of G minus Omega: there is no duality gap.
• Remark 5.4 (p. 18): With rho = -lambda and sigma = delta_0, minus the primal value is the Delsarte constant D_G(X,Omega), so D_G(X,Omega) is the infimum of the s with s delta_0 - lambda = nu - kappa for such nu and kappa.