arXiv subjects

Mita D. Ramabulana

Publications and source records attributed to Mita D. Ramabulana.

4 recordsLinked to original sources

arXiv2026

Duality for Delsarte's extremal problem on compact Gelfand pairs

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. 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.

math.CA↗

arXiv2026

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

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.

math.FA↗

arXiv2025

Delsarte-type extremal problems and convolution roots on homogeneous spaces

For a locally compact group $G$ and compact subgroup $K$, we consider a Delsarte-type extremal problem for $G$-invariant positive definite kernels on the homogeneous space $G/K$, generalising a certain Turán problem for isotropic positive definite kernels on the unit sphere $\mathbb{S}^d$ in $\mathbb{R}^{d+1}$. We exploit a correspondence between $G$-invariant kernels on $G/K$ and $K$-bi-invariant functions on $G$ to show that the Delsarte-type problem on a homogeneous space is equivalent to a Delsarte-type problem for $K$-bi-invariant functions on its group $G$ of transformations. We use this correspondence to show the existence of an extremal function for the Delsarte problem on the homogeneous space. In the case where $(G,K)$ is a compact Gelfand pair, we show the existence of $K$-bi-invariant convolution roots for positive definite $K$-bi-invariant functions, consequently obtaining the existence of a $G$-invariant convolution root for $G$-invariant positive definite kernels.

math.CA↗

arXiv2024

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

A unifying framework for some extremal problems on locally compact Abelian groups is considered, special cases of which include the Delsarte and Turán 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.

math.CA↗