arXiv subjects

Marcell Gaál

Publications and source records attributed to Marcell Gaál.

11 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↗

arXiv2020

Minimal energy point systems on the unit circle and the real line

In this paper, we investigate discrete logarithmic energy problems in the unit circle. We study the equilibrium configuration of $n$ electrons and $n-1$ pairs of external protons of charge $+1/2$. It is shown that all the critical points of the discrete logarithmic energy are global minima, and they are the solutions of certain equations involving Blaschke products. As a nontrivial application, we refine a recent result of Simanek, namely, we prove that any configuration of $n$ electrons in the unit circle is in stable equilibrium (that is, they are not just critical points but are of minimal energy) with respect to an external field generated by $n-1$ pairs of protons.

math.CA↗

arXiv2019

A note on real operator monotone functions

In this paper we initiate the study of real operator monotonicity for functions of tuples of operators, which are multivariate structured maps with a functional calculus called free functions that preserve the order between real parts (or Hermitian parts) of bounded linear Hilbert space operators. We completely characterize such functions on open convex free domains in terms of ordinary operator monotone free functions on self-adjoint domains. Further assuming the more stringent free holomorphicity, we prove that all such functions are affine linear with completely positive nonconstant part. This problem has been proposed by David Blecher at the biannual OTOA conference held in Bangalore in December 2016.

math.FA↗

arXiv2019

On the existence of an extremal function in the delsarte extremal problem

This paper is concerned with a Delsarte type extremal problem. Denote by $\mathcal{P}(G)$ the set of positive definite continuous functions on a locally compact abelian group $G$. We consider the function class, which was originally introduced by Gorbachev, \begin{multline*} \mathcal{G}(W, Q)_G = \left\{ f \in \mathcal{P}(G) \cap L^1(G) ~ : \right. ~ \left. f(0) = 1, ~ \text{supp}f_+ \subseteq W,~ \text{supp}\hat{f} \subseteq Q \right\} \end{multline*} where $W\subseteq G$ is closed and of finite Haar measure and $Q\subseteq \hat{G}$ is compact. We also consider the related Delsarte type problem of finding the extremal quantity \begin{equation*} \mathcal{D}(W,Q)_G = \sup \left\{ \int_{G} f(g) dλ_G(g) ~ : ~ f \in \mathcal{G}(W,Q)_G\right\}. \end{equation*} The main objective of the current paper is to prove the existence of an extremal function for the Delsarte type extremal problem $\mathcal{D}(W,Q)_G$. The existence of the extremal function has recently been established by Berdysheva and Révész in the most immediate case where $G=\mathbb{R}^d$. So the novelty here is that we consider the problem in the general setting of locally compact abelian groups. In this way our result provides a far reaching generalization of the former work of Berdysheva and Révész.

math.CA↗

arXiv2019

Integral comparisons of nonnegative positive definite functions on LCA groups

In this paper we investigate the following questions. Let $μ, ν$ be two regular Borel measures of finite total variation. When do we have a constant $C$ satisfying $$\int f dν\le C \int f dμ$$ whenever $f$ is a continuous nonnegative positive definite function? How the admissible constants $C$ can be characterized, and what is their optimal value? We first discuss the problem in locally compact abelian groups. Then we make further specializations when the Borel measures $μ, ν$ are both either purely atomic or absolutely continuous with respect to a reference Haar measure. In addition, we prove a duality conjecture posed in our former paper.

math.FA↗

arXiv2018

On a class of determinant preserving maps for finite von Neumann algebras

Let $\mathscr{R}$ be a finite von Neumann algebra with a faithful tracial state $τ$ and let $Δ$ denote the associated Fuglede-Kadison determinant. In this paper, we characterize all unital bijective maps $ϕ$ on the set of invertible positive elements in $\mathscr{R}$ which satisfy $$Δ(ϕ(A)+ϕ(B)) = Δ(A+B).$$ We show that any such map originates from a $τ$-preserving Jordan $*$-automorphism of $\mathscr{R}$ (either $*$-automorphism or $*$-anti-automorphism in the more restrictive case of finite factors). In establishing the aforementioned result, we make crucial use of the solutions to the equation $Δ(A + B) = Δ(A) + Δ(B)$ in the set of invertible positive operators in $\mathscr{R}$. To this end, we give a new proof of the inequality $$Δ(A+B) \ge Δ(A) + Δ(B),$$ using a generalized version of the Hadamard determinant inequality and conclude that equality holds for invertible $B$ if and only if $A$ is a nonnegative scalar multiple of $B$.

math.OA↗

arXiv2017

On certain generalized isometries of the special orthogonal group

In this paper we explore the structure of certain generalized isometries of the special orthogonal group $SO(n)$ which are transformations that leave any member of a large class of generalized distance measures invariant. This gives us a far-reaching generalization of the former structural result of T. Abe, S. Akiyama and O. Hatori concerning $c$-spectral isometries of the special orthogonal group.

math.FA↗

arXiv2017

On isometry groups of self-adjoint traceless and skew-symmetric matrices

This note is concerned with isometries on the spaces of self-adjoint traceless matrices. We compute the group of isometries with respect to any unitary similarity invariant norm. This completes and extends the result of Nagy on Schatten $p$-norm isometries. Furthermore, we point out that our proof techniques could be applied to obtain an old result concerning isometries on skew-symmetric matrices.

math.FA↗

arXiv2017

Maps on positive operators preserving Rényi type relative entropies and maximal $f$-divergences

In this paper we deal with two quantum relative entropy preserver problems on the cones of positive (either positive definite or positive semidefinite) operators. The first one is related to a quantum Rényi relative entropy like quantity which plays an important role in classical-quantum channel decoding. The second one is connected to the so-called maximal $f$-divergences introduced by D. Petz and M. B. Ruskai who considered this quantity as a generalization of the usual Belavkin-Staszewski relative entropy. We emphasize in advance that all the results are obtained for finite dimensional Hilbert spaces.

math.FA↗

arXiv2015

Transformations on density operators and on positive definite operators preserving the quantum Rényi divergence

In a certain sense we generalize the recently introduced and extensively studied notion called quantum Rényi divergence (in another name, sandwiched Rényi relative entropy) and describe the structures of corresponding symmetries. More precisely, we characterize all transformations on the set of density operators which leave our new general quantity invariant and also determine the structure of all bijective transformations on the cone of positive definite operators which preserve the quantum Rényi divergence.

math.FA↗