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Hicham Zoubeir

Publications and source records attributed to Hicham Zoubeir.

14 recordsLinked to original sources
arXiv2026

Probability measure annihilating all finite-dimensional subspaces

We propose in this short note a prime numbers-based method for constructing probability measures on infinite-dimensional Banach spaces annihilating all finite-dimensional subspaces, supplementing the methods of construction of Gaussian measures and infinite-product-type probability measures. This new method confirms that probability measures with this property are generic amongst probability measures that are not supported on finite-dimensional subspaces. In the process, we show the existence of an uncountable measurable family of independent vectors having the cardinality of the continuum in any infinite-dimensional Banach space.

math.FA↗
arXiv2026

Two Hornich-Hlawka-type and Gram matrix-based inequalities

We establish two inequalities in real inner product spaces. The first is a multiplicative strengthening of the classical Hornich-Hlawka inequality: for all vectors $x, y, z$ in a real inner product space $H$ \[ \|x\|\,\|y\| + \|z\|\,\|x+y+z\| \;\geq\; \|x+z\|\,\|y+z\|. \] We provide a complete characterization of the equality cases in terms of the linear dependence of $x,y,z$, and explicit conditions on their Gram matrix, showing in particular that equality occurs only in flat (at most two-dimensional) configurations. We also show that this inequality implies the classical Hornich-Hlawka inequality, thereby establishing a strict hierarchy between the two. The second result is a parametric inequality derived from the positive semidefiniteness of Gram matrices: for all $x,y,z \in H$ and $α, β, γ\in \mathbb{R}$, \[ α^2\|x\|^2\langle y,z\rangle^2 + β^2\|y\|^2\langle x,z\rangle^2 + γ^2\|z\|^2\langle x,y\rangle^2 + 2(αβ+ αγ+ βγ)\langle x,y\rangle\langle x,z\rangle\langle y,z\rangle \;\geq\; 0. \] Optimizing over the parameters yields sharp inequalities relating the pairwise inner products and norms of three vectors, which can be viewed as reverse inequalities to the Gram determinant inequality $\det G \geq 0$. As a special case, this recovers and strengthens the classical Cauchy-Schwarz inequality.

math.CA↗
arXiv2025

Almost sure linear independence of absolutely continuous Hilbert space-valued random vectors with respect to a special class of Hilbert space probability measures

This note examines the implications of randomly selecting vectors from an infinite-dimensional Hilbert space on linear independence, assuming that for all $k$, the first $k$ vectors follow an absolutely continuous law with respect to a probability measure. It demonstrates that no constraints on the random dimension of their span are necessary, provided that all finite-dimensional vector subspaces are considered negligible with respect to the Hilbert space probability measure.

math.FA↗
arXiv2025

Invertibility in partially ordered nonassociative rings

Invertibility is important in ring theory because it enables division and facilitates solving equations. Moreover, (nonassociative) rings can be endowed with an extra ''structure'' such as order and topology allowing more richness in the theory. The two main theorems of this article are contributions to invertibility in the context of partially ordered nonassociative rings \textit{and} Hausdorff sequentially Cauchy-complete weak-quasi-topological nonassociative rings. Specifically, the first theorem asserts that the interval $]0,1]$ in any suitable partially ordered nonassociative ring consists entirely of invertible elements. The second theorem asserts that if $f$ is a suitably generalized concept of seminorm from a nonassociative ring to a partially ordered nonassociative ring endowed with Frink's interval topology, then under certain conditions, the subset of elements such that $f(1-a) < 1$ consists entirely of invertible elements. Part of the assumption of the second theorem is that of Hausdorff sequential Cauchy-completeness of the first ring under the topology induced by the seminorm $f$ (which takes values in a partially ordered nonassociative ring endowed with Frink's interval topology). Frink's interval topology is an example of a coarse locally-convex $T_1$ topology. Moreover, to our knowledge, the topology induced by a seminorm into a partially ordered nonassociative ring has never been introduced. Some additional original facts, such as the fact that the topology on a nonassociative ring $R_1$ induced by a norm into a totally ordered associative division ring $R_2$ endowed with Frink's interval topology (or equivalently, with the order topology, since the order of $R_2$ is total) is a Hausdorff locally convex quasi-topological group with an additional separate continuity property of the product, are dealt with in the second section ''Preliminaries''.

math.AC↗
arXiv2023

On *-fusion frames for Hilbert C*-modules

Our main goal in this paper, is to generalize to Hilbert C*-modules the concept of fusion frames. Indeed we introduce the notion of *ñfusion frames associated to weighted sequences of orthogonally complemented submodules of a Hilbert C*-module, and prove for such a *-fusion frames some fundamental results.

math.GM↗
arXiv2020

On the representation and the uniform polynomial approximation of polyanalytic functions of Gevrey type on the unit disk

In this paper we define Gevrey polyanalytic classes of order N on the unit disk D and we obtain for these classes a characteristic expansion into N-analytic polynomials on suitable neighborhoods of D. As an application of our main theorem, we perform for the Gevrey polyanalytic classes of order N on the unit disk D, an analogue to E. M. Dyn'kin's theorem. We also derive, for these classes, their characteristic degree of the best uniform approximation on D by N-analytic polynomials.

math.CV↗
arXiv2019

Solvability in Gevrey classes of some linear functional equations

In this paper, we associate to each positive number k a new class of endomorphisms of the sheaf of germs of holomorphic functions on [-1,1] and prove the solvability in the Gevrey class G_k([-1,1]) of some linear functional equations related to endomorphisms.

math.CV↗