Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.06114

Exactness of weighted exponential systems with a defect

Ivan Arefev·Andrei V. Semenov
View original source ↗Read source document ↗Export metadata ↓

Abstract

Let $w \colon (0,1) \to \mathbb{R}_{+}$ be a weight. We prove that for an arbitrary Schauder basis $\{e^{i λ_n t}\}_{n \in \mathbb{Z}}$ in $L^2(0,1)$ and an arbitrary lacunary defect set $A \subset \mathbb{Z}$ the system $\{w(t) r_n(t)\}_{n \in \mathbb{Z} \setminus A}$ is always complete and never minimal for any weight $w$ satisfying a natural decay condition, which is sharp on the exponential scale. Moreover, we establish a simple combinatorial criterion for systems of the form $ \{ w(t) e^{2πi n t} \}_{n \in \mathbb{Z} \setminus A} $ to be complete and minimal in $L^2(0,1)$ for an arbitrary weight $w$ and a finite defect set $A \subset \mathbb{Z}$.

Explore related subjects

math.FAmath.CAmath.CO

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Ivan Arefev, Andrei V. Semenov. 2026-10-05. Exactness of weighted exponential systems with a defect. https://arxiv.org/abs/2610.06114

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Operations on $\mathscr{M}\otimes_{\mathcal{A}(U)}\mathrm{Alt}^{\bullet}(U,M)$. Stokes theorem for locally convex linear space valued forms. Divergence theorem for locally convex linear space valued vector fields

We extend to $\mathscr{M}\otimes_{\mathcal{A}(U)}\mathfrak{T}_{\bullet}^{\bullet}(U,M)$ the usual operations defined on $\mathfrak{T}_{\bullet}^{\bullet}(U,M)$, then by employing the properties of the projective tensor product of locally convex spaces, we generalize to $\mathscr{M}\otimes_{\mathcal{A}(U)}\mathrm{Alt}^{\bullet}(U,M)$ the usual wedge product, insertion operator and exterior differential defined on $\mathrm{Alt}^{\bullet}(U,M)$ and establish their properties. Here $M$ is a smooth finite dimensional manifold, $U$ an open submanifold of $M$, $\mathcal{A}(U)$ the ring of smooth maps on $U$, $\mathfrak{T}_{\bullet}^{\bullet}(U,M)$ the $\mathcal{A}(U)$-module of smooth tensor fields of $M$ defined on $U$, $\mathrm{Alt}^{\bullet}(U,M)$ the $\mathcal{A}(U)$-module of smooth alternating tensor fields of $M$ defined on $U$, while $\mathscr{M}$ is a $\mathcal{A}(U)$-module. $\mathscr{M}$ depending by the operation might be either general, or one of the following function spaces: $\mathscr{L}_{c}^{1}(U,G,λ)$, $\mathscr{L}_{loc}^{1}(U,G,λ)$, $\mathcal{B}^{p}(U,G)$ with $G$ a Hausdorff locally convex space. This framenwork permits to construct the divergence of a $G$-valued $p$-times continuously differentiable vector field, to define the weak integral of $G$-valued compactly supported scalarly integrable maximal forms, to associate a $G$-valued measure with any locally integrable $G$-valued form, to establish a Stokes type theorem for $G$-valued compactly supported $p$-times continuously differentiable forms and to obtain a divergence type theorem for $G$-valued $p$-times continously differentiable vector fields.

math.FA↗
arXiv2026

The Grothendieck Constant is Strictly Larger than Davie-Reeds' Bound

The Grothendieck constant $K_{G}$ is a fundamental quantity in functional analysis, with important connections to quantum information, combinatorial optimization, and the geometry of Banach spaces. Despite decades of study, the value of $K_{G}$ is unknown. The best known lower bound on $K_{G}$ was obtained independently by Davie and Reeds in the 1980s. In this paper we show that their bound is not optimal. We prove that $K_{G} \ge K_{DR} + 10^{-12}$, where $K_{DR}$ denotes the Davie-Reeds lower bound. Our argument is based on a perturbative analysis of the Davie-Reeds operator. We show that every near-extremizer for the Davie-Reeds problem has $Ω(1)$ weight on its degree-3 Hermite coefficients, and therefore introducing a small cubic perturbation increases the integrality gap of the operator.

math.FA↗
arXiv2026

Higher-order differentiability of Korevaar--Schoen energy forms and energy measures

In this paper, we investigate the differentiability of Korevaar--Schoen $p$-energy forms and the associated $p$-energy measures. We obtain higher-order derivatives by virtue of explicit realizations of Korevaar--Schoen $p$-energy forms and the associated $p$-energy measures as subsequential pointwise limits of certain double integrals.

math.FA↗