Related papers: Upper and lower estimates for schauder frames and …

In this paper it is proved that an irreducible weight module with finite-dimensional weight spaces over the Schr\"{o}dinger-Virasoro algebras is a highest/lowest weight module or a uniformly bounded module. Furthermore, indecomposable…

We show that every infinite-dimensional Banach space with separable dual admits an equivalent norm which is weakly locally uniformly rotund but not locally uniformly rotund.

We establish that the summability of the series $\sum\varepsilon_n$ is the necessary and sufficient criterion ensuring that every $(1+\varepsilon_n)$ Markushevich basis in a separable Hilbert space is a Riesz basis. Further we show that if…

We prove a sampling theorem for infinite-dimensional Paley-Wiener spaces on graphs which allows for stable frame reconstruction. We prove that all sampling sets for a fixed Paley-Wiener space are complements of lambda-sets (i.e. sets where…

In this paper, we begin by constructing global linear maps on (n-2)-dimensional subspaces, derived from the local continuity of linear transformations among central sections of a convex body. Using these linear maps, we subsequently…

We prove the existence of tight frames whose elements lie on an arbitrary ellipsoidal surface within a real or complex separable Hilbert space H, and we analyze the set of attainable frame bounds. In the case where H is real and has finite…

We prove that if every element $u$ in a Hilbert space $H$ admits a representation as unconditionally convergent series $$u=\sum_{k=1}^\infty \langle u, y_k\rangle x_k,$$ then there exist nonzero scalars $\{\alpha_k\}_{k=1}^\infty$ such that…

It is known that any separable Banach space with BAP is a complemented subspace of a Banach space with a basis. We show that every operator with bounded approximation property, acting from a separable Banach space, can be factored through a…

Motivated from two decades old famous Feichtinger conjectures for frames, $R_\varepsilon$-conjecture and Weaver's conjecture for Hilbert spaces (and their solution by Marcus, Spielman, and Srivastava), we formulate Feichtinger conjectures…

We study two related quantities which generalize the concept of upper Banach density of a set to two measurable subsets of the plane. The first of them allows us to generalize a classic result on sufficiently large distances realized in a…

Given a Banach space $X$ and a real number $\alpha\ge 1$, we write: (1) $D(X)\le\alpha$ if, for any locally finite metric space $A$, all finite subsets of which admit bilipschitz embeddings into $X$ with distortions $\le C$, the space $A$…

A theorem of Delorme states that every unitary representation of a connected Lie group with nontrivial reduced first cohomology has a finite-dimensional subrepresentation. More recently Shalom showed that such a property is inherited by…

A boundary for a Banach space is a subset of the dual unit sphere with the property that each element of the Banach space attains its norm on an element of that subset. Trivially, the pointwise convergence with respect to such a boundary is…

We develop a novel and unifying setting for phase retrieval problems that works in Banach spaces and for continuous frames and consider the questions of uniqueness and stability of the reconstruction from phaseless measurements. Our main…

Let $B_Y$ denote the unit ball of a normed linear space $Y$. A symmetric, bounded, closed, convex set $A$ in a finite dimensional normed linear space $X$ is called a {\it sufficient enlargement} for $X$ if, for an arbitrary isometric…

This paper considers explicit constructions of Auerbach bases in separable Banach spaces. Answering the question of A. Pe{\l}czy{\'n}ski, we prove by construction the existence of Auerbach basis in arbitrary subspace of $c_0$ of finite…

Motivated by the Lyapunov convexity theorem in infinite dimensions, we extend the convexity of the integral of a decomposable set to separable Banach spaces under the strengthened notion of nonatomicity of measure spaces, called…

In this paper we deal with those Banach spaces $Z$ which satisfy the Mazur--Ulam property, namely that every surjective isometry $\Delta$ from the unit sphere of $Z$ to the unit sphere of any Banach space $Y$ admits an unique extension to a…

We show that every Banach space $X$ containing an isomorphic copy of $c_0$ has an infinite equilateral set and also that if $X$ has a bounded biorthogonal system of size $\alpha$ then it can be renormed so as to admit an equilateral set of…

Existing concentration bounds for bounded vector-valued random variables include extensions of the scalar Hoeffding and Bernstein inequalities. While the latter is typically tighter, it requires knowing a bound on the variance of the random…