P. Gavruta
We prove the Feichtinger Conjecture for a class of Bessel sequences of unit norm vectors in a Hilbert space. Also, we prove that every Bessel sequence of unit vectors in a Hilbert space can be partitioned into finitely many uniformly separated sequences.
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In this paper we study the Feichtinger Conjecture in frame theory, which was recently shown to be equivalent to the 1959 Kadison-Singer Problem in $C^{*}$-Algebras. We will show that every bounded Bessel sequence can be decomposed into two subsets each of which is an arbitrarily small perturbation of a sequence with a finite orthogonal decomposition. This construction is then used to answer two open problems concerning the Feichtinger Conjecture: 1. The Feichtinger Conjecture is equivalent to the conjecture that every unit norm Bessel sequence is a finite union of frame sequences. 2. Every uni…
We prove two new equivalences of the Feichtinger conjecture that involve reproducing kernel Hilbert spaces. We prove that if for every Hilbert space, contractively contained in the Hardy space, each Bessel sequence of normalized kernel functions can be partitioned into finitely many Riesz basic sequences, then a general bounded Bessel sequence in an arbitrary Hilbert space can be partitioned into finitely many Riesz basic sequences. In addition, we examine some of these spaces and prove that for these spaces bounded Bessel sequences of normalized kernel functions are finite unions of Riesz bas…
We discuss the equivalence of the Feichtinger Conjecture with a weaker variant and we show its connection with a conjecture of Agler--McCarthy--Seip concerning complete Nevanlinna--Pick reproducing kernel spaces. Some new examples of sequences of normalized reproducing kernels satisfying the Feichtinger Conjecture are consequently obtained.
We sharpen the constant in the $KS_2$ conjecture of Weaver \cite{We}, which was validated by Marcus, Spielman, and Srivastava \cite{MSS} in their solution of the Kadison--Singer problem. We then apply this result to prove optimal asymptotic bounds on the size of partitions in the Feichtinger conjecture.
We obtain two results concerning the Feichtinger conjecture for systems of normalized reproducing kernels in the model subspace $K_Θ= H^2\ominus ΘH^2$ of the Hardy space $H^2$, where $Θ$ is an inner function. First, we verify the Feichtinger conjecture for the kernels $ \tilde k_{λ_n} = k_{λ_n}/\|k_{λ_n}\|$ under the assumption that $\sup_n |Θ(λ_n)|<1$. Secondly, we prove the Feichtinger conjecture in the case where $Θ$ is a one-component inner function, meaning that the set $\{z:|Θ(z)|<\varepsilon\}$ is connected for some $\varepsilon\in(0,1)$.
Selector form of Weaver's conjecture, Feichtinger's conjecture, and frame sparsification
本文推广了Marcus等人解决Kadison-Singer问题的概率结果，建立了Weaver和Feichtinger猜想的选择子形式，并应用于框架稀疏化与离散化，改进了现有结果。
We show that any weakly separated Bessel system of model spaces in the Hardy space on the unit disc is a Riesz system and we highlight some applications to interpolating sequences of matrices. This will be done without using the recent solution of the Feichtinger conjecture, whose natural generalization to multi-dimensional model sub-spaces of $\mathrm{H}^2$ turns out to be false.
We prove that if a Bessel sequence in a Hilbert space, that is indexed by a countably infinite group in an invariant manner, can be partitioned into finitely many Riesz basic sequences, then each of the sets in the partition can be chosen to be syndetic. We then apply this result to prove that if a Fourier frame for a measurable subset of a higher dimensional cube can be partitioned into Riesz basic sequences, then each subset can be chosen to be a syndetic subset of the corresponding higher dimensional integer lattice. Both of these results follow from a result about syndetic pavings of eleme…
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 for p-approximate Schauder frames, $R_\varepsilon$-conjecture, Weaver's conjectures and Akemann-Weaver conjectures for Banach spaces. We also formulate conjectures on p-approximate Schauder frames based on the results of Casazza for frames for Hilbert spaces. We state conjectures and problems for p-approximate Schauder frames based on fundamental inequality f…
The Feichtinger conjecture for exponentials asserts that the following property holds for every fat Cantor subset B of the circle group: the set of restrictions to B of exponential functions can be covered by Riesz sets. In their seminal paper on the Kadison-Singer problem, Bourgain and Tzafriri proved that this property holds if the characteristic function of B has Sobolev regularity. Their probability based proof does not explicitly construct a Riesz cover. They also showed how to construct fat Cantor sets whose characteristic functions have Sobolev regularity. However, these fat Cantor sets…
Based on the truth of Feichtinger conjecture by Marcus, Spielman and Srivastava \textit{[Ann. of Math. (2), 2015]} and from the localized version by Gröchenig \textit{[Adv. Comput. Math., 2003]}, we introduce the notion of localization of approximate Schauder frames (ASFs) and approximate Riesz sequences (ARSs). We show that localized bounded below ASFs are finite unions of ARSs.
The Kadison-Singer problem asks: does every pure state on the diagonal sublgebra of the C*-algebra of bounded operators on a separable infinite dimensional Hilbert space admit a unique extension? A yes answer is equivalent to several open conjectures including Feichtinger's: every bounded frame is a finite union of Riesz sequences. We consider the special case: Feichtinger's conjecture for exponentials and prove that the set of projections onto a measurable subset of the circle group of the set of exponential functions equals a union of a finite number of Reisz sequences if and only if there e…
The Feichtinger Conjecture, if true, would have as a corollary that for each set $E\subset \T$ and $Λ\subset \Z$, there is a partition $Λ_1,...,Λ_N$ of $\Z$ such that for each $1\le i \le N$, $\{\exp(2πi xλ): λ\in Λ_i\}$ is a Riesz sequence. In this paper, sufficient conditions on sets $E\subset \T$ and $Λ\subset \R$ are given so that $\{\exp(2πi xλ) 1_E: λ\in Λ\}$ can be uniformly partitioned into Riesz sequences.
We prove that every unconditionally summable sequence in a Hilbert space can be factorized as the product of a square summable scalar sequence and a Bessel sequence. Some consequences on the representation of unconditionally convergent multipliers are obtained, thus providing positive answers to a conjecture by Balazs and Stoeva in some particular cases.
Weighted frames, weighted lower semi frames and unconditionally convergent multipliers
本文研究序列在加权后成为框架或下半框架的条件，提出相应猜想并给出若干成立条件，同时证明两个关于无条件收敛乘子与权重平移的旧猜想之间的等价性，并揭示其与新猜想的关系。
Weighted discrete Hilbert transforms $(a_n)_n \mapsto \sum_n a_n v_n/(z-γ_n)$ from $\ell^2_v$ to a weighted $L^2$ space are studied, with $Γ=(γ_n)$ a sequence of distinct points in the complex plane and $v=(v_n)$ a corresponding sequence of positive numbers. In the special case when $|γ_n|$ grows at least exponentially, bounded transforms of this kind are described in terms of a simple relative to the Muckenhoupt $(A_2)$ condition. The special case when $z$ is restricted to another sequence $Λ$ is studied in detail; it is shown that a bounded transform satisfying a certain admissibility condit…
Frame-normalizable Sequences
本文研究可分希尔伯特空间中的框架可规范化序列，给出其充要条件与扰动定理，验证了Balazs-Stoeva猜想对Bessel可规范化序列成立，并部分解答了正规算子相关迭代系统的框架性公开问题。
We formulate a quantitative finite-dimensional conjecture about frame multipliers and prove that it is equivalent to Conjecture 1 in [SB2]. We then present solutions to the conjecture for certain classes of frame multipliers. In particular, we prove that there is a universal constant $κ>0$ so that for all $C,β>0$ and $N\in\mathbb{N}$ the following is true. Let $(x_j)_{j=1}^N$ and $(f_j)_{j=1}^N$ be sequences in a finite dimensional Hilbert space which satisfy $\|x_j\|=\|f_j\|$ for all $1\leq j\leq N$ and $$\Big\|\sum_{j=1}^N \varepsilon_j\langle x,f_j\rangle x_j\Big\|\leq C\|x\|, \qquad\textrm…
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