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 unit norm Bessel sequence is a finite union of sets each of which is $ω$-independent for $\ell_2$-sequences.
In this paper we show that the Fourier transform induces an isomorphism between the coorbit spaces defined by Feichtinger and Gröchenig of the mixed, weighted Lebesgue spaces $L_{v}^{p,q}$ with respect to the quasi-regular representation of a semi-direct product $\mathbb{R}^{d}\rtimes H$ with suitably chosen dilation group $H$, and certain decomposition spaces $\mathcal{D}\left(\mathcal{Q},L^{p},\ell_{u}^{q}\right)$ (essentially as introduced by Feichtinger and Gröbner), where the localized ,,parts`` of a function are measured in the $\mathcal{F}L^{p}$-norm. This equivalence is useful in sever…
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…
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…
We generalize Feichtinger and Kaiblinger's theorem on linear deformations of uniform Gabor frames to the setting of a locally compact abelian group $G$. More precisely, we show that Gabor frames over lattices in the time-frequency plane of $G$ with windows in the Feichtinger algebra are stable under small deformations of the lattice by an automorphism of $G \times \widehat{G}$. The topology we use on the automorphisms is the Braconnier topology. We characterize the groups in which the Balian--Low theorem for the Feichtinger algebra holds as exactly the groups with noncompact identity component…
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 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.
In this work we show that if the frame property of a Gabor frame with window in Feichtinger's algebra and a fixed lattice only depends on the parity of the window, then the lattice can be replaced by any other lattice of the same density without losing the frame property. As a byproduct we derive a generalization of a result of Lyubarskii and Nes, who could show that any Gabor system consisting of an odd window function from Feichtinger's algebra and any separable lattice of density $\frac{n+1}{n}$, $n \in \mathbb{N}_+$, cannot be a Gabor frame for the Hilbert space of square-integrable functi…
We provide explicit criteria for wavelets to give rise to frames and atomic decompositions in ${\rm L}^2(\mathbb{R}^d)$, but also in more general Banach function spaces. We consider wavelet systems that arise by translating and dilating the mother wavelet, with the dilations taken from a suitable subgroup of ${\rm GL}(\mathbb{R}^d)$, the so-called {\em dilation group}.The paper provides a unified approach that is applicable to a wide range of dilation groups, thus giving rise to new atomic decompositions for homogeneous Besov spaces in arbitrary dimensions, but also for other function spaces s…
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…
Center for Data Science, New York University（纽约大学数据科学中心）；Department of Mathematics, Bar-Ilan University（巴伊兰大学数学系）；Department of Mathematics, ETH Zürich（苏黎世联邦理工学院数学系）
[L. Gavruta, Frames for Operators, Appl. comput. Harmon. Anal. 32(2012), 139-144] introduced a special kind of frames, named $K$-frames, where $K$ is an operator, in Hilbert spaces, is significant in frame theory and has many applications. In this paper, first of all, we have introduced the notion of approximative $K$-atomic decomposition in Banach spaces. We gave two characterizations regarding the existence of approximative $K$-atomic decompositions in Banach spaces. Also some results on the existence of approximative $K$-atomic decompositions are obtained. We discuss several methods to cons…
We construct generalised shift-invariant systems of functions of several real variables for anisotropic Besov spaces that can be generated by the decomposition method using any given expansive matrix and establish the conditions on those systems under which they will constitute Banach frames or sets of atoms for the anisotropic homo- or inhomogeneous Besov spaces.
Let $Λ$ be a lattice in a second countable, locally compact abelian group $G$ with annihilator $Λ^{\perp} \subseteq \widehat{G}$. We investigate the validity of the following statement: For every $η$ in the Feichtinger algebra $S_0(G)$, the Gabor system $\{ M_τ T_λ η\}_{λ\in Λ, τ\in Λ^{\perp}}$ is not a frame for $L^2(G)$. When $G = \mathbb{R}$ and $Λ= α\mathbb{Z}$, this statement is a variant of the Balian-Low theorem. Extending a result of R. Balan, we show that whether the statement generalizes to $(G,Λ)$ is equivalent to the nontriviality of a certain vector bundle over the compact space $…
A new construction of decomposition smoothness spaces of homogeneous type is considered. The smoothness spaces are based on structured and flexible decompositions of the frequency space $\mathbb{R}^d\backslash\{0\}$. We construct simple adapted tight frames for $L_2(\mathbb{R}^d)$ that can be used to fully characterise the smoothness norm in terms of a sparseness condition imposed on the frame coefficients. Moreover, it is proved that the frames provide a universal decomposition of tempered distributions with convergence in the tempered distributions modulo polynomials. As an application of th…
We investigate the problem of constructing sparse time-frequency representations with flexible frequency resolution, studying the theory of nonstationary Gabor frames in the framework of decomposition spaces. Given a painless nonstationary Gabor frame, we construct a compatible decomposition space and prove that the nonstationary Gabor frame forms a Banach frame for the decomposition space. Furthermore, we show that the decomposition space norm can be completely characterized by a sparseness condition on the frame coefficients and we prove an upper bound on the approximation error occurring wh…
We prove that a Schauder frame for any separable Banach space is shrinking if and only if it has an associated space with a shrinking basis, and that a Schauder frame for any separable Banach space is shrinking and boundedly complete if and only if it has a reflexive associated space. To obtain these results, we prove that the upper and lower estimate theorems for finite dimensional decompositions of Banach spaces can be extended and modified to Schauder frames. We show as well that if a separable infinite dimensional Banach space has a Schauder frame, then it also has a Schauder frame which i…
$\newcommand{mc}[1]{\mathcal{#1}}$ $\newcommand{D}{\mc{D}(\mc{Q},L^p,\ell_w^q)}$ We present a framework for the construction of structured, possibly compactly supported Banach frames and atomic decompositions for decomposition spaces. Such a space $\D$ is defined using a frequency covering $\mc{Q}=(Q_i)_{i\in I}$: If $(φ_i)_{i}$ is a suitable partition of unity subordinate to $\mc{Q}$, then $\Vert g\Vert_{\D}:=\left\Vert\left(\Vert\mc{F}^{-1}(φ_i\hat{g})\Vert_{L^p}\right)_{i}\right\Vert_{\ell_w^q}$. We assume $\mc{Q}=(T_iQ+b_i)_{i}$, with $T_i\in{\rm GL}(\Bbb{R}^d),b_i\in\Bbb{R}^d$. Given a pr…
We derive an extension of the Walnut-Daubechies criterion for the invertibility of frame operators. The criterion concerns general reproducing systems and Besov-type spaces. As an application, we conclude that $L^2$ frame expansions associated with smooth and fast-decaying reproducing systems on sufficiently fine lattices extend to Besov-type spaces. This simplifies and improves recent results on the existence of atomic decompositions, which only provide a particular dual reproducing system with suitable properties. In contrast, we conclude that the $L^2$ canonical frame expansions extend to m…
We turn a given filter bank into a filtering scheme that provides perfect reconstruction, synthesis is the adjoint of the analysis part (so-called unitary filter banks), all filters have equal norm, and the essential features of the original filter bank are preserved. Unitary filter banks providing perfect reconstruction are induced by tight generalized frames, which enable signal decomposition using a set of linear operators. If, in addition, frame elements have equal norm, then the signal energy is spread through the various filter bank channels in some uniform fashion, which is often more s…
Classical and non classical Besov and Triebel-Lizorkin spaces with complete range of indices are developed in the general setting of Dirichlet space with a doubling measure and local scale-invariant Poincaré inequality. This leads to Heat kernel with small time Gaussian bounds and Hölder continuity, which play a central role in this article. Frames with band limited elements of sub-exponential space localization are developed, and frame and heat kernel characterizations of Besov and Triebel-Lizorkin spaces are established. This theory, in particular, allows to develop Besov and Triebel-Lizorki…
We introduce almost diagonal matrices in the setting of (anisotropic) discrete homogeneous Triebel-Lizorkin type spaces and homogeneous modulation spaces, and it is shown that the class of almost diagonal matrices is closed under matrix multiplication. We then connect the results to the continuous setting and show that the "change of frame" matrix for a pair of time-frequency frames, with suitable decay properties, is almost diagonal. As an application of this result, we consider a construction of compactly supported frame expansions for homogeneous decomposition spaces of Triebel-Lizorkin typ…
In this paper we consider series expansions via a frame and a non-frame and the possibilities for interchange of the two sequences, both in the Hilbert and Banach space setting. First we give a characterization of frame-related concepts in Banach spaces (atomic decompositions, Banach frames, $X_d$-Riesz bases, $X_d$-frames, $X_d$-Bessel sequences, and sequences satisfying the lower $X_d$-frame condition). We also determine necessary and sufficient conditions for operators to preserve the type of the concepts listed above. Then we discuss differences and relationships between expansions in a Ba…
We propose a simple abstract version of Calderon--Zygmund theory, which is applicable to spaces with exponential volume growth, and then show that amenable Lie groups can be treated within this framework.
Full Gabor frames, its existence problem, and a non-uniform Balian-Low type theorem
School of Mathematical Sciences and LPMC, Nankai University（南开大学数学科学学院）；Institute for Advanced Study in Mathematics, Harbin Institute of Technology（哈尔滨工业大学数学学院）
The construction of frames for a Hilbert space H can be equated to the decomposition of the frame operator as a sum of positive operators having rank one. This realization provides a different approach to questions regarding frames with particular properties and motivates our results. We find a necessary and sufficient condition under which any positive finite-rank operator B can be expressed as a sum of rank-one operators with norms specified by a sequence of positive numbers {c_i}. Equivalently, this result proves the existence of a frame with B as it's frame operator and with vector norms g…
