Stevan Pilipović, Diana T. Stoeva
We analyze the construction of a sequence space $\widetildeΘ$, resp. a sequence of sequence spaces, in order to have $\{g_i\}$ as a $\widetildeΘ$-frame or Banach frame for a Banach space $X$, resp. pre-$F$-frame or $F$-frame for a Fréchet space $X_F=\cap_{s\in {\mathbb N}_0} X_s$, where $\{X_s\}_{s\in {\mathbb N}_0}$ is a sequence of Banach spaces.
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It is known in Hilbert space frame theory that a Bessel sequence can be expanded to a frame. Contrary to Hilbert space situation, using a result of Casazza and Christensen, we show that there are Banach spaces and approximate Bessel sequences which can not be expanded to approximate Schauder frames. We characterize Banach spaces in which one can expand approximate Bessel sequences to approximate Schauder frames.
Total sequences and the lower frame inequality
本文证明希尔伯特空间中任意全序列经缩放后满足下框架不等式，解决了Kulikov在MathOverflow提出的问题。
Fusion frames, and, more generally, operator-valued frame sequences are generalizations of classical frames, which are today a standard notion when redundant, yet stable sequences are required. However, the question of stability of duals with respect to perturbations has not been satisfactorily answered. In this paper, we quantitatively measure this stability by considering the associated deviations of the canonical and alternate dual sequences from the original ones. It is proven that operator-valued frame sequences are indeed stable in this sense. Along the way, we also generalize existing d…
This note is a survey and collection of results, as well as presenting some original research. For Bessel sequences and frames, the analysis, synthesis and frame operators as well as the Gram matrix are well-known, bounded operators. We investigate these operators for arbitrary sequences, which in general lead to possibly unbounded operators. We characterize various classes of sequences in terms of these operators and vice-versa. Finally, we classify these sequences by operators applied on orthonormal bases.
An unbounded operator theory approach to lower frame and Riesz-Fischer sequences
该研究采用无界算子理论方法，研究下框架与Riesz-Fischer序列的关联，拓展广义框架算子与Gram矩阵思路，填补合成算子分类空白，证明规范对偶性质并推广框架结论，还重新定义了适用于任意算子的唯一伴随算子。
Let $\{f_n:n\in\mathbb{N}\}$ be a $J$-frame for a Krein space ${\textbf{\textit{K}}}$ and $P_M$ be a $J$-orthogonal projection from ${\textbf{\textit{K}}}$ onto a subspace $M$. In this article we find sufficient conditions under which $\{P_M(f_n):n\in\mathbb{N}\}$ is a $J$-frame for $P_M\textbf{\textit{K}}$ and $\{(I-P_M)f_n\}_{n\in{\mathbb{N}}}$ is a $J$-frame for $(I-P_M)\textbf{\textit{K}}$. We also introduce $J$-frame sequence for a Krein space ${\textbf{\textit{K}}}$ and study some properties of $J$-frame sequence analogues to Hilbert space frame theory.
Frame related sequences in tensor product of Hilbert spaces
该研究从代数角度探讨可分希尔伯特空间张量积中的框架及相关序列，刻画张量序列的下半框架与Riesz-Fischer性质并推广至无限张量积，还分析算子张量和对相关序列的作用。
Sequences that do frame reconstruction
本文研究了可进行放松框架重建的序列，通过非可逆算子定义并分类了此类序列，同时提供了稳定性结果和基底分类。
The paper studies finite extensions of Bessel sequences in infinite-dimensional Hilbert spaces. We provide a characterization of Bessel sequences that can be extended to frames by adding finitely many vectors. We also characterize frames that can be converted to Parseval frames by finite-dimensional perturbations. Finally, some results on excesses of frames and near-Riesz bases are derived.
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.
Given a discrete group and a unitary representation on a Hilbert space $\mathcal{H}$, we prove that the notions of operator Bracket map and Gramian coincide on a dense set of $\mathcal{H}$. As a consequence, combining this result with known frame theory, we can recover in a simple way all previous Bracket characterizations of Riesz and frame sequences generated by a single element under a unitary representation.
Two Bessel sequences are orthogonal if the composition of the synthesis operator of one sequence with the analysis operator of the other sequence is the 0 operator. We characterize when two Bessel sequences are orthogonal when the Bessel sequences have the form of translates of a finite number of functions in $\ltwod$. The characterizations are applied to Bessel sequences which have an affine structure, and a quasi-affine structure. These also lead to characterizations of superframes. Moreover, we characterize perfect reconstruction, i.e. duality, of subspace frames for translation invariant (…
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.
The construction of finite tight Gabor frames plays an important role in many applications. These applications include significant ones in signal and image processing. We explore when constant amplitude zero autocorrelation (CAZAC) sequences can be used to generate tight Gabor frames. The main theorem uses Janssen's representation and the zeros of the discrete periodic ambiguity function to give necessary and sufficient conditions for determining whether any Gabor frame is tight. The relevance of the theorem depends significantly on the construction of examples. These examples are necessarily …
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…
A counter-example by Xiangchun Xiao, Guoping Zhao and Guorong Zhou in [J. Pseudo-Differ. Oper. Appl., (2021) 12:60] is incorrect. Further, there is no new characterization of weaving for g-frames by Xiangchun Xiao et al. in [2]. Xiangchun Xiao et al. in [2] gave a wrong interpretation about a characterization of weaving g-frames in separable Hilbert spaces which already proved by Deepshikha, Vashisht and Verma in [1].
Frame-related Sequences in Chains and Scales of Hilbert Spaces
本文研究Hilbert空间标度中框架相关性质（完备性、半框架性）在不同空间之间的传播规律，证明上半框架性质向更大空间保持、下半框架性质向更小空间保持，并得出非平凡标度中同一序列不能同时成为两个空间的框架这一否定性结论。
Outer product frames are important objects in Hilbert space frame theory. But very little is known about them. In this paper, we make the first detailed study of the family of outer product frames induced directly by vector sequences. We are interested in both the quantitative attributs of these outer product sequences (in particular, their Riesz and frame bounds), as well as their independence and spanning properties. We show that Riesz sequences of vectors yield Riesz sequences of outer products with the same (or better) Riesz bounds. Equiangular tight frames are shown to produce Riesz seque…
On the Phase Retrievable Sequences
该文研究希尔伯特空间上的相位可恢复序列，给出其刻画方式、多种构造方法，并探究了g-框架与框架的相位恢复性质。
It is known that the tensor product of two sequences, in the tensor product of two separable Hilbert spaces, is a frame if and only if each component of that product is a frame. This paper proposes a sort of generalization of the aforementioned result by dealing with sequences S that are finite minimal sums of tensor products of a finite number of sequences. We prove that S is a Bessel sequence if and only if it is a sum for which each term is the tensor product of Bessel sequences. We also state necessary conditions for S to be a frame. For dimensions higher than one, we deduce several result…
Characterization of Frame-Related Sequences on Semi-Hilbert Spaces
本文针对半希尔伯特空间，研究稠定可闭算子的约化加权伴随，构造框架型系统并刻画AB-贝塞尔序列、AB-框架等，导出AB-框架的道格拉斯型值域刻画。
Given a finite sequence of vectors $\mathcal F_0$ in $\C^d$ we characterize in a complete and explicit way the optimal completions of $\mathcal F_0$ obtained by adding a finite sequence of vectors with prescribed norms, where optimality is measured with respect to majorization (of the eigenvalues of the frame operators of the completed sequence). Indeed, we construct (in terms of a fast algorithm) a vector - that depends on the eigenvalues of the frame operator of the initial sequence $\cF_0$ and the sequence of prescribed norms - that is a minimum for majorization among all eigenvalues of fra…
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 prove that several results of lineability/spaceability in the framework of sequence spaces are valid in a stricter sense.
In this paper we introduce the concept of von Neumann-Schatten Bessel multipliers and obtain some of their characterizations. Finally, special attention is devoted to the study of invertible Hilbert--Schmidt frame multipliers. These results are not only of interest in their own right, but also they pave the way for obtaining some new results for diagonalization of matrices in finite dimensional setting as well as for dual $g$-frames. In particular, we show that a $g$-frame is uniquely determined by the set of its $g$-frames.
We give necessary and sufficient conditions for a subfamily of regularly spaced translates of a function to form a frame (resp. a Riesz basis) for its span. One consequence is that ifthetranslates are taken only from a subset of the natural numbers, then this family is a frame if and only if it is a Riesz basis. We also consider arbitrary sequences of translates and show that for sparse sets, having an upper frame bound is equivalent to the family being a frame sequence. Finally, we use the fractional Hausdorff dimension to identify classes of exact frame sequences.
Counting function estimates for coherent frames and Riesz sequences
该研究针对相干框架与Riesz序列点集的计数函数渐近性给出多种估计，既恢复了一般幺模顺从群下的必要密度条件，又在多项式增长群的相干系统附加局部化条件下得到更精确的结果。
The notion of multipliers in Hilbert space was introduced by Schatten in 1960 using orthonormal sequences and was generalized by Balazs in 2007 using Bessel sequences. This was extended to Banach spaces by Rahimi and Balazs in 2010 using p-Bessel sequences. In this paper, we further extend this by considering Lipschitz functions. On the way we define frames for metric spaces which extends the notion of frames and Bessel sequences for Banach spaces. We show that when the symbol sequence converges to zero, the multiplier is a Lipschitz compact operator. We study how the variation of parameters i…
Gabor frames have interested many mathematicians and physicists due to their potential applications in time-frequency analysis, in particular, signal processing. A Gabor system is a collection of vectors which is obtained by applying modulation and shift operators to non-zero functions in signal spaces. In many applications, for example, signal processing related to Gabor systems, the corresponding shifts may not be uniform. That is, the set associated with shifts may not be a group under usual addition. We analyze discrete vector-valued nonuniform Gabor frames (DVNUG frames, in short) in disc…
Generalized frame operator, lower semi-frames and sequences of translates
本文定义了 Hilbert 空间中与半双线性形式关联的广义框架算子 $T_\xi$，研究了其自伴性等性质，并在 $L^2(\mathbb{R})$ 函数整数平移序列背景下给出了其显式表达式及下半框架的刻画。
Bessel sequences in Hilbert $C^{\ast}$-modules
本文研究Hilbert C*-模中Bessel序列的算子形式，提出紧致紧框架概念，并给出其编织理论。
This paper is a contribution to the theory of dynamical sampling. Our purpose is twofold. We first consider representations of sequences in a Hilbert space in terms of iterated actions of a bounded linear operator. This generalizes recent results about operator representations of frames, and is motivated by the fact that only very special frames have such a representation. As our second contribution we give a new proof of a construction of a special class of frames that are proved by Aldroubi et al. to be representable via a bounded operator. Our proof is based on a single result by Shapiro \&…
Let $\hil$ be a finite dimensional (real or complex) Hilbert space and let $\{a_i\}_{i=1}^\infty$ be a non-increasing sequence of positive numbers. Given a finite sequence of vectors $\f$ in $\hil$ we find necessary and sufficient conditions for the existence of $r\in \NN\cup\{\infty\}$ and a Bessel sequence $\g$ in $\hil$ such that $\cF\cup\cG$ is a tight frame for $\hil$ and $\|g_i\|^2=a_i$ for $1\leq i\leq r$. Moreover, in this case we compute the minimum $r\in \NN\cup\{\infty\}$ with this property. Using recent results on the Schur-Horn theorem, we also obtain a not so optimal but algorith…
We study the construction of exponential frames and Riesz sequences for a class of fractal measures on ${\mathbb R}^d$ generated by infinite convolution of discrete measures using the idea of frame towers and Riesz-sequence towers. The exactness and overcompleteness of the constructed exponential frame or Riesz sequence is completely classified in terms of the cardinality at each level of the tower. Using a version of the solution of the Kadison-Singer problem, known as the $R_ε$-conjecture, we show that all these measures contain exponential Riesz sequences of infinite cardinality. Furthermor…
Controlled frames have been recently introduced in Hilbert spaces to improve the numerical efficiency of interactive algorithms for inverting the frame operator. In this paper, unlike the cross-Gram matrix of two different sequences which is not always a diagnostic tool, we define the controlled-Gram matrix of a sequence as a practical implement to diagnose that a given sequence is a controlled Bessel, frame or Riesz basis. Also, we discuss the cases that the operator associated to controlled Gram matrix will be bounded, invertible, Hilbert-Schmidt or a trace-class operator. Similar to standar…
Frames and weak frames for unbounded operators
本文将Găvruţa的K-框架和原子系统概念推广到无界稠定算子，分别基于非Bessel序列和Bessel序列建立两种弱框架分解方式，以刻画算子值域。
The conditions for sequences $\{f_{k}\}_{k=1}^{\infty}$ and $\{g_{k}\}_{k=1}^{\infty}$ being Bessel sequences, frames or Riesz bases, can be expressed in terms of the so-called cross-Gram matrix. In this paper we investigate the cross-Gram operator, $G$, associated to the sequence $\{\langle f_{k}, g_{j}\rangle\}_{j, k=1}^{\infty}$ and sufficient and necessary conditions for boundedness, invertibility, compactness and positivity of this operator are determined depending on the associated sequences. We show that invertibility of $G$ is not possible when the associated sequences are frames but n…
In this paper, we give some sufficient conditions under which perturbations preserve Hilbert frames and near-Riesz bases. Similar results are also extended to frame sequences, Riesz sequences and Schauder frames. It is worth mentioning that some of our perturbation conditions are quite different from those used in the previous literatures on this topic.
Exponential frames and Riesz sequences at the critical density
本文研究了在临界密度下指数系统形成框架或里斯序列的条件，并证明了重复点集在该密度下自动成为里斯基底。
In this note we investigate the operators associated with frame sequences in a Hilbert space $H$, i.e., the synthesis operator $T:\ell ^{2}(\mathbb{N}) \to H$, the analysis operator $T^{\ast}:H\to $ $% \ell ^{2}(\mathbb{N}) $ and the associated frame operator $S=TT^{\ast}$ as operators defined on (or to) the whole space rather than on subspaces. Furthermore, the projection $P$ onto the range of $T$, the projection $Q$ onto the range of $T^{\ast}$ and the Gram matrix $G=T^{\ast}T$ are investigated. For all these operators, we investigate their pseudoinverses, how they interact with each other, …
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。
In this paper we give a combinatorial characterization of tight fusion frame (TFF) sequences using Littlewood-Richardson skew tableaux. The equal rank case has been solved recently by Casazza, Fickus, Mixon, Wang, and Zhou. Our characterization does not have this limitation. We also develop some methods for generating TFF sequences. The basic technique is a majorization principle for TFF sequences combined with spatial and Naimark dualities. We use these methods and our characterization to give necessary and sufficient conditions which are satisfied by the first three highest ranks. We also gi…
A Kaczmarz algorithm for sequences of projections, infinite products, and applications to frames in IFS $L^{2}$ spaces
本文提出了一种新的递归迭代方案，用于处理无限维和非交换设置中的问题，如算子谱理论、优化、大规模稀疏系统、迭代函数系统（IFS）和分形调和分析。
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