Karlheinz Gröchenig, Eugenia Malinnikova
We prove a strong uncertainty principle for Riesz bases in L^2(R^d) and show that the orthonormal basis constructed by Bourgain possesses the optimal phase-space localization.
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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。
In this paper, we give new characterizations of modular Riesz bases in Hilbert $C^*$-modules. We prove that modular Riesz bases share many properties with Riesz bases in Hilbert spaces. Moreover we show that there are also important differences; for example, there exist exact frames that are not modular Riesz bases.
In this note we prove that if two Riesz-Fischer sequences in a separable Hilbert space $H$ are biorthogonal and one of them is complete in $H$, then both sequences are Riesz bases for $H$. This complements a recent result by D. T. Stoeva where the same conclusion holds if one replaces the phrase ``Riesz-Fischer sequences'' by ``Bessel sequences''.
In this article, we introduce and study Riesz bases in a separable quaternionic Hilbert spaces. Some results on Riesz bases in a separable quaternionic Hilbert spaces are proved. It is also proved that a Riesz basis in a separable quaternionic Hilbert space a frame for the quaternionic Hilbert space. Riesz sequences are defined and equivalence of a Riesz basis and a complete Riesz sequence in a separable quaternionic Hilbert space is proved.
Pseudo- Riesz Bases
该研究推广Holub提出的近Riesz基概念，提出包含非框架序列的伪Riesz基，证明其保留近Riesz基的诸多优良性质并建立基本扰动结果。
The paper is devoted to continuous frames and Riesz bases in Hilbert C*-modules. we define a continuous Riesz basis for Hilbert C*-modules and give some results about them.
Exploring K-Riesz and K-g-Riesz bases in the context of Hilbert pro-C*-modules
本文研究 pro-C*-Hilbert 模中的 K-Riesz 基与 K-g-Riesz 基，分别借助下有界算子、有界满射算子及相应正交基刻画其结构，并揭示二者关系。
Let $ 1\leq p< \infty$ and let $ψ\in L^{p}(\R^d)$. We study $p-$Riesz bases of quasi shift invariant spaces $V^p(ψ;Y)$.
Near-Riesz and pseudo-Riesz bases of Hilbert C*-modules and semi-C*-Fredholm operators
在Hilbert C*-模中研究伪Riesz序列与伪Riesz基，证明Bessel序列为伪Riesz序列当且仅当伴随合成算子是上半C*-Fredholm，并引入伪Riesz-Weyl序列概念，应用于扰动研究。
In this paper, we introduce orthonoramal and Riesz bases for g-fusion frames and will show that the weights have basic roles. Next, we prove an effective theorem between frames and g-fusion frames by using an operator. Finally, perturbations of g-fusion frames will be presented.
A bounded, Riemann integrable and measurable set $K\subset \mathbb{R}^d$, which fulfills \[\sum\limits_{γ\inΓ}\mathbb{1}_K(x-γ)=k\text{ almost everywhere, $x\in\mathbb{R}^d$}\] for a lattice $Γ\subset\mathbb{R}^d$ is called $k$-tiling. If $K\subset\mathbb{R}^d$ is $k$-tiling $L^2(K)$ will admit a Riesz basis of exponentials. We use this result to construct generalized Riesz wavelet bases of $L^2(\mathbb{R}^2)$, arising from the action of suitable subsets of the affine group. One example of our construction is the first known shearlet Riesz basis.
We study Riesz bases/Riesz sequences of reproducing kernels in the model space $K_θ$ in connection with the corresponding Schur--Nevanlinna parameters and functions. In particular, we construct inner functions with given Schur--Nevanlinna parameters at a given sequence $Λ$ such that the corresponding systems of projections of reproducing kernels in the model space are complete/non complete. Furthermore, we give a compactness criterion for Hankel operators with symbol $θ\overline B$, where $θ$ is an inner function and $B$ is an interpolating Blaschke product and use this criterion to describe R…
The notions of Bessel sequence, Riesz-Fischer sequence and Riesz basis are generalized to a rigged Hilbert space $\D[t] \subset \H \subset \D^\times[t^\times]$. A Riesz-like basis, in particular, is obtained by considering a sequence $\{ξ_n\}\subset \D$ which is mapped by a one-to-one continuous operator $T:\D[t]\to\H[\|\cdot\|]$ into an orthonormal basis of the central Hilbert space $\H$ of the triplet. The operator $T$ is, in general, an unbounded operator in $\H$. If $T$ has a bounded inverse then the rigged Hilbert space is shown to be equivalent to a triplet of Hilbert spaces.
Weaving frames have potential applications in wireless sensor networks that require distributed processing of signal under different frames. In this paper, we study some new properties of weaving generalized frames (or $g$-frames) and weaving generalized orthonormal bases (or $g$-orthonormal bases). It is shown that a $g$-frame and its dual $g$-frame are woven. The inter-relation of optimal $g$-frame bounds and optimal universal $g$-frame bounds is studied. Further, we present a characterization of weaving $g$-frames. Illustrations are given to show the difference in properties of weaving gene…
Given an orthonormal basis $ {\mathcal V}= \{v_j\} _{j\in N}$ in a separable Hilbert space $H$ and a set of unit vectors $ {\mathcal B}=\{w_j\}_{j\in N}$, we consider the sets $ {\mathcal B}_N$ obtained by replacing the vectors $v_1, ...,\, v_N$ with vectors $w_1,\, ...,\, w_N$. We show necessary and sufficient conditions that ensure that the sets $ {\mathcal B}_N$ are Riesz bases of $H$ and we estimate the frame constants of the $ {\mathcal B}_N$. Then, we prove conditions that ensure that $ {\mathcal B}$ is a Riesz basis. Applications to the construction of exponential bases on domains of $ …
G-frames are generalized frames which include ordinary frames, bounded invertible linear operators, as well as many recent generalizations of frames, e.g., bounded quasi-projectors and frames of subspaces. G-frames are natural generalizations of frames and provide more choices on analyzing functions from frame expansion coefficients. We give characterizations of g-frames and prove that g-frames share many useful properties with frames. We also give generalized version of Riesz bases and orthonormal bases. As an application, we get atomic resolutions for bounded linear operators.
It is well known that a sequence in a Hilbert space is a Riesz basis if and only if it is a complete Bessel sequence with biorthogonal sequence which is also a complete Bessel sequence. Here we prove that the completeness of one (any one) of the biorthogonal sequences can be omitted in the characterization.
In this note a review, some considerations and new results about maps with values in a distribution space and domain in a $σ$-finite measure space $X$, are obtained. In particular, it is a survey about Bessel, frames and bases (in particular Riesz and Gel'fand bases) in a distribution space. In this setting, the Riesz-Fischer maps and semi-frames are defined and new results about them are attained. Some example in tempered distributions space are examined.
B-Frames, B-Riesz bases, and Their tensor products
本文定义了对偶b-框架和b-里斯基，证明了两序列的张量积为b-框架（或b-里斯基）的充要条件，建立了b-框架与g-框架的对应关系并提出了由b-框架诱导框架的构造方法。
Linear combinations of exponentials $e^{iλ_kt}$ in the case where the distance between some points $λ_k$ tends to zero are studied. D. Ullrich has proved the basis property of the divided differences of exponentials in the case when the groups consist of equal number of points all of them are close enough to integers. We have generalized this result for groups with arbitrary number of close points and obtained a full description of Riesz bases of exponential divided differences. The application to a control problem is presented.
We study functions generating Gabor Riesz bases on the integer lattice. The classical Balian-Low theorem restricts the simultaneous time and frequency localization of such functions. We obtain a quantitative estimate that extends both this result and other related theorems.
In this paper we investigate a new notion of bases in Hilbert spaces and similar to fusion frame theory we introduce fusion bases theory in Hilbert spaces. We also introduce a new definition of fusion dual sequence associated with a fusion basis and show that the operators of a fusion dual sequence are continuous projections. Next we define the fusion biorthogonal sequence, Bessel fusion basis, Hilbert fusion basis and obtain some characterizations of them. we study orthonormal fusion systems and Riesz fusion bases for Hilbert spaces. we consider the stability of fusion bases under small pertu…
An algebraic Riccati equation for linear operators is studied, which arises in systems theory. For the case that all involved operators are unbounded, the existence of infinitely many selfadjoint solutions is shown. To this end, invariant graph subspaces of the associated Hamiltonian operator matrix are constructed by means of a Riesz basis with parentheses of generalised eigenvectors and two indefinite inner products. Under additional assumptions, the existence and a representation of all bounded solutions is obtained. The theory is applied to Riccati equations of differential operators.
Operators for matrix-valued Riesz bases over LCA groups
本文研究了LCA群上矩阵值里斯基底的构造方法，提出了一些算子类，并证明了自共轭算子与里斯基底映射之间的关系。
Riesz Bases in Krein Spaces
本文研究 Krein 空间中 Riesz 基的定义与性质，引入双正交序列和 Gram 矩阵，给出 Riesz 基成为 Bessel 序列及 Gram 矩阵有界可逆的等价条件。
We construct a family of spline affine Riesz bases, i.e. sequences of dilations and translations generated by the special spline functions $ψ_m$, and we prove that their Riesz bounds are independent of $m$. We put $ψ_0=χ$ as the Haar step function and every following function $ψ_{m+1}$ is obtained by integrating the previous function $ψ_m$ with consequent antiperiodization. We give a representation of the spline $ψ_m$ as a finite sum of Rademacher chaos series and we use a notion of a simple Walsh spectrum of a function in connection with orthogonality of affine systems.
Riesz bases of exponentials for multi-tiling measures
本文研究由具有指数 Riesz 基的测度平移拼接而成的多重铺砌测度，给出其容许指数 Riesz 基的充要条件，并据此证明正方形边界不存在一类自然候选形式的指数 Riesz 基。
This work is devoted to the study of Bessel and Riesz systems of the type $\big\{L_γ\mathsf{f}\big\}_{γ\in Γ}$ obtained from the action of the left regular representation $L_γ$ of a discrete non abelian group $Γ$ which is a semidirect product, on a function $\mathsf{f}\in \ell^2(Γ)$. The main features about these systems can be conveniently studied by means of a simple matrix-valued function $\mathbf{F}(ξ)$. These systems allow to derive sampling results in principal $Γ$-invariant spaces, i.e., spaces obtained from the action of the group $Γ$ on a element of a Hilbert space. Since the systems …
In this paper, we study the change of spectrum and the existence of Riesz bases of specific classes of $n\times n$ unbounded operator matrices, called: diagonally and off-diagonally generalized subordinate block operator matrices. An application to a $n\times n$ Gribov operator matrix acting on a sum of Bargmann spaces, illustrates the abstract results. As example, we consider a particular Gribov operator matrix by taking special values of the real parameters of Pomeron.
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…
The main purpose of this paper is to give a vector lattice version of a Theorem by Burkholder about convergence of martingales. The proof is based on a vector lattice analogue of Austin's sample function theorem, proved recently by Grobler, Labuschagne and Marraffa and on a new characterization of elements of the sup-completion of a universally complete vector lattice which do not belong to the space.
Riesz sequences near the critical density
本文在度量测度空间的再生核希尔伯特空间中构造了Beurling密度任意接近临界密度的Riesz序列，适用于指数系统、非局部化Gabor系统及幂零李群相干系统，方法基于框架冗余度和Weaver猜想的选择子形式。
We describe the radial Fock type spaces which possess Riesz bases of normalized reproducing kernels and which are (are not) isomorphic to de Branges spaces in terms of the weight functions.
A Gabor system generated by a window function $g\in L^2(\mathbb{R}^d)$ and a separable set $Λ\times Γ\subset \mathbb{R}^{2d}$ is the collection of time-frequency shifts of $g$ given by $\mathcal G(g, Λ\times Γ) = \left\{ e^{2πi ξ\cdot t}g(t-x)\right\}_{ (x,ξ)\in Λ\times Γ}$. One of the fundamental problems in Gabor analysis is to characterize all windows and time-frequency sets that generate a Gabor frame or Gabor orthonormal basis. The case of Gabor orthonormal bases generated by characteristic functions $g=χ_Ω$ has been solved by Han and Wang. In this paper, we build on these results and obt…
We provide a necessary and sufficient condition to ensure that a multi-tile $Ω$ of $R^d$ of positive measure (but not necessarily bounded) admits a structured Riesz basis of exponentials for $ L^{2}(Ω)$. New examples are given and this characterization is generalized to abstract locally compact abelian groups.
For p-subordinate perturbations of unbounded normal operators, the change of the spectrum is studied and spectral criteria for the existence of a Riesz basis with parentheses of root vectors are established. A Riesz basis without parentheses is obtained under an additional a priori assumption on the spectrum of the perturbed operator. The results are applied to two classes of block operator matrices.
We consider a regular indefinite Sturm-Liouville problem with two self-adjoint boundary conditions affinely dependent on the eigenparameter. We give sufficient conditions under which the root vectors of this Sturm-Liouville problem can be selected to form a Riesz basis of a corresponding weighted Hilbert space.
Let S be a bounded, Riemann measurable set in R^d, and L be a lattice. By a theorem of Fuglede, if S tiles R^d with translation set L, then S has an orthogonal basis of exponentials. We show that, under the more general condition that S multi-tiles R^d with translation set L, S has a Riesz basis of exponentials. The proof is based on Meyer's quasicrystals.
We consider a regular indefinite Sturm-Liouville problem with two self-adjoint boundary conditions, one being affinely dependent on the eigenparameter. We give sufficient conditions under which a basis of each root subspace for this Sturm-Liouville problem can be selected so that the union of all these bases constitutes a Riesz basis of a corresponding weighted Hilbert space.
Weaving Riesz Bases
本文研究可分 Hilbert 空间中的编织框架，通过简化问题为基获得唯一刻画，建立适用于 Fourier 矩阵的向量重构充要条件，并证明该结论在无限维 Riesz 基及平移编织 Riesz 基中同样成立。
Aldroubi has shown how one can construct any frame $\gtu$ starting with one frame $\ftu $,using a bounded operator $U$ on $l^2(N)$. We study the overcompleteness of the frames in terms of properties of $U$. We also discuss perturbation of frames in the sense that two frames are ``close'' if a certain operator is compact. In this way we obtain an equivalence relation with the property that members of the same equivalence class have the same overcompleteness. On the other hand we show that perturbation in the Paley-Wiener sense does not have this property. \\ Finally we construct a frame which i…
Let $S$ be the union of finitely many disjoint intervals on the real line. Suppose that there are two real numbers $α, β$ such that the length of each interval belongs to $Z α+ Z β$. We use quasicrystals to construct a discrete set of real frequencies such that the corresponding system of exponentials is a Riesz basis in the space $L^2(S)$.
In this note we discuss notions of convolutions generated by biorthogonal systems of elements of a Hilbert space. We develop the associated biorthogonal Fourier analysis and the theory of distributions, discuss properties of convolutions and give a number of examples.
We study a capacity theory based on a definition of a Riesz potential in metric spaces with a doubling measure. In this general setting, we study the basic properties of the Riesz capacity, including monotonicity, countable subadditivity and several convergence results. We define a modified version of the Hausdorff measure and provide lower bound and upper bound estimates for the capacity in terms of the modified Hausdorff content.
We prove the existence of sampling sets and interpolation sets near the critical density, in Paley Wiener spaces of a locally compact abelian (LCA) group G . This solves a problem left by Gröchenig, Kutyniok, and Seip in the article: `Landau's density conditions for LCA groups ' (J. of Funct. Anal. 255 (2008) 1831-1850). To achieve this result, we prove the existence of universal Riesz bases of characters for L2(Omega), provided that the relatively compact subset Omega of the dual group of G satisfies a multi-tiling condition. This last result generalizes Fuglede's theorem, and extends to LCA …
We consider systems of exponentials with frequencies belonging to simple quasicrystals in $\mathbb{R}^d$. We ask if there exist domains $S$ in $\mathbb{R}^d$ which admit such a system as a Riesz basis for the space $L^2(S)$. We prove that the answer depends on an arithmetical condition on the quasicrystal. The proof is based on the connection of the problem to the discrepancy of multi-dimensional irrational rotations, and specifically, to the theory of bounded remainder sets. In particular it is shown that any bounded remainder set admits a Riesz basis of exponentials. This extends to several …
In this work, we prove that certain L^2-unbounded transformations of orthogonal wavelet bases generate vaguelets. The L^2-unbounded functions involved in the transformations are assumed to be quasi-homogeneous at high frequencies. We provide natural examples of functions which are not quasi-homogeneous and for which the resulting transformations are not vaguelets. We also address the related question of whether the considered family of functions is a Riesz basis in L^2(R). The Riesz property could be deduced directly from the results available in the literature or, as we outline, by using the …
Biframes and some of their properties
本文在希尔伯特空间中引入双框架概念，研究其基本性质与分类，并定义 b-Riesz 基，证明其构成所有 Riesz 基的真子集。
Generalized Sobolev-Orlicz spaces based on the Riesz fractional gradient as interpolation and potential spaces
本文证明基于Riesz分数阶梯度的Musielak-Orlicz函数分数阶Sobolev空间与广义Orlicz框架下的Bessel位势空间一致，并识别为复插值空间，进而获得空间的结构性质。
The location of the spectrum and the Riesz basis property of well-posed homogeneous infinite-dimensional linear port-Hamiltonian systems on a 1D spatial domain are studied. It is shown that the Riesz basis property is equivalent to the fact that system operator generates a strongly continuous group. Moreover, in this situation the spectrum consists of eigenvalues only, located in a strip parallel to the imaginary axis and they can decomposed into finitely many sets having each a uniform gap.
We consider three special and significant cases of the following problem. Let D be a (possibly unbounded) set of finite Lebesgue measure in R^d. Find conditions on D for which the standard exponential basis on the unit cube of R^d is a frame, a Riesz sequence, or a Riesz basis on L^2(D).
Continuous biframes in Hilbert Spaces
该研究提出希尔伯特空间中连续双框架的概念，探究其以双框架算子为核心的基本性质，并引入新型连续双框架-Riesz基，丰富了希尔伯特空间的框架理论。
In a paper from 1996, D. Jerison and C. Kenig among other results provided a $H^{1/2}$ regularity result for the Dirichlet problem for the Laplace equation in Lipschitz domains. In this article, we adopt a Hilbertian approach to construct two Riesz bases for $L^2(\partial Ω),$ which will allow to find in a different way some of the results of D. Jerison and C. Kenig, and G. Savaré (1998) about the regularity issue of the Laplace equation.
Local form subordination without a power decay and a criterion of Riesz basesness
本文提出无需幂衰减的局部形式次级条件，用于证明扰动算子特征系统具有Riesz基性质。
Given a Hilbert space and the generator of a strongly continuous group on this Hilbert space. If the eigenvalues of the generator have a uniform gap, and if the span of the corresponding eigenvectors is dense, then these eigenvectors form a Riesz basis (or unconditional basis) of the Hilbert space. Furthermore, we show that none of the conditions can be weakened. However, if the eigenvalues (counted with multiplicity) can be grouped into subsets of at most $K$ elements, and the distance between the groups is (uniformly) bounded away from zero, then the spectral projections associated to the gr…
We look at the time-frequency localisation of generators of lattice Gabor systems. For a generator of a Riesz basis, this localisation is described by the classical Balian-Low theorem. We establish Balian-Low type theorems for complete and minimal Gabor systems with a frame-type approximation property. These results describe how the best possible localisation of a generator is limited by the degree of control over the coefficients in approximations given by the system, and provide a continuous transition between the classical Balian-Low conditions and the corresponding conditions for generator…
Riesz bases of exponentials for convex polytopes with symmetric faces
本文证明对任意中心对称且各维面均中心对称的凸多面体，其 $L^2$ 空间中存在指数函数Riesz基，该结果在 $d>1$ 的所有维度上均为首次建立。
In this paper, we consider the time-frequency localization of the generator of a principal shift-invariant space on the real line which has additional shift-invariance. We prove that if a principal shift-invariant space on the real line is translation-invariant then any of its orthonormal (or Riesz) generators is non-integrable. However, for any $n\ge2$, there exist principal shift-invariant spaces on the real line that are also $\nZ$-invariant with an integrable orthonormal (or a Riesz) generator $ϕ$, but $ϕ$ satisfies $\int_{\mathbb R} |ϕ(x)|^2 |x|^{1+ε} dx=\infty$ for any $ε>0$ and its Four…
We consider the trigonometric-like system of piecewise linear functions introduced recently by Daubechies, DeVore, Foucart, Hanin, and Petrova. We provide an alternative proof that this system forms a Riesz basis of $L_2([0,1])$ based on the Gershgorin theorem. We also generalize this system to higher dimensions $d>1$ by a construction, which avoids using (tensor) products. As a consequence, the functions from the new Riesz basis of $L_2([0,1]^d)$ can be easily represented by neural networks. Moreover, the Riesz constants of this system are independent of $d$, making it an attractive building …
Inducing Riesz and orthonormal bases in $L^2$ via composition operators
本文研究了通过复合算子在 $L^2$ 空间诱导 Riesz 基的问题，刻画了能保持 Riesz 基的可微映射 $h$ 的 Jacobi 行列式有界特性，并指出双射神经网络可用于构建具有良好逼近性质的 Riesz 基。
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