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A Riesz basis for a separable Hilbert space H is the image of an orthonormal basis under a bounded, linear and bijective operator acting on H. Equivalently, it is an exact frame that shares the properties of a basis for H. Let G be a σ -compact and metrizable locally compact abelian group, and s and r be positive integers. It is illustrated that the image of a matrix-valued orthonormal basis under a bijective, bounded and linear operator acting on the matrix-valued signal space L 2 (G , ℂ s × r) may not be a frame, hence not a basis of the space L 2 (G , ℂ s × r). We introduce a notion of matrix-valued Riesz basis in the space L 2 (G , ℂ s × r) , where the adjointability of a bounded linear operator in the definition of Riesz basis with respect to the matrix-valued inner product plays a crucial role. We establish the existence of matrix-valued Riesz bases of the space L 2 (G , ℂ s × r). Extending results for standard Riesz bases of separable Hilbert spaces, we give necessary and sufficient conditions, and a characterization of matrix-valued Riesz bases of the space L 2 (G , ℂ s × r). [ABSTRACT FROM AUTHOR]

We establish that the summability of the series ∑ ε n is the necessary and sufficient criterion ensuring that every (1 + ε n) -bounded Markushevich basis in a separable Hilbert space is a Riesz basis. Further we show that if n ε n → ∞ , then in ℓ 2 there exists a (1 + ε n) -bounded Markushevich basis that under any permutation is non-equivalent to a Schauder basis. We extend this result to any separable Banach space. Finally we provide examples of Auerbach bases in 1-symmetric separable Banach spaces whose no permutations are equivalent to any Schauder basis or (depending on the space) any unconditional Schauder basis. [ABSTRACT FROM AUTHOR]

We study the topological properties of max-plus convex sets in an Archimedean Riesz space E with respect to the topology and the max-plus structure associated to a given order unit u; the definition of max-plus convex sets is algebraic and we do not assume that E has an a priori given topological structure. To a given unit u one can associate . two equivalent norms on E one of which, denoted ||·||u, is classical, the other ||·||hu is introduced here following a previous unpublished work of Stéphane Gaubert on the geodesic structure of finite dimensional max-plus; it is shown that the distance Dhu on E associated to ||·||hu is a geodesic distance, called the Hilbert affine distance associated to u, for which max-plus convex sets in E are precisely the geodesically closed sets. Under suitable assumptions, we establish max-plus versions of some fixed points and con- tinuous selection theorems that are well known for linear convex sets and we show that hyperspaces of compact max-plus convex sets are Absolute Retracts. We formulate a max-plus version of the Knaster-Kuratowski-Mazurkiewicz Lemma from which, following A. Granas and J. Dugundji, all of the consequences of the classical KKM Lemma can be derived in a max-plus version. P. de la Harpe showed that the interior of the standard simplex Δn equipped with the classical Hilbert metric-defined by the cross- ration of four appropriate points is isometric to a finite dimensional normed space. We give an explicit proof of that result: the norm space in question is Rn with the Hilbert affine norm ||·||hu with respect to u = (1,..., 1). [ABSTRACT FROM AUTHOR]

We consider special class of C0-groups from [12], whose generators are unbounded, have pure point imaginary spectrum and corresponding dense and minimal family of eigenvectors, which however does not form a Schauder basis. We obtain two-sided estimates for norms of C0-groups from this class and thus prove that these C0-groups have linear growth. Moreover we show that C0-groups from the considered class do not have any maximal asymptotics. This means that the fastest growing orbits do not exist. [ABSTRACT FROM AUTHOR]

The theory of algebraic frames for a Hilbert space H is a generalization of the theory of frames and generalized frames. The paper applies the theory of unbounded operators to define the dual of algebraic frames with densely defined unbounded analysis operators. It is shown that every algebraic frame has an algebraic dual frame, and if an algebraic frame has a nonzero redundancy, then it is not Riesz-type. An example of an algebraic frame with finite redundancy is constructed which is not a Riesz-type algebraic frame. Finally, for a lower bounded analytic frame, the discreteness of its indexing measure space and the uniqueness of its algebraic dual are studied and shown to be interrelated. [ABSTRACT FROM AUTHOR]

Some results of fuzzy frames on fuzzy Hilbert spaces from the point of view of Bag and Samanta are proved. In this paper, we define dual fuzzy frames and fuzzy Riesz bases and establish some fundamental results via dual fuzzy frames and fuzzy Ries bases. Next, we investigate the relation between fuzzy frame and fuzzy Riesz bases. [ABSTRACT FROM AUTHOR]

The first part of the present paper is an enhancement of the paper (Foulis et al. in Order 33:311-332, 2016). A new type of congruences on generalized pseudoeffect algebras (GPEAs), called R1-congruences, is introduced, which is in one-to-one correspondence with normal R1-ideals. The notion of Riesz congruences is reconsidered, and they are defined as congruences which are in one-to-one correspondence with normal Riesz ideals. In upward directed GPEAs, in particular in pseudoeffect algebras, both these types of congruences as well as ideals coincide. Conditions under which congruences and ideals in a GPEA P may be extended to a γ-unitization of U of P are clarified. In the last part of the paper, subcentral and central ideals in GPEAs and their relations to subdirect and direct decompositions are studied. [ABSTRACT FROM AUTHOR]

Abstract Two frames { ϕ i } i ∈ I and { ψ i } i ∈ I for a separable Hilbert space H are woven if there are positive constants A ≤ B such that for every subset σ ⊂ I , the family { ϕ i } i ∈ σ ∪ { ψ i } i ∈ σ c is a frame for H with frame bounds A , B. Bemrose et al. introduced weaving frames in separable Hilbert spaces and observed that weaving frames have potential applications in signal processing. Motivated by this, and the recent work of Balan in the direction of application of vector-valued frames (or superframes) in signal processing, we study vector-valued weaving frames. In this paper, first we give some fundamental properties of vector-valued weaving frames. It is shown that if a family of vector-valued frames is woven, then the corresponding family of frames for atomic spaces is woven, but the converse is not true. We present a technique for the construction of vector-valued woven frames from given woven frames for atomic spaces. Necessary and sufficient conditions for vector-valued weaving Riesz sequences are given. Several numerical examples are given to illustrate the results. [ABSTRACT FROM AUTHOR]

A sequence {φn} in a Hilbert space H with inner product <·,·> is called a generalized Riesz system if there exist an ONB e={en} in H and a densely defined closed operator T in H with densely defined inverse such that {en}⊂D(T)∩D((T-1)⁎) and Ten=φn, n=0,1,⋯, and (e,T) is called a constructing pair for {φn} and T is called a constructing operator for {φn}. The main purpose of this paper is to investigate under what conditions the ordered set Cφ of all constructing operators for a generalized Riesz system {φn} has maximal elements, minimal elements, the largest element, and the smallest element in order to find constructing operators fitting to each of the physical applications. [ABSTRACT FROM AUTHOR]

We construct some new Riesz bases and consider the stability of them. The investigation is based on the stability of Riesz bases of cosines and sines in the Hilbert space L2[0,π]. [ABSTRACT FROM AUTHOR]

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 standard frames, we present an explicit structure for controlled Riesz bases and show that every (U , C) -controlled Riesz basis { f k } k = 1 ∞ is in the form { U − 1 C M e k } k = 1 ∞ , where M is a bijective operator on H. Furthermore, we propose an equivalent accessible condition to the sequence { f k } k = 1 ∞ being a (U , C) -controlled Riesz basis. [ABSTRACT FROM AUTHOR]

In Archimedean vector lattices bands can be introduced via three different coinciding notions. First, they are order closed ideals. Second, they are precisely those ideals which equal their double disjoint complements. The third concept is that of an ideal which contains the supremum of any of its bounded subsets, provided the supremum exists in the vector lattice. We investigate these three notions and their relationships in the more general setting of Archimedean pre-Riesz spaces. We introduce the notion of a supremum closed ideal, which is related to the third aforementioned notion in vector lattices. We show that for a directed ideal I in a pervasive pre-Riesz space with the Riesz decomposition property these three concepts coincide, provided the double disjoint complement of I is directed. In pervasive pre-Riesz spaces every directed band is supremum closed and every supremum closed directed ideal I equals its double disjoint complement, provided the double disjoint complement of I is directed. In general, in Archimedean pre-Riesz spaces the three notions differ. For this we provide appropriate counterexamples. [ABSTRACT FROM AUTHOR]

Let (xα) be a net in a locally solid vector lattice (X,τ); we say that (xα) is unbounded τ-convergent to a vector x∈X if |xα-x|∧w→τ0 for all w∈X+. In this paper, we study general properties of unbounded τ-convergence (shortly uτ-convergence). uτ-convergence generalizes unbounded norm convergence and unbounded absolute weak convergence in normed lattices that have been investigated recently. We introduce uτ-topology and briefly study metrizability and completeness of this topology. [ABSTRACT FROM AUTHOR]

The Kato Theorem on similarity for sequences of projections in a Hilbert space is extended to the case when both sequences consist of nonselfadjoint projections. Passing to subspaces, this leads to stability theorems for Riesz bases of subspaces, at least one of which is finite dimensional, and for arbitrary vector Riesz bases. The following is proved as an application. If {φn}∞ n=1 is a Riesz basis and |θn| ≤ C for large n, where the constant C depends only on {φn}∞ n=1, then {φn + θnφn+1}∞ n=1 also forms a Riesz basis. [ABSTRACT FROM AUTHOR]

We present a description of the relationship between the Sobolev spaces W21(ℝ) and W22(ℝ) and the Hilbert space ℓ2. Let Y be a finite or countable set of points on ℝ and let d ≔ inf {|y′ − y′′|, y′, y′′ ∈ Y, y′ ≠ y′′}. By using this relationship, we prove that if d = 0, then the systems of δ -functions {δ(x − yj), yj ∈ Y} and their derivatives {δ′(x − yj), yj ∈ Y} do not form Riesz bases in the closures of their linear spans in the Sobolev spaces W2−1(ℝ) and W2−2(ℝ) but, on the contrary, form the indicated bases in the case where d > 0. We also present the description of Friedrichs and Krein extensions and demonstrate their transversality. Moreover, the construction of a basis boundary triple and the description of all nonnegative self-adjoint extensions of the operator A′ are presented. [ABSTRACT FROM AUTHOR]

Let K be a compact group, and let ρ be a representation of K on a Hilbert space 𝓗ρ. We classify invariant subspaces of 𝓗ρ in terms of range functions, and investigate frames of the form {ρ(ξ)fi}ξ∈K,i∈I. This is done first in the setting of translation invariance, where K is contained in a larger group G and ρ is left translation on 𝓗ρ = L2(G). For this case, our analysis relies on a new, operator-valued version of the Zak transform. For more general representations, we develop a calculational system known as a bracket to analyze representation structures and frames with a single generator. Several applications are explored. Then we turn our attention to frames with multiple generators, giving a duality theorem that encapsulates much of the existing research on frames generated by finite groups, as well as classical duality of frames and Riesz sequences. [ABSTRACT FROM AUTHOR]

In this paper, firstly, in order to establish our main techniques we give a direct proof for the existence of the dilations for pairs of dual group frames. Then we focus on proving the uniqueness of such dilations in certain sense of similarity and giving an operator parameterization of the dilations of all pairs of dual group frames for a given group frame. We show that the operators which transform different dilations are of special structured lower triangular. [ABSTRACT FROM AUTHOR]

In this paper we introduce a family of embeddings jn, n ≥ 2, of a near vector space into a vector space. We give some examples for such embeddings and show that jn's invariant metric on a near vector space S defines an isometry on the vector space Rn(S), and if S is a near vector lattice then jn's are join preserving on the vector lattices Rn(S). Finally, we will find applications of this embeddings in Hilbert spaces, filtrations, martingales, and metric spaces. [ABSTRACT FROM AUTHOR]

A convolution in a Hilbert space is defined, and its basic properties are studied. [ABSTRACT FROM AUTHOR]

The equation which describes the small vibrations of a nonhomogeneous damped string can be rewritten as an abstract Cauchy problem for the densely defined closed operator iA. We prove that the set of root vectors of the operator A forms a basis of subspaces in a certain Hilbert space H. Furthermore, we give the rate of convergence for the decomposition with respect to this basis. In the second main result we show that with additional assumptions the set of root vectors of the operator A is a Riesz basis for H. [ABSTRACT FROM AUTHOR]

The primary goal of our article is to implement some standard spin geometry techniques related to the study of Dirac and Laplace operators on Dirac vector bundles into the multidimensional theory of Hilbert space operators. The transition from spin geometry to operator theory relies on the use of Clifford environments, which essentially are Clifford algebra augmentations of unital complex C*-algebras that enable one to set up counterparts of the geometric Bochner-Weitzenböck and Bochner-Kodaira-Nakano curvature identities for systems of elements of a C*--algebra. The so derived self-commutator identities in conjunction with Bochner's method provide a natural motivation for the definitions of several types of seminormal systems of operators. As part of their study, we single out certain spectral properties, introduce and analyze a singular integral model that involves Riesz transforms, and prove some self-commutator inequalities. [ABSTRACT FROM AUTHOR]

In this paper we introduce general theories of semi-regular biorthogonal pairs, generalized Riesz bases and its physical applications. Here we deal with biorthogonal sequences {Øn} and {ψn} in a Hilbert space H, with domains D(Ø) = {x ∊ H; ∑k=0∞|(x|Øk)|² < ∞} and D(ψ) = {x ∊ H; ∑k=0∞|(x|ψk)|² < ∞} and linear spans DØ ≡ Span{Øn} and Dψ ≡ Span{ψn}. A biorthogonal pair ({Øn}, {ψn}) is called regular if both DØ and Dψ are dense in H, and it is called semi-regular if either DØ and D(Ø) or Dψ and D(ψ) are dense in H. In a previous paper [H. Inoue, J. Math. Phys. 57, 083511 (2016)], we have shown that if ({Øn}, {ψn}) is a regular biorthogonal pair then both {Øn} and {ψn} are generalized Riesz bases defined in the work of Inoue and Takakura [J. Math. Phys. 57, 083505 (2016)]. Here we shall show that the same result holds true if the pair is only semi-regular by using operators TØ,e, Te,Ø, Tψ,e, and Te,ψ defined by an orthonormal basis e in H and a biorthogonal pair ({Øn}, {ψn}). Furthermore, we shall apply this result to pseudo-bosons in the sense of the papers of Bagarello [J. Math. Phys. 51, 023531 (2010); J. Phys. A 44, 015205 (2011); Phys. Rev. A 88, 032120 (2013); and J. Math. Phys. 54, 063512 (2013)]. [ABSTRACT FROM AUTHOR]

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 attributes of these outer product sequences (in particular, their Riesz and frame bounds) and 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 sequences with optimal Riesz bounds for outer products. We provide constructions of frames which produce Riesz outer product bases with “good” Riesz bounds. We show that the family of unit norm frames which yield independent outer product sequences is open and dense (in a Euclidean-analytic sense) within the topological space ⊗ i = 1 M S N − 1 where M is less than or equal to the dimension of the space of symmetric operators on H N ; that is to say, almost every frame with such a bound on its cardinality will induce a set of independent outer products. Thus, this would mean that finding the necessary and sufficient conditions such that the induced outer products are dependent is a more interesting question. For the coup de grâce, we give a full analytic and geometric classification of such sequences which produce dependent outer products. [ABSTRACT FROM AUTHOR]

G-frames and g-Riesz frames as generalized frames in Hilbert spaces have been studied by many authors in recent years. The super Hilbert space has a certain advantage compared with the Hilbert space in the field of studying quantum mechanics. In this paper, for super Hilbert space H⊕K, the definitions of a g-Riesz frame and minimal g-complete are put forward; also a characterization of g-Riesz frames is obtained. In particular, we generalize them to general super Hilbert space L1⊕L2⊕⋯⊕Ln. Finally, a conclusion of the stability of a g-Riesz frame for the super Hilbert space is given. [ABSTRACT FROM AUTHOR]

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 perturbations. We also generalized a result of Paley-Wiener [16] to the situation of fusion basis. [ABSTRACT FROM AUTHOR]

In this article, the orthogonal projection and the Riesz representation theorem are mainly formalized. In the first section, we defined the norm of elements on real Hilbert spaces, and defined Mizar functor RUSp2RNSp, real normed spaces as real Hilbert spaces. By this definition, we regarded sequences of real Hilbert spaces as sequences of real normed spaces, and proved some properties of real Hilbert spaces. Furthermore, we defined the continuity and the Lipschitz the continuity of functionals on real Hilbert spaces. Referring to the article [15], we also defined some definitions on real Hilbert spaces and proved some theorems for defining dual spaces of real Hilbert spaces. As to the properties of all definitions, we proved that they are equivalent properties of functionals on real normed spaces. In Sec. 2, by the definitions [11], we showed properties of the orthogonal complement. Then we proved theorems on the orthogonal decomposition of elements of real Hilbert spaces. They are the last two theorems of existence and uniqueness. In the third and final section, we defined the kernel of linear functionals on real Hilbert spaces. By the last three theorems, we showed the Riesz representation theorem, existence, uniqueness, and the property of the norm of bounded linear functionals on real Hilbert spaces. We referred to [36], [9], [24] and [3] in the formalization. [ABSTRACT FROM AUTHOR]

We discuss the existence and stability of Riesz bases of exponential type of L² (T) for special domains T ⊂ R² called trapezoids, and we construct exponential bases on the finite union of rectangles with the same height. We also generalize our main theorems in dimension d ≤ 3. [ABSTRACT FROM AUTHOR]

We generalize the characterizations of the direct sum decomposition of a Hilbert space in terms of the angle $$R$$ and, using this result, present another proof of the equality concerning the norms of the projections $$P$$ and $$I-P$$ . As an application to abstract frame theory, we show that the ranges of the analysis operators of oblique dual frame sequences satisfy the the angle condition. [ABSTRACT FROM AUTHOR]

The article focuses on the Dirac operator (Lp), a mathematical equation. Topics discussed include the mathematical theorems, the eigenvectors, and the Riez vector spaces. Also mentioned are the Hilbert space, quadratic equation formulas, and the several geometric properties of Dirac operator including minimality, completeness, and Riesz basis property.

We show that any finite union of intervals supports a Riesz basis of exponentials. [ABSTRACT FROM AUTHOR]

In this paper, we develop the basic stability theory for g-frames and g-Riesz bases in Hilbert C*-module. Also, we study stability of the dual g-frames in Hilbert C*-module. We extend the Casazza-Christensen perturbation theorem to g-frames in Hilbert C*-module but this is not valid for g-Riesz bases. We prove some characterizations of g-frames and g-Riesz bases in Hilbert C*-module. [ABSTRACT FROM AUTHOR]

Neutral type systems considered in infinite-dimensional Hilbert space are analyzed for exact controllability characterization. The approach is based on the problem of moments using a Riesz basis of eigenvectors. The duality with observability is investigated. A criterion of exact observability is deduced. [ABSTRACT FROM AUTHOR]

In the present paper, we deal with the perturbed operatorwhereis a closed densely defined linear operator on a separable Hilbert spaceXwith domainwhileare linear operators onXwith the same domainand satisfying a specific growing inequality. The basic idea here is to investigate under sufficient conditions assuring the existence of a Riesz basis of eigenvectors of the operator. The obtained results are of importance for application to a nonself-adjoint problem describing the radiation of a vibrating structure in a light fluid and to a nonself-adjoint Gribov operator in Bargmann space. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper we present a family of analysis and synthesis systems of operators with frame-like properties for the range of a bounded operator on a separable Hilbert space. This family of operators is called a Θ-g-frame, where Θ is a bounded operator on a Hilbert space. Θ-g-frames are a generalization of g-frames, which allows to reconstruct elements from the range of Θ. In general, range of Θ is not a closed subspace. We also construct new Θ-g-frames by considering Θ-g-frames for its components. We further study Riesz decompositions for Hilbert spaces, which are a generalization of the notion of Riesz bases. We define the coefficient operators of a Riesz decomposition and we will show that these coefficient operators are continuous projections. We obtain some results about stability of Riesz decompositions under small perturbations. [ABSTRACT FROM AUTHOR]

The aim of this paper is to give a functional analytic proof of the Lebesgue--Darst decomposition theorem [1]. We show that the decomposition of a nonnegative valued additive set function into absolutely continuous and singular parts with respect to another derives from the Riesz orthogonal decomposition theorem employed in a corresponding Hilbert space. [ABSTRACT FROM AUTHOR]

Some necessary and suffcient conditions are given for the existence of a G-positive (G-repositive) solution of adjointable operator equations AX = C,AXA(*) = C and AXB = C over Hilbert C*-modules, respectively. Moreover, the expressions of these general G-positive (G-repositive) solutions are also derived. Some of the findings of this paper extend some known results in the literature. [ABSTRACT FROM AUTHOR]

We characterize random signals which can be linearly determined by their samples. This problem is related to the question of the representation of random variables by means of a countable Riesz basis. We study different representations for processes which are linearly determined by a countable Riesz basis. This concerns the representation of continuous time processes by means of discrete samples. [ABSTRACT FROM AUTHOR]

It is observed that the Dirichlet ring admits a representation in an infinite-dimensional matrix algebra. The resulting matrices are subsequently used in the construction of nonorthogonal Riesz bases in a separable Hilbert space. This framework enables custom design of a plethora of bases with interesting features. Remarkably, the representation of signals in any one of these bases is numerically implementable via fast algorithms. [ABSTRACT FROM AUTHOR]

Given the g-orthonormal basis for Hilbert space H, we characterize the g-frames, normalized tight g-frames, and g-Riesz bases in terms of the g-preframe operators. Then we consider the transformations of g-frames, normalized tight g-frames, and g-Riesz bases, which are induced by operators and characterize them in terms of the operators. Finally, we discuss the sums and g-dual frames of g-frames by applying the results of characterizations. [ABSTRACT FROM AUTHOR]

One of the essential applications of frames is that they lead to expansions of vectors in the underlying Hilbert space in terms of the frame elements. In this decomposition, dual frames have a key role. G-frames, introduced by Sun, cover many other recent generalizations of frames. In this paper, we give some characterizations of dual g-frames. Moreover, we prove that if two g-frames are close to each other, then we can find duals of them which are close to each other. [ABSTRACT FROM AUTHOR]

G-frames are generalizations of ordinary frames for Hilbert spaces. In the present paper we study frames, and operators on a special separable Hilbert C* -module, B(H,K), where H and K are Hilbert spaces, and we prove that every g-frame for H is a frame for B(H,K) and vice versa. Also, we derive some relationships between g-Riesz bases for H and Riesz bases in B(H,K) . Similar results for orthogonal bases will be discussed. [ABSTRACT FROM AUTHOR]

We introduce the modular continuous g-Riesz basis to improve one existing result for continuous g-Riesz basis in Hilbert C*- modules, and then we study the equivalency relations between continuous g-frames in Hilbert C*-modules, and, in particular, we obtain two necessary and sufficient conditions under which two continuous g-frames are similar. Finally, we generalize a stability result for alternate duals of g-frames in Hilbert spaces to alternate duals of continuous g-frames in Hilbert C*-modules. [ABSTRACT FROM AUTHOR]

In this paper we study fusion frames and g-frames for the tensor products and direct sums of Hilbert spaces. We show that the tensor product of a finite number of g-frames (resp. fusion frames, g-Riesz bases) is a g-frame (resp. fusion frame, g-Riesz basis) for the tensor product space and vice versa. Moreover we obtain some important results in tensor products and direct sums of g-frames, fusion frames, resolutions of the identity and duals. [ABSTRACT FROM AUTHOR]

In this paper we study the relationship between g-frames for super Hilbert space H ⊕ K and g-frames for H and K and frames for H, K and H ⊕ K . Also the concepts of disjoint g-frames and strong complementary g-frame are introduced and studied. [ABSTRACT FROM AUTHOR]

Given a U-invariant sampling scheme on a Hilbert space \cal H. This paper characterizes atomic subspaces \cal A of \cal H such that every signal x\in\cal A can be reconstructed from its generalized samples acquired with the U-invariant sampling scheme. If signal recovery is possible a linear filter is derived which reconstructs the signal from the samples. Moreover, the paper gives necessary and sufficient conditions on a sequence of sampling functions such that it forms a pseudoframe or pseudo-Riesz basis for a given atomic subspace \cal A. [ABSTRACT FROM PUBLISHER]

Abstract: We introduce a simple and efficient method to reconstruct an element of a Hilbert space in terms of an arbitrary finite collection of linearly independent reconstruction vectors, given a finite number of its samples with respect to any Riesz basis. As we establish, provided the dimension of the reconstruction space is chosen suitably in relation to the number of samples, this procedure can be implemented in a completely numerically stable manner. Moreover, the accuracy of the resulting approximation is determined solely by the choice of reconstruction basis, meaning that reconstruction vectors can be readily tailored to the particular problem at hand. An important example of this approach is the accurate recovery of a piecewise analytic function from its first few Fourier coefficients. Whilst the standard Fourier projection suffers from the Gibbs phenomenon, by reconstructing in a piecewise polynomial basis we obtain an approximation with root-exponential accuracy in terms of the number of Fourier samples and exponential accuracy in terms of the degree of the reconstruction. Numerical examples illustrate the advantage of this approach over other existing methods. [Copyright &y& Elsevier]

In this paper we construct infinitely many selfadjoint solutions of the control algebraic Riccati equation using invariant subspaces of the associated Hamiltonian. We do this under the assumptions that the system operator is normal and has compact inverse and that the Hamiltonian possesses a Riesz basis of invariant subspaces. [ABSTRACT FROM AUTHOR]