In respect of b-linear functional, Riesz representation theorem in n-Hilbert space have been proved. We define b-sesquilinear functional in n-Hilbert space and establish the polarization identities. A generalized form of the Schwarz inequality in n-Hilbert space is being discussed. Finally, a generalized version of Riesz representation theorem with respect to b-sesquilinear functional in n-Hilbert space have been developed.
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 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.
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.
The fractional Hilbert transforms plays an important role in optics and signal processing. In particular the analytic signal proposed by Gabor has as a key component the Hilbert transform. The higher dimensional Hilbert transform is the Riesz-Hilbert transform which was used by Felsberg and Sommer to construct the monogenic signal. We will construct fractional and quaternionic fractional Riesz-Hilbert transforms based on a eigenvalue decomposition. We will prove properties of these transformations such as shift and scale invariance, orthogonality and the semigroup property. Based on the fracti…
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.
This note concerns a further study about Riesz-Fischer maps, already introduced by the author in a recent work, that is a notion that extends to the spaces of distributions the sequences that are known as Riesz-Fischer sequences. In particular it is proved a characterizing inequality that have as consequence the existence of the continuous inverse of the synthesis operator.
We prove that a Hilbert space frame $\fti$ contains a Riesz basis if every subfamily $\ftj , J \subseteq I ,$ is a frame for its closed span. Secondly we give a new characterization of Banach spaces which do not have any subspace isomorphic to $c_0$. This result immediately leads to an improvement of a recent theorem of Holub concerning frames consisting of a Riesz basis plus finitely many elements.
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…
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.
We give a simple and short proof of the classical Lebesgue decomposition theorem of measures via the Riesz orthogonal decomposition theorem of Hilbert spaces. The tools we employ are elementary Hilbert space techniques.
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…
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 shall say that a bounded linear operator $T$ acting on a Banach space $X$ admits a generalized Kato-Riesz decomposition if there exists a pair of $T$-invariant closed subspaces $(M,N)$ such that $X=M\oplus N$, the reduction $T_M$ is Kato and $T_N$ is Riesz. In this paper we define and investigate the generalized Kato-Riesz spectrum of an operator. For $T$ is said to be generalized Drazin-Riesz invertible if there exists a bounded linear operator $S $ acting on $X$ such that $TS=ST$, $STS=S$, $ TST-T$ is Riesz. We investigate generalized Drazin-Riesz invertible operators and also, characteri…
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…
In this paper, we prove the Spanne-type boundedness of the generalized Riesz potential operator from the one generalized weighted local Morrey spaces to the another one, and from the generalized weighted local Morrey spaces to the weak generalized weighted local Morrey spaces. We also prove the Adams-type boundedness of the operator from the weighted spaces to the another one, and from the weighted spaces to the weak weighted spaces.
Pseudo-Hermitian Hamiltonians have recently become a field of wide investigation. Originally, the Generalized Riesz Systems (GRS) have been introduced as an auxiliary tool in this theory. In contrast, the current paper, GRSs are analysed in terms of basis theory. The relationship between semi-regular sequences and GRSs is provided. Various characterizations of GRSs are discussed.
We develope a local theory for frames on finite dimensional Hilbert spaces. In particular, a bounded frame on a finite dimensional Hilbert space contains a subset which is a good Riesz basis for a percentage (arbitrarily close to one) of the space. We also construct a normalized frame for a Hilbert space which contains a subset which is a Schauder basis for H but does not contain any subset which is a Riesz basis for H.
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 consider a Riesz $ϕ$-variation for functions $f$ defined on the real line when $φ:Ω\times[0,\infty)\to[0,\infty)$ is a generalized $Φ$-function. We show that it generates a quasi-Banach space and derive an explicit formula for the modular when the function $f$ has bounded variation. The resulting $BV$-type energy has previously appeared in image restoration models. We generalize and improve previous results in the variable exponent and Orlicz cases and answer a question regarding the Riesz--Medvedev variation by Appell, Banaś and Merentes [\emph{Bounded Variation and Around}, Studies in Non…
We prove that the operator norm on weighted Lebesgue space L2(w) of the commutators of the Hilbert, Riesz and Beurling transforms with a BMO function b depends quadratically on the A2-characteristic of the weight, as opposed to the linear dependence known to hold for the operators themselves. It is known that the operator norms of these commutators can be controlled by the norm of the commutator with appropriate Haar shift operators, and we prove the estimate for these commutators. For the shift operator corresponding to the Hilbert transform we use Bellman function methods, however there is n…
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.
Let $A\in\mathcal{B}(X)$, $B\in\mathcal{B}(Y)$ and $C\in\mathcal{B}(Y,X)$ where $X$ and $Y$ are infinite Banach or Hilbert spaces. Let $M_{C}=\begin{pmatrix} A & C\cr 0 & B \end{pmatrix}$ be $2\times 2$ upper triangular operator matrix acting on $X\oplus Y$. In this paper, we consider some necessary and sufficient conditions for $M_{C}$ to be generalized Drazin-Riesz invertible. Furthermore, the set $\bigcap_{C\in \mathcal{B}(Y,X)}σ_{gDR}(M_{C})$ will be investigated and their relation between $\bigcap_{C\in \mathcal{B}(Y,X)}σ_{b}(M_{C})$ will be studied, where $σ_{gDR}(M_{C})$ and $σ_{b}(M_{C…
We consider generalized Orlicz-Morrey spaces $M_{Φ,φ}(\Rn)$ including their weak versions $WM_{Φ,φ}(\Rn)$. In these spaces we prove the boundedness of the Riesz potential from $M_{Φ,φ_1}(\Rn)$ to $M_{Ψ,φ_2}(\Rn)$ and from $M_{Φ,φ_1}(\Rn)$ to $WM_{Ψ,φ_2}(\Rn)$. As applications of those results, the boundedness of the commutators of the Riesz potential on generalized Orlicz-Morrey space is also obtained. In all the cases the conditions for the boundedness are given either in terms of Zygmund-type integral inequalities on $(φ_{1},φ_{2})$, which do not assume any assumption on monotonicity of $φ_{…
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…
Let $Ω\subset \mathbb{R}^n$ be an unbounded open set. We consider the generalized weighted Morrey spaces $\mathcal{M}^{p(\cdot),φ}_ω(Ω)$ and the vanishing generalized weighted Morrey spaces $V\mathcal{M}^{p(\cdot),φ}_ω(Ω)$ with variable exponent $p(x)$ and a general function $φ(x,r)$ defining the Morrey-type norm. The main result of this paper are the boundedness of Riesz potential and its commutators on the spaces $\mathcal{M}^{p(\cdot),φ}_ω(Ω)$ and $V\mathcal{M}^{p(\cdot),φ}_ω(Ω)$. This result generalizes several existing results for Riesz potential and its commutators on Morrey type spaces.…
In this paper we introduce generalized grand Morrey spaces in the framework of quasimetric measure spaces, in the spirit of the so-called grand Lebesgue spaces. We prove a kind of reduction lemma which is applicable to a variety of operators to reduce their boundedness in generalized grand Morrey spaces to the corresponding boundedness in Morrey spaces, as a result of this application, we obtain the boundedness of the Hardy-Littlewood maximal operator as well as the boundedness of Calderón-Zygmund potential type operators. Boundedness of Riesz type potential operators are also obtained in the …
The purpose of this article is twofold. First of all, the notion of $(D, E)$-quasi basis is introduced for a pair $(D, E)$ of dense subspaces of Hilbert spaces. This consists of two biorthogonal sequences $\{ φ_n \}$ and $\{ ψ_n \}$ such that $\sum_{n=0}^\infty \ip{x}{φ_n}\ip{ψ_n}{y}=\ip{x}{y}$ for all $x \in D$ and $y \in E$. Secondly, it is shown that if biorthogonal sequences $\{ φ_n \}$ and $\{ ψ_n \}$ form a $(D ,E)$-quasi basis, then they are generalized Riesz systems. The latter play an interesting role for the construction of non-self-adjoint Hamiltonians and other physically relevant …
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.
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.
This paper continues the study of infinite dimensional bicomplex Hilbert spaces introduced in previous articles on the topic. Besides obtaining a Best Approximation Theorem, the main purpose of this paper is to obtain a bicomplex analogue of the Riesz-Fischer Theorem. There are many statements of the Riesz-Fischer (R-F) Theorem in the literature, some are equivalent, some are consequences of the original versions. The one referred to in this paper is the R-F Theorem which establishes that the spaces $l^2$ is the canonical model space.
In this paper we have some new results on sums of Hilbert space frames and Riesz bases. We also have a correction for some results in "S. Obeidat et al., Sums of Hilbert space frames, J. Math. Anal. Appl. 351 (2009) 579-585."
The class of Sturm-Liouville operators on the space of square integrable functions on a finite interval is considered. According to the Riesz-spectral property, the self-adjointness and the positivity of such unbounded linear operators on that space, a class of Hilbert spaces constructed as the domains of the positive (in particular, fractional) powers of any Sturm-Liouville operator is considered. On these spaces, it is shown that any Sturm-Liouville operator is a Riesz-spectral operator that possesses the same eigenvalues as the original ones, associated to rescaled eigenfunctions. This cons…
This paper considers different facets of the interplay between reproducing kernel Hilbert spaces (RKHS) and stable analysis/synthesis processes: First, we analyze the structure of the reproducing kernel of a RKHS using frames and reproducing pairs. Second, we present a new approach to prove the result that finite redundancy of a continuous frame implies atomic structure of the underlying measure space. Our proof uses the RKHS structure of the range of the analysis operator. This in turn implies that all the attempts to extend the notion of Riesz basis to general measure spaces are fruitless si…
In this manuscript, the concept of dual and approximate dual for continuous frames in Hilbert spaces will be introduced. Some of its properties will be studied. Also, the relations between two continuous Riesz bases in Hilbert spaces will be clarified through examples.
We consider generalized Hausdorff operators with positive definite and permutable perturbation matrices on Lebesgue spaces and prove that such operators are not Riesz operators provided they are non-zero.
We prove that supermodularity is a necessary condition for the generalized Hardy- Littlewood and Riesz rearrangement inequalities. We also show the necessity of the monotonicity of the kernels involved in the Riesz{type integral.
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.
We characterize Riesz frames and frames with the subframe property and use this to answer most of the questions from the literature concerning these properties and their relationships to the projection methods etc.
In this paper, we introduce a discrete Riesz transforms associated with the non-symmetric trigonometric Heckman-Opdam polynomials of type $A_1$. We prove that they can be extended to a bounded operators on $\ell^p(\mathbb{Z})$, $1<p<\infty$.
Some sharp results related to the convergence of means and families of operators generated by the generalized Bochner-Riesz kernels are obtained. The exact order of approximation of functions by these methods via $K$-functional (or its realization in the case of the space $L_p$, $0<p<1$) is derived.
Weaving frames are powerful tools in wireless sensor networks and pre-processing signals. In this paper, we introduce the concept of weaving for g-frames in Hilbert spaces. We first give some properties of weaving g-frames and present two necessary conditions in terms of frame bounds for weaving g-frames. Then we study the properties of weakly woven g-frames and give a sufficient condition for weaving g-frames. It is shown that weakly woven is equivalent to woven. Two sufficient conditions for weaving g-Riesz bases are given. And a weaving equivalent of an unconditional g-basis for weaving g-R…
In this paper, the mean value formula depends on the Bessel generalized shift operator corresponding to the solutions of the boundary value problem related to the Bessel operator are studied. In addition to, Riesz Bessel transforms related to the Bessel operators are studied. Since Bessel generalized shift operator is translation operator corresponding to the Bessel operator, we construct a family of RBxj by using Bessel generalized shift operator. Finally, we analysis weighted inequalities involving Riesz Bessel transforms .
We study totally bounded subsets in weighted variable exponent amalgam and Sobolev spaces. Moreover, this paper includes several detailed generalized results of some compactness criterions in these spaces.
In this paper, we investigate the convergence of the Bochner-Riesz means on some Sobolev type spaces including $L^p$-Sobolev spaces $(p\geq 1)$ and $H^q$-Sobolev spaces $(0<q<1)$. The relation between the smoothness imposed on functions and the rate of almost everywhere convergence of the generalized Bochner-Riesz means is given.
For the solution of operator equations, Stevenson introduced a definition of frames, where a Hilbert space and its dual are {\em not} identified. This means that the Riesz isomorphism is not used as an identification, which, for example, does not make sense for the Sobolev spaces $H_0^1(Ω)$ and $H^{-1}(Ω)$. In this article, we are going to revisit the concept of Stevenson frames and introduce it for Banach spaces. This is equivalent to $\ell^2$-Banach frames. It is known that, if such a system exists, by defining a new inner product and using the Riesz isomorphism, the Banach space is isomorph…
Let $T$ be a bounded quaternionic normal operator on a right quaternionic Hilbert space $\mathcal{H}$. We show that $T$ can be factorized in a strongly irreducible sense, that is, for any $δ>0$ there exist a compact operator $K$ with $\|K\|< δ$, a partial isometry $W$ and a strongly irreducible operator $S$ on $\mathcal{H}$ such that \begin{equation*} T = (W+K) S. \end{equation*} We illustrate our result with an example. We also prove a quaternionic version of the Riesz decomposition theorem and as a consequence, show that if the spherical spectrum of a bounded quaternionic operator (need not …
The goal of the paper is to obtain analogs of the sampling theorems and of the Riesz-Boas interpolation formulas which are relevant to the Discrete Hilbert and Kak-Hilbert transforms in $l^{2}$.
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 …
