341 research outputs found

    Weighted projections and Riesz frames

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    Let H\mathcal{H} be a (separable) Hilbert space and {ek}k≥1\{e_k\}_{k\geq 1} a fixed orthonormal basis of H\mathcal{H}. Motivated by many papers on scaled projections, angles of subspaces and oblique projections, we define and study the notion of compatibility between a subspace and the abelian algebra of diagonal operators in the given basis. This is used to refine previous work on scaled projections, and to obtain a new characterization of Riesz frames.Comment: 23 pages, to appear in Linear Algebra and its Application

    Linear combinations of generators in multiplicatively invariant spaces

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    Multiplicatively invariant (MI) spaces are closed subspaces of L2(Ω,H)L^2(\Omega,\mathcal{H}) that are invariant under multiplications of (some) functions in L∞(Ω)L^{\infty}(\Omega). In this paper we work with MI spaces that are finitely generated. We prove that almost every linear combination of the generators of a finitely generated MI space produces a new set on generators for the same space and we give necessary and sufficient conditions on the linear combinations to preserve frame properties. We then apply what we prove for MI spaces to system of translates in the context of locally compact abelian groups and we obtain results that extend those previously proven for systems of integer translates in L2(Rd)L^2(\mathbb{R}^d).Comment: 13 pages. Minor changes have been made. To appear in Studia Mathematic

    Invariances of Frame Sequences under Perturbations

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    This paper determines the exact relationships that hold among the major Paley–Wiener perturbation theorems for frame sequences. It is shown that major properties of a frame sequence such as excess, deficit, and rank remain invariant under Paley–Wiener perturbations, but need not be preserved by compact perturbations. For localized frames, which are frames with additional structure, it is shown that the frame measure function is also preserved by Paley–Wiener perturbations

    Continuous K-g-fusion frames in Hilbert spaces

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    This paper aims at introducing the concept of c-K-g-fusion frames, which are generalizations of K-g-fusion frames, proving some new results on c-K-g-fusion frames in Hilbert spaces, defining duality of c-K-g-fusion frames and characterizing the kinds of the duals, and discussing the perturbation of c-K-g-fusion frames.Publisher's Versio

    Operator-Valued Frames for the Heisenberg Group

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    A classical result of Duffin and Schaeffer gives conditions under which a discrete collection of characters on R\mathbb{R}, restricted to E=(−1/2,1/2)E = (-1/2, 1/2), forms a Hilbert-space frame for L2(E)L^2(E). For the case of characters with period one, this is just the Poisson Summation Formula. Duffin and Schaeffer show that perturbations preserve the frame condition in this case. This paper gives analogous results for the real Heisenberg group HnH_n, where frames are replaced by operator-valued frames. The Selberg Trace Formula is used to show that perturbations of the orthogonal case continue to behave as operator-valued frames. This technique enables the construction of decompositions of elements of L2(E)L^2(E) for suitable subsets EE of HnH_n in terms of representations of HnH_n

    Multipliers for p-Bessel sequences in Banach spaces

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    Multipliers have been recently introduced as operators for Bessel sequences and frames in Hilbert spaces. These operators are defined by a fixed multiplication pattern (the symbol) which is inserted between the analysis and synthesis operators. In this paper, we will generalize the concept of Bessel multipliers for p-Bessel and p-Riesz sequences in Banach spaces. It will be shown that bounded symbols lead to bounded operators. Symbols converging to zero induce compact operators. Furthermore, we will give sufficient conditions for multipliers to be nuclear operators. Finally, we will show the continuous dependency of the multipliers on their parameters.Comment: 17 page
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