129 research outputs found

    Frames, semi-frames, and Hilbert scales

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    Given a total sequence in a Hilbert space, we speak of an upper (resp. lower) semi-frame if only the upper (resp. lower) frame bound is valid. Equivalently, for an upper semi-frame, the frame operator is bounded, but has an unbounded inverse, whereas a lower semi-frame has an unbounded frame operator, with bounded inverse. For upper semi-frames, in the discrete and the continuous case, we build two natural Hilbert scales which may yield a novel characterization of certain function spaces of interest in signal processing. We present some examples and, in addition, some results concerning the duality between lower and upper semi-frames, as well as some generalizations, including fusion semi-frames and Banach semi-frames.Comment: 27 pages; Numerical Functional Analysis and Optimization, 33 (2012) in press. arXiv admin note: substantial text overlap with arXiv:1101.285

    Unconditional convergence and invertibility of multipliers

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    In the present paper the unconditional convergence and the invertibility of multipliers is investigated. Multipliers are operators created by (frame-like) analysis, multiplication by a fixed symbol, and resynthesis. Sufficient and/or necessary conditions for unconditional convergence and invertibility are determined depending on the properties of the analysis and synthesis sequences, as well as the symbol. Examples which show that the given assertions cover different classes of multipliers are given. If a multiplier is invertible, a formula for the inverse operator is determined. The case when one of the sequences is a Riesz basis is completely characterized.Comment: 31 pages; changes to previous version: 1.) the results from the previous version are extended to the case of complex symbols m. 2.) new statements about the unconditional convergence and boundedness are added (3.1,3.2 and 3.3). 3.) the proof of a preliminary result (Prop. 2.2) was moved to a conference proceedings [29]. 4.) Theorem 4.10. became more detaile

    Frames for the solution of operator equations in Hilbert spaces with fixed dual pairing

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    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 H01(Ω)H_0^1(\Omega) and H−1(Ω)H^{-1}(\Omega). In this article, we are going to revisit the concept of Stevenson frames and introduce it for Banach spaces. This is equivalent to ℓ2\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 isomorphic to a Hilbert space. In this article, we deal with the contrasting setting, where H\mathcal H and H′\mathcal H' are not identified, and equivalent norms are distinguished, and show that in this setting the investigation of ℓ2\ell^2-Banach frames make sense.Comment: 23 pages; accepted for publication in 'Numerical Functional Analysis and Optimization

    Weighted frames, weighted lower semi frames and unconditionally convergent multipliers

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    In this paper we ask when it is possible to transform a given sequence into a frame or a lower semi frame by multiplying the elements by numbers. In other words, we ask when a given sequence is a weighted frame or a weighted lower semi frame and for each case we formulate a conjecture. We determine several conditions under which these conjectures are true. Finally, we prove an equivalence between two older conjectures, the first one being that any unconditionally convergent multiplier can be written as a multiplier of Bessel sequences by shifting of weights, and the second one that every unconditionally convergent multiplier which is invertible can be written as a multiplier of frames by shifting of weights. We also show that these conjectures are also related to one of the newly posed conjectures.Comment: 14 page

    GG-dual Frames in Hilbert C∗C^{*}-module Spaces

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    In this paper, we introduce the concept of gg-dual frames for Hilbert C∗C^{*}-modules, and then the properties and stability results of gg-dual frames  are given.  A characterization of gg-dual frames, approximately dual frames and dual frames of a given frame is established. We also give some examples to show that the characterization of gg-dual frames for Riesz bases in Hilbert spaces is not satisfied in general Hilbert C∗C^*-modules
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