1,524 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

    Riesz-like bases in rigged Hilbert spaces

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    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 \{\xi_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 TT is, in general, an unbounded operator in \H. If TT has a bounded inverse then the rigged Hilbert space is shown to be equivalent to a triplet of Hilbert spaces

    Heisenberg modules as function spaces

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    Let Δ\Delta be a closed, cocompact subgroup of G×G^G \times \widehat{G}, where GG is a second countable, locally compact abelian group. Using localization of Hilbert C∗C^*-modules, we show that the Heisenberg module EΔ(G)\mathcal{E}_{\Delta}(G) over the twisted group C∗C^*-algebra C∗(Δ,c)C^*(\Delta,c) due to Rieffel can be continuously and densely embedded into the Hilbert space L2(G)L^2(G). This allows us to characterize a finite set of generators for EΔ(G)\mathcal{E}_{\Delta}(G) as exactly the generators of multi-window (continuous) Gabor frames over Δ\Delta, a result which was previously known only for a dense subspace of EΔ(G)\mathcal{E}_{\Delta}(G). We show that EΔ(G)\mathcal{E}_{\Delta}(G) as a function space satisfies two properties that make it eligible for time-frequency analysis: Its elements satisfy the fundamental identity of Gabor analysis if Δ\Delta is a lattice, and their associated frame operators corresponding to Δ\Delta are bounded.Comment: 24 pages; several changes have been made to the presentation, while the content remains essentially unchanged; to appear in Journal of Fourier Analysis and Application

    Basic definition and properties of Bessel multipliers

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    This paper introduces the concept of Bessel multipliers. These operators are defined by a fixed multiplication pattern, which is inserted between the Analysis and synthesis operators. The proposed concept unifies the approach used for Gabor multipliers for arbitrary analysis/synthesis systems, which form Bessel sequences, like wavelet or irregular Gabor frames. The basic properties of this class of operators are investigated. In particular the implications of summability properties of the symbol for the membership of the corresponding operators in certain operator classes are specified. As a special case the multipliers for Riesz bases are examined and it is shown that multipliers in this case can be easily composed and inverted. Finally the continuous dependence of a Bessel multiplier on the parameters (i.e. the involved sequences and the symbol in use) is verified, using a special measure of similarity of sequences.Comment: 15 pages; Paper was cut from 27 to 15 pages and got a new titl

    Gabor Duality Theory for Morita Equivalent C∗C^*-algebras

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    The duality principle for Gabor frames is one of the pillars of Gabor analysis. We establish a far-reaching generalization to Morita equivalent C∗C^*-algebras where the equivalence bimodule is a finitely generated projective Hilbert C∗C^*-module. These Hilbert C∗C^*-modules are equipped with some extra structure and are called Gabor bimodules. We formulate a duality principle for standard module frames for Gabor bimodules which reduces to the well-known Gabor duality principle for twisted group C∗C^*-algebras of a lattice in phase space. We lift all these results to the matrix algebra level and in the description of the module frames associated to a matrix Gabor bimodule we introduce (n,d)(n,d)-matrix frames, which generalize superframes and multi-window frames. Density theorems for (n,d)(n,d)-matrix frames are established, which extend the ones for multi-window and super Gabor frames. Our approach is based on the localization of a Hilbert C∗C^*-module with respect to a trace.Comment: 36 page
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