Frames and generalized shift-invariant systems

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    Abstract

    With motivation from the theory of Hilbert-Schmidt operators we review recent topics concerning frames in L 2 (R) and their duals. Frames are generalizations of orthonormal bases in Hilbert spaces. As for an orthonormal basis, a frame allows each element in the underlying Hilbert space to be written as an unconditionally convergent infinite linear combination of the frame elements; however, in contrast to the situation for a basis, the coefficients might not be unique. We present the basic facts from frame theory and the motivation for the fact that most recent research concentrates on tight frames or dual frame pairs rather than general frames and their canonical dual. The corresponding results for Gabor frames and wavelet frames are discussed in detail.
    Original languageEnglish
    Title of host publicationOPERATOR THEORY : ADVANCES AND APPLICATIONS : Pseudo-Differential Operators and Related Topics
    VolumeVolume 164
    PublisherBirkhäuser Verlag
    Publication date2004
    Pages193-209
    ISBN (Print)37-64-37513-2
    Publication statusPublished - 2004
    EventInternational Conference on Pseudo-Differential Operators and Related Topics - Vaxjo Univ, Vaxjo, SWEDEN
    Duration: 1 Jan 2004 → …

    Conference

    ConferenceInternational Conference on Pseudo-Differential Operators and Related Topics
    CityVaxjo Univ, Vaxjo, SWEDEN
    Period01/01/2004 → …

    Cite this

    Christensen, O. (2004). Frames and generalized shift-invariant systems. In OPERATOR THEORY : ADVANCES AND APPLICATIONS: Pseudo-Differential Operators and Related Topics (Vol. Volume 164, pp. 193-209). Birkhäuser Verlag.