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Pseudo-Differential Operators and Related Topics pp 193–209Cite as

Frames and Generalized Shift-Invariant Systems

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Part of the book series: Operator Theory: Advances and Applications ((OT,volume 164))

Abstract

With motivation from the theory of Hilbert-Schmidt operators we review recent topics concerning frames in L2(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.

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Author information

Authors and Affiliations

  1. Department of Mathematics, Technical University of Denmark, Building 303, Denmark

    Ole Christensen

Authors
  1. Ole Christensen
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Editor information

Editors and Affiliations

  1. Dipartimento di Matematica, Università di Torino, Via Carlo Alberto, 10, 10123, Torino, Italy

    Paolo Boggiatto  & Luigi Rodino  & 

  2. School of Mathematics and Systems Engineering, Växjö University, SE-351 95, Växjö, Sweden

    Joachim Toft

  3. Department of Mathematics and Statistics, York University, 4700 Keele Street, Toronto, Ontario, M3J 1P3, Canada

    M. W. Wong

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© 2006 Birkhäuser Verlag Basel/Switzerland

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Christensen, O. (2006). Frames and Generalized Shift-Invariant Systems. In: Boggiatto, P., Rodino, L., Toft, J., Wong, M.W. (eds) Pseudo-Differential Operators and Related Topics. Operator Theory: Advances and Applications, vol 164. Birkhäuser Basel. https://doi.org/10.1007/3-7643-7514-0_14

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