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Localization of Frames, Banach Frames, and the Invertibility of the Frame Operator

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Abstract

We introduce a new concept to describe the localization of frames. In our main result we show that the frame operator preserves this localization and that the dual frame possesses the same localization property. As an application we show that certain frames for Hilbert spaces extend automatically to Banach frames. Using this abstract theory, we derive new results on the construction of nonuniform Gabor frames and solve a problem about non-uniform sampling in shift-invariant spaces.

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  1. Department of Mathematics, The University of Connecticut, Storrs, CT 06269-3009, USA

    Karlheinz Gröchenig

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  1. Karlheinz Gröchenig
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Correspondence to Karlheinz Gröchenig.

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Gröchenig, K. Localization of Frames, Banach Frames, and the Invertibility of the Frame Operator. J. Fourier Anal. Appl. 10, 105–132 (2004). https://doi.org/10.1007/s00041-004-8007-1

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  • DOI: https://doi.org/10.1007/s00041-004-8007-1

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