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A Guide to Localized Frames and Applications to Galerkin-Like Representations of Operators

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Frames and Other Bases in Abstract and Function Spaces

Part of the book series: Applied and Numerical Harmonic Analysis ((ANHA))

Abstract

This chapter offers a detailed survey on intrinsically localized frames and the corresponding matrix representation of operators. We re-investigate the properties of localized frames and the associated Banach spaces in full detail. We investigate the representation of operators using localized frames in a Galerkin-type scheme. We show how the boundedness and the invertibility of matrices and operators are linked and give some sufficient and necessary conditions for the boundedness of operators between the associated Banach spaces.

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Notes

  1. 1.

    Note that those “frame-related” operators can be defined as possibly unbounded operators for any sequence [6].

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Acknowledgements

The first author was in part supported by the START project FLAME Y551-N13 of the Austrian Science Fund (FWF) and the DACH project BIOTOP I-1018-N25 of Austrian Science Fund (FWF). The second author acknowledges the support of the FWF-project P 26273-N25. P.B. wishes to thank NuHAG for the hospitality as well as the availability of its webpage. He also thanks Dominik Bayer, Gilles Chardon, Stephan Dahlke, Helmut Harbrecht, Wolfgang Kreuzer, Michael Speckbacher, and Diana Stoeva for related interesting discussions.

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Authors and Affiliations

  1. Acoustics Research Institute, Austrian Academy of Sciences, Wohllebengasse 12-14, Vienna, 1040, Austria

    Peter Balazs

  2. Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, Vienna, 1090, Austria

    Karlheinz Gröchenig

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  1. Peter Balazs
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  2. Karlheinz Gröchenig
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Correspondence to Peter Balazs .

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Editors and Affiliations

  1. Department of Mathematics, Temple University, Philadelphia, Pennsylvania, USA

    Isaac Pesenson

  2. School of Mathematics and Statistics, The University of New South Wales, Sydney, New South Wales, Australia

    Quoc Thong Le Gia

  3. Department of Mathematics, The Graduate Center, CUNY, New York, New York, USA

    Azita Mayeli

  4. Institute of Mathematical Sciences, Claremont Graduate University, Claremont, California, USA

    Hrushikesh Mhaskar

  5. Department of Mathematics, City University of Hong Kong, Kowloon Tong, Hong Kong

    Ding-Xuan Zhou

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Balazs, P., Gröchenig, K. (2017). A Guide to Localized Frames and Applications to Galerkin-Like Representations of Operators. In: Pesenson, I., Le Gia, Q., Mayeli, A., Mhaskar, H., Zhou, DX. (eds) Frames and Other Bases in Abstract and Function Spaces. Applied and Numerical Harmonic Analysis. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-55550-8_4

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