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Continuous Frames and the Kadison-Singer Problem

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Coherent States and Their Applications

Part of the book series: Springer Proceedings in Physics ((SPPHY,volume 205))

Abstract

In this paper we survey a recent progress on continuous frames inspired by the solution of the Kadison-Singer problem [26] by Marcus, Spielman, and Srivastava [29]. We present an extension of Lyapunov’s theorem for discrete frames due to Akemann and Weaver [2] and a similar extension for continuous frames by the author [10]. We also outline a solution of the discretization problem, which was originally posed by Ali, Antoine, and Gazeau [4], and recently solved by Freeman and Speegle [22].

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Acknowledgements

The author was partially supported by the NSF grant DMS-1665056 and by a grant from the Simons Foundation #426295. The author is grateful for useful comments of the referees and for an inspiring conversation with Hans Feichtinger.

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Authors and Affiliations

  1. Department of Mathematics, University of Oregon, Eugene, OR, 97403-1222, USA

    Marcin Bownik

  2. Institute of Mathematics, Polish Academy of Sciences, ul. Wita Stwosza 57, 80-952, Gdańsk, Poland

    Marcin Bownik

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  1. Marcin Bownik
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Correspondence to Marcin Bownik .

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Editors and Affiliations

  1. Institut de recherche en mathématique et physique, Université catholique de Louvain, Louvain-la-Neuve, Belgium

    Jean-Pierre Antoine

  2. Dipartimento di Energia, Ingegneria dell’Informazione e Modelli Matematici, Palermo, Italy

    Fabio Bagarello

  3. Laboratoire APC, Paris Diderot University, Paris, France

    Jean-Pierre Gazeau

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Bownik, M. (2018). Continuous Frames and the Kadison-Singer Problem. In: Antoine, JP., Bagarello, F., Gazeau, JP. (eds) Coherent States and Their Applications. Springer Proceedings in Physics, vol 205. Springer, Cham. https://doi.org/10.1007/978-3-319-76732-1_4

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