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Global Symmetric Approximation of Frames

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Abstract

We solve the problem of best approximation by Parseval frames to an arbitrary frame in a subspace of an infinite dimensional Hilbert space. We explicitly describe all the solutions and we give a criterion for uniqueness. Our proof relies on the geometric structure of the set of all Parseval frames quadratically close to a given frame. In the process we show that its connected components can be parametrized by using the notion of index of a pair of projections, and we prove existence and uniqueness results of best approximation by Parseval frames restricted to these connected components.

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Acknowledgements

I am grateful to Jorge Antezana for several helpful discussions. Also I would like to thank the referees for their constructive comments. This research was supported by Grants CONICET (PIP 2016 112201), FCE-UNLP (11X681) and ANPCyT (2015 1505).

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Authors and Affiliations

  1. Departamento de Matemática & Centro de Matemática La Plata, FCE-UNLP, Calles 50 y 115, (1900), La Plata, Argentina

    Eduardo Chiumiento

  2. Instituto Argentino de Matemática, ‘Alberto P. Calderón’, CONICET, Saavedra 15 3er. piso, (1083), Buenos Aires, Argentina

    Eduardo Chiumiento

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  1. Eduardo Chiumiento
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Correspondence to Eduardo Chiumiento.

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Communicated by Hans G. Feichtinger.

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Chiumiento, E. Global Symmetric Approximation of Frames. J Fourier Anal Appl 25, 1395–1423 (2019). https://doi.org/10.1007/s00041-018-9632-4

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  • DOI: https://doi.org/10.1007/s00041-018-9632-4

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