Abstract
As a starting point we assume to have a continuous frame in a Hilbert space with respect to a measure space. This frame inherits a unitary structure from a unitary representation of a locally compact abelian group in the Hilbert space. In this setting we state a continuous sampling result for the range space of the associated analysis frame operator. The data sampling are functions also defined by using the underlying unitary structure. The result is illustrated by using continuous frames in Paley–Wiener and shift-invariant spaces generated by translates of fixed functions. A sampling strategy working only for discrete abelian groups is also discussed.
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Acknowledgements
This work has been supported by the grant MTM2017-84098-P from the Spanish Ministerio de Economía, Industria y Competitividad.
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Communicated by Christopher Heil.
The authors are very pleased to dedicate this paper to Professor F. H. Szafraniec on the occasion of his 80th birthday. Professor Szafraniec’s research and mentorship have, over the years, inspired and influenced many mathematicians throughout the world; we are fortunate to be two of these mathematicians.
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García, A.G., Muñoz-Bouzo, M.J. A note on continuous stable sampling. Adv. Oper. Theory 5, 994–1013 (2020). https://doi.org/10.1007/s43036-020-00061-x
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DOI: https://doi.org/10.1007/s43036-020-00061-x