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Existence and exactness of exponential Riesz sequences and frames for fractal measures

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Abstract

We study the construction of exponential frames and Riesz sequences for a class of fractal measures on ℝd generated by infinite convolution of discrete measures using the idea of frame towers and Riesz-sequence towers. The exactness and overcompleteness of the constructed exponential frame or Riesz sequence is completely classified in terms of the cardinality at each level of the tower. Using a version of the solution of the Kadison-Singer problem, known as the Rϵ-conjecture, we show that all these measures contain exponential Riesz sequences of infinite cardinality. Furthermore, when the measure is the middle-third Cantor measure, or more generally for self-similar measures with no-overlap condition, there are always exponential Riesz sequences of maximal possible Beurling dimension.

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Acknowledgements

The authors thank the referee for his/her valuable suggestion. They would also like to thank the referee and Professor Peter Casazza for pointing out how Lemma 5.2 can also be deduced from a weaker version of the Bourgain-Tzafriri theorem. This work was partially supported by a grant from the Simons Foundation (#228539 to Dorin Dutkay).

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Authors and Affiliations

  1. Department of Mathematics, University of Central Florida, 4000 Central Florida Blvd., P.O. Box 161364, Orlando, FL, 32816-1364, USA

    Dorin Ervin Dutkay

  2. Department of Mathematics, UC Davis, One Shields Ave, Davis, CA, 95616, USA

    Shahram Emami

  3. Department of Mathematics, San Francisco State University, 1600 Holloway Avenue, San Francisco, CA, 94132, USA

    Chun-Kit Lai

Authors
  1. Dorin Ervin Dutkay
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  2. Shahram Emami
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  3. Chun-Kit Lai
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Correspondence to Chun-Kit Lai.

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Dutkay, D.E., Emami, S. & Lai, CK. Existence and exactness of exponential Riesz sequences and frames for fractal measures. JAMA 143, 289–311 (2021). https://doi.org/10.1007/s11854-021-0156-5

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  • DOI: https://doi.org/10.1007/s11854-021-0156-5

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