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On the translates of general dyadic systems on \({{\mathbb {R}}}\)

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Abstract

Many techniques in harmonic analysis use the fact that a continuous object can be written as a sum (or an intersection) of dyadic counterparts, as long as those counterparts belong to an adjacent dyadic system. Here we generalize the notion of adjacent dyadic system and explore when it occurs, leading to some new and perhaps surprising classifications. In particular, we show that every dyadic grid is determined by two parameters, the shift and the location; moreover two dyadic grids form an adjacent dyadic system if and only if their shifts and locations satisfy readily verifiable conditions.

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Authors and Affiliations

  1. Department of Mathematics, Purdue University, 150 N. University St., West Lafayette, IN, 47907, USA

    Theresa C. Anderson

  2. Department of Mathematics, University of Wisconsin, Madison, 480 Lincoln Dr., Madison, WI, 53705, USA

    Bingyang Hu

  3. School of Operations Research and Information Engineering, Cornell University, 206 Rhodes Hall, Ithaca, NY, 14850, USA

    Liwei Jiang

  4. Department of Mathematics, Pennsylvania State University, University Park, McAllister Building, Pollock Road, University Park, PA, 16802, USA

    Connor Olson

  5. Department of Statistics, University of Washington, Box 354322, Seattle, WA, 98195-4322, USA

    Zeyu Wei

Authors
  1. Theresa C. Anderson
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  2. Bingyang Hu
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  3. Liwei Jiang
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  4. Connor Olson
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  5. Zeyu Wei
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Correspondence to Theresa C. Anderson.

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Communicated by Loukas Grafakos.

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This project was partially supported by NSF DMS 1502464, NSF DMS 1600458, an NSF RTG in Analysis and Applications, NSF grant 1500182, and an NSF Graduate Research Fellowship (for T. C. Anderson)

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Anderson, T.C., Hu, B., Jiang, L. et al. On the translates of general dyadic systems on \({{\mathbb {R}}}\). Math. Ann. 377, 911–933 (2020). https://doi.org/10.1007/s00208-019-01951-z

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  • DOI: https://doi.org/10.1007/s00208-019-01951-z

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