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On the existence of multiresolution analysis for framelets

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Abstract

We show that a compactly supported tight framelet comes from an MRA if the intersection of all dyadic dilations of the space of negative dilates, which is defined as the shift-invariant space generated by the negative scales of a framelet, is trivial. We also construct examples of (non-tight) framelets, which are arbitrarily close to tight frame framelets, such that the corresponding space of negative dilates is equal to the entire space L 2ℝ.

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Authors and Affiliations

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

    Marcin Bownik

  2. Mathematical Institute, University of Wrocław, 2/4, Pl. Grunwaldzki, 50-384, Wrocław, Poland

    Ziemowit Rzeszotnik

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

    Ziemowit Rzeszotnik

Authors
  1. Marcin Bownik
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  2. Ziemowit Rzeszotnik
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Corresponding author

Correspondence to Marcin Bownik.

Additional information

The first author was partially supported by the NSF grant DMS–0441817

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Bownik, M., Rzeszotnik, Z. On the existence of multiresolution analysis for framelets. Math. Ann. 332, 705–720 (2005). https://doi.org/10.1007/s00208-005-0645-3

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  • DOI: https://doi.org/10.1007/s00208-005-0645-3

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