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Wavelets Centered on a Knot Sequence: Theory, Construction, and Applications

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An Erratum to this article was published on 21 February 2015

Abstract

We develop a general notion of orthogonal wavelets ‘centered’ on an irregular knot sequence. We present two families of orthogonal wavelets that are continuous and piecewise polynomial. We develop efficient algorithms to implement these schemes and apply them to a data set extracted from an ocelot image. As another application, we construct continuous, piecewise quadratic, orthogonal wavelet bases on the quasi-crystal lattice consisting of the \(\tau \)-integers where \(\tau \) is the golden ratio. The resulting spaces then generate a multiresolution analysis of \(L^2(\mathbf {R})\) with scaling factor \(\tau \).

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Acknowledgments

The research of JSG was partially supported by NSF Grant DMS-0500641 and the research of DPH was partially supported by NSF Grant DMS-1109266. We thank the referees for their careful reading and thoughtful suggestions.

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Authors and Affiliations

  1. Birmingham, AL, USA

    Bruce W. Anderson

  2. Nashville, TN, USA

    Derek O. Bruff & Douglas P. Hardin

  3. Atlanta, GA, USA

    Jeffrey S. Geronimo

Authors
  1. Bruce W. Anderson
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  2. Derek O. Bruff
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  3. Jeffrey S. Geronimo
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  4. Douglas P. Hardin
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Corresponding author

Correspondence to Jeffrey S. Geronimo.

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Communicated by Gitta Kutyniok.

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Anderson, B.W., Bruff, D.O., Geronimo, J.S. et al. Wavelets Centered on a Knot Sequence: Theory, Construction, and Applications. J Fourier Anal Appl 21, 509–553 (2015). https://doi.org/10.1007/s00041-014-9375-9

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  • DOI: https://doi.org/10.1007/s00041-014-9375-9

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