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The Construction of Wavelet Sets

  • John J. Benedetto
  • Robert L. Benedetto
Chapter
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)

Abstract

Sets Ω in d-dimensional Euclidean space are constructed with the property that the inverse Fourier transform of the characteristic function 1 Ω of the set Ω is a single dyadic orthonormal wavelet. The iterative construction is characterized by its generality, its computational implementation, and its simplicity. The construction is transported to the case of locally compact abelian groups G with compact open subgroups H. The best known example of such a group is \(G = {\mathbb{Q}}_{p}\), the field of p-adic rational numbers (as a group under addition), which has the compact open subgroup \(H = {\mathbb{Z}}_{p}\), the ring of p-adic integers. Fascinating intricacies arise. Classical wavelet theories, which require a non-trivial discrete subgroup for translations, do not apply to G, which may not have such a subgroup. However, our wavelet theory is formulated on G with new group theoretic operators, which can be thought of as analogues of Euclidean translations. As such, our theory for G is structurally cohesive and of significant generality. For perspective, the Haar and Shannon wavelets are naturally antipodal in the Euclidean setting, whereas their analogues for G are equivalent.

Keywords

Multiresolution Analysis Tight Frame Compact Abelian Group Wavelet Theory Coset Representative 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Notes

Acknowledgements

The first named author gratefully acknowledges support from ONR Grant N0001409103 and MURI-ARO Grant W911NF-09-1-0383. He is also especially appreciative of wonderful mathematical interaction through the years, on the Euclidean aspect of this topic, with Professors Larry Baggett, David Larson, and Kathy Merrill, and for more recent invaluable technical contributions by Dr. Christopher Shaw. The second named author gratefully acknowledges support from NSF DMS Grant 0901494.

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Authors and Affiliations

  1. 1.Department of MathematicsUniversity of MarylandCollege ParkUSA

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