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Wavelets and Multiscale Analysis pp 17–56Cite as

The Construction of Wavelet Sets

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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.

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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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  1. Department of Mathematics, University of Maryland, College Park, MD, 20742, USA

    John J. Benedetto

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  1. , Department of Mathematical Sciences, DePaul University, North Kenmore Avenue 2320, Chicago, 60614-3250, Illinois, USA

    Jonathan Cohen

  2. , Department of Mathematical Sciences, DePaul University, North Kenmore Avenue 2320, Chicago, 60614-3250, Illinois, USA

    Ahmed I. Zayed

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Benedetto, J.J., Benedetto, R.L. (2011). The Construction of Wavelet Sets. In: Cohen, J., Zayed, A. (eds) Wavelets and Multiscale Analysis. Applied and Numerical Harmonic Analysis. Birkhäuser Boston. https://doi.org/10.1007/978-0-8176-8095-4_2

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