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Crystallographic Haar Wavelets

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Abstract

(Γ,a)-crystallographic multiwavelets are a finite set of functions \(\Psi= \{ \psi ^{i}\}_{i=1}^{L}\), which generate an orthonormal basis, a Riesz basis or a Parseval frame for L 2(ℝd), under the action of a crystallographic group Γ, and powers of an appropriate expanding affine map a, taking the place of the translations and dilations in classical wavelets respectively. Associated crystallographic multiresolution analysis of multiplicity n ((Γ,a)-MRA) are defined in a natural way. A complete characterization of scaling function vectors which generates Haar type (Γ,a)-MRA’s in terms of (Γ,a)-multireptiles is given. Examples of (Γ,a)-MRA crystallographic wavelets of Haar type in dimension 2 and 3 are provided.

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Authors and Affiliations

  1. Departamento de Matemática, Universidad Nacional de Mar del Plata, 7600, Mar del Plata, Argentina

    Alfredo L. González & María del Carmen Moure

Authors
  1. Alfredo L. González
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  2. María del Carmen Moure
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Correspondence to María del Carmen Moure.

Additional information

Communicated by Karlheinz Gröchenig.

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González, A.L., Moure, M.d.C. Crystallographic Haar Wavelets. J Fourier Anal Appl 17, 1119–1137 (2011). https://doi.org/10.1007/s00041-011-9175-4

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  • DOI: https://doi.org/10.1007/s00041-011-9175-4

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