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Construction of multivariate compactly supported orthonormal wavelets

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Abstract

We propose a constructive method to find compactly supported orthonormal wavelets for any given compactly supported scaling function φ in the multivariate setting. For simplicity, we start with a standard dilation matrix 2I2×2 in the bivariate setting and show how to construct compactly supported functions ψ1,. . .,ψn with n>3 such that {2kψj(2kx−ℓ,2ky−m), k,ℓ,m∈Z, j=1,. . .,n} is an orthonormal basis for L2(ℝ2). Here, n is dependent on the size of the support of φ. With parallel processes in modern computer, it is possible to use these orthonormal wavelets for applications. Furthermore, the constructive method can be extended to construct compactly supported multi-wavelets for any given compactly supported orthonormal multi-scaling vector. Finally, we mention that the constructions can be generalized to the multivariate setting.

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Authors and Affiliations

  1. Department of Mathematics, The University of Georgia, Athens, GA, 30602, USA

    Ming-Jun Lai

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  1. Ming-Jun Lai
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Correspondence to Ming-Jun Lai.

Additional information

Communicated by Y. Xu

Dedicated to Professor Charles A. Micchelli on the occasion of his 60th birthday

Mathematics subject classifications (2000)

42C15, 42C30.

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Lai, MJ. Construction of multivariate compactly supported orthonormal wavelets. Adv Comput Math 25, 41–56 (2006). https://doi.org/10.1007/s10444-004-7643-y

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  • DOI: https://doi.org/10.1007/s10444-004-7643-y

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