Construction of multivariate compactly supported orthonormal wavelets

Author： Lai Ming-Jun  

Publisher： Springer Publishing Company

ISSN： 1019-7168

Source： Advances in Computational Mathematics, Vol.25, Iss.1-3, 2006-07, pp. : 41-56

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Abstract

We propose a constructive method to find compactly supported orthonormal wavelets for any given compactly supported scaling function &phis; 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 &phis;. 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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