Characterization of Smoothness of Multivariate Refinable Functions in Sobolev Spaces

Jia, Rong-Qing

doi:10.1090/S0002-9947-99-02185-6

Characterization of Smoothness of Multivariate Refinable Functions in Sobolev Spaces
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Trans. Amer. Math. Soc. 351 (1999), 4089-4112 Request permission

Abstract:

Wavelets are generated from refinable functions by using mul tiresolution analysis. In this paper we investigate the smoothness properties of multivariate refinable functions in Sobolev spaces. We characterize the optimal smoothness of a multivariate refinable function in terms of the spectral radius of the corresponding transition operator restricted to a suitable finite dimensional invariant subspace. Several examples are provided to illustrate the general theory.

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