Regularity of anisotropic refinable functions

This paper presents a detailed regularity analysis of multivariate refinable functions with general dilation matrices, with emphasis on the anisotropic case. In the univariate setting, the smoothness of refinable functions is well understood by means of the matrix approach. In the multivariate setting, this approach has been extended only to the special case of isotropic refinement with the dilation matrix all of whose eigenvalues are equal in the absolute value. The general anisotropic case has resisted to be fully understood: the matrix approach can determine whether a refinable function belongs to C(Rs) or Lp(Rs), 1≤p<∞, but its Hölder regularity remained mysteriously unattainable.We show how to compute the Hölder regularity in C(Rs) or Lp(Rs), 1≤p<∞. In the anisotropic case, our expression for the exact Hölder exponent of a refinable function reflects the impact of the variable moduli of the eigenvalues of the corresponding dilation matrix. In the isotropic case, our results reduce to the well-known facts from the literature. We also analyze the higher regularity of anisotropic refinable functions. We illustrate our results with several examples.