Computing the Smoothness Exponent of a Symmetric Multivariate Refinable Function

Abstract
Smoothness and symmetry are two important properties of a refinable function. It is known that the Sobolev smoothness exponent of a refinable function can be estimated by computing the spectral radius of a certain finite matrix which is generated from a mask. However, the increase of dimension and the support of a mask tremendously increase the size of the matrix and therefore make the computation very expensive. In this paper, we shall present a simple and efficient algorithm for the numerical computation of the smoothness exponent of a symmetric refinable function with a general dilation matrix. By taking into account the symmetry of a refinable function, our algorithm greatly reduces the size of the matrix and enables us to numerically compute the Sobolev smoothness exponents of a large class of symmetric refinable functions. Step-by-step numerically stable algorithms are given. To illustrate our results by performing some numerical experiments, we construct a family of dyadic interpolatory masks in an...

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