Smoothness of multivariate refinable functions with infinitely supported masks (Q979044)

Let \(a\) be an infinitely supported mask with polynomial decay, that is, \(a\in B_\mu\) for a positive integer \(\mu\), where \(\|a\|_{B_\mu}:=\sum_{\alpha\in \mathbb{Z}^s} |a(\alpha)|(1+|\alpha|)^\mu<\infty\). This paper studies the Sobolev smoothness exponent of the standard \(M\)-refinable function \(\phi\), which is defined by \(\hat \phi(\xi):=\prod_{j=1}^\infty \hat a((M^T)^{-j}\xi)\). Under the assumption that the dilation matrix \(M\) is isotropic and \(\phi\in \mathcal{L}_2(\mathbb{R}^s)\) (that is, \(\sum_{\alpha\in \mathbb{Z}^s} |\phi(\cdot-\alpha)|\in L_2(\mathbb{T}^s)\)), if \(\{\phi(\cdot-\alpha)\}_{\alpha\in \mathbb{Z}^s}\) forms a Riesz sequence in \(L_2(\mathbb{R}^s)\) and the mask \(a\in B_\mu\) has polynomial decay, a characterization of the Sobolev smoothness exponent \(\nu(\phi)\) (which is defined at the end of Section 1) is given in Theorem 3.2. This generalizes several results in the literature from finitely supported masks to infinitely supported masks with polynomial decay.