Spectral Analysis of the Transition Operator and Its Applications to Smoothness Analysis of Wavelets

Abstract

This paper investigates spectral properties of the transition operator associated to a multivariate vector refinement equation and their applications to the study of smoothness of the corresponding refinable vector of functions. Let $\Phi=(\phi_1,\ldots,\phi_r)^T$ be an $r \times 1$ vector of compactly supported functions in $L_2(\Rs)$ satisfying $\,\Phi = \sum_{\ga\in\Zs} a(\ga) \Phi({M\kern .1em \cdot}-\ga)$, where M is an expansive integer matrix. The smoothness of $\Phi$ is measured by the Sobolev critical exponent $\gl (\Phi) := \sup \bigl\{ \gl: \int_{\Rs} |\hat \phi_j(\xi)|^2(1+|\xi|^\gl)^2\,d\xi<\infty, \, 1 \le j\le r \bigr\}$. Suppose %the mask $a$ is finitely supported, i.e., %the set $M$ is similar to %a diagonal matrix $\hbox{diag}(\gs_1,\ldots,\gs_s)$ with $|\gs_1|=\cdots=|\gs_s|$ and $\supp a := \{\ga\in\Zs: a(\ga) \ne 0\}$ is finite. For $\mu=(\mu_1,\ldots,\mu_s)\in\NN_0^s$, define $\gs^{-\mu} := \gs_1^{-\mu_1} \cdots \gs_s^{-\mu_s}$. Let $A := \sum_{\ga\in\Zs} a(\ga)/|\det M|$ and $b(\ga) := \sum_{\gb\in\Zs} \overline{a(\gb)} \otimes a(\ga+\gb)/|\det M|$, $\ga\in\Zs$, where $\,\otimes\,$ denotes the (right) Kronecker product. Suppose that the highest total degree of polynomials reproduced by $\Phi$ is $k-1$ and $\spec(A)$ (the spectrum of $A$) is $\{\eta_1,\eta_2, \ldots,\eta_r\}$ with $\eta_1=1$ and $\eta_j \ne 1, 2\le j\le r$. Set $$ E_k := \{ \eta_j \overline{\gs^{-\mu}}, \overline{\eta_j} \gs^{-\mu}: |\mu|