Vector cascade algorithms and refinable function vectors in Sobolev spaces
Journal of Approximation Theory
Convergence rates of vector cascade algorithms in Lp
Journal of Approximation Theory
Construction for a class of interpolation multiscaling functions with dilation factor a≥3
Computers & Mathematics with Applications
On stable refinable function vectors with arbitrary support
Journal of Approximation Theory
Interpolatory quad/triangle subdivision schemes for surface design
Computer Aided Geometric Design
Convergence rates of vector cascade algorithms in Lp
Journal of Approximation Theory
Smoothness of multivariate refinable functions with infinitely supported masks
Journal of Approximation Theory
A blind watermarking scheme using new nontensor product wavelet filter banks
IEEE Transactions on Image Processing
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We consider the smoothness of solutions of a system of refinement equations written in the form $$\phi = \sum\nolimits_{\ga\in\ZZ} a(\ga)\phi({2\,\cdot}-\ga),$$ where the vector of functions $\phi=(\phi_1,\ldots,\phi_r)^T$ is in $(L_p(\mbox{\smallBbb R}))^r$ and a is a finitely supported sequence of r X r matrices called the refinement mask. We use the generalized Lipschitz space $\hbox{Lip}^*(\nu, L_p(\mbox{\smallBbb R}))$, $\nu 0$, to measure smoothness of a given function. Our method is to relate the optimal smoothness, $\nu_p(\phi)$, to the $p$-norm joint spectral radius of the block matrices $A_\gep$, $\gep = 0, 1$, given by $A_\gep = ( a(\gep+2\ga-\gb) )_{\ga,\gb}$, when restricted to a certain finite dimensional common invariant subspace $V$. Denoting the $p$-norm joint spectral radius by $\rho_p(A_0|_V,A_1|_V)$, we show that $\nu_p(\phi) \ge 1/p - \log_2 \rho_p(A_0|_V, A_1|_V) $ with equality when the shifts of $\phi_1,\ldots,\phi_r$ are stable and the invariant subspace is generated by certain vectors induced by difference operators of sufficiently high order. This allows an effective use of matrix theory. Also the computational implementation of our method is simple.When p = 2, the optimal smoothness is also given in terms of the spectral radius of the transition matrix associated with the refinement mask. To illustrate the theory, we give a detailed analysis of two examples where the optimal smoothness can be given explicitly. We also apply our methods to the smoothness analysis of multiple wavelets. These examples clearly demonstrate the applicability and practical power of our approach.