Some Properties Of Refinement Equations With Infinitely Supported Masks

In this thesis, we investigate the solutions of refinement equations of the form where the vector of functionsφ= (φ>1,…,φr)T is in (L2(Rs))r, a=:(a(a))a∈zs is an infinitely supported sequence of r x r matrices called refinement mask and M is an s x s integer matrix such that limn→∞M-n= 0. To investigate the solutions of refinement equations, we introduce the linear operator Qa on (L2(Rs))r given by The iterative scheme (Qanf)n=1,2…, is called a vector subdivision scheme or vector cascade algorithm.In chapter 1, we will firstly present the background and current status of refine-ment equations associated with an infinitely supported mask and a general dilation matrix. Then, the main results of this thesis are stated.In chapter 2, we investigate the vector cascade algorithms associated with an exponentially decaying mask and a general dilation matrix. A necessary and suffi-cient condition is provided to guarantee the sequence (Qanf)n=1,2…to converge in the L2-norm. As an application, we also characterize the biorthogonal multiple refinable functions. Let a be finitely supported andφ= (φ1,...,φr)T be a L2-solution of the refinement equation with linearly independent shifts. If the refinement mask satisfies some solvable conditions, then there exists another refinable vector (p= (φ1,...,φr) of compactly supported functions in L2(RS) such that (φis dual toφ.In chapter 3, we investigate the convergence rates of (Qa,Mnφ0)n=1,2,... in LP(RS) with r= 1 and mask a being infinitely supported. It is proved that under some conditions on the initial functionφ0 and (φ, Qa,Mnφ0 converges in LP(RS) with an exponential rate. In chapter 4, we characterize the optimal smoothness of multivariate refin-able functions with infinitely supported masks and an isotropic dilation matrix. In addition, for a given infinitely supported mask a, we provide an estimate of the smoothness of the refinable function by calculating the spectral radius of the restricted transition operator associated with the truncated mask aN.