1.  This is a collection of Matlab 5 routines developed for
    Sobolev and Holder smoothness computations of refinable functions. 

     Download the archive regsoft.tar or regsoft.zip from 
         http://www.math.umsl.edu/~jiang 
     Create a directory and archive there: 
          tar -xvf regsoft.tar
      
        or unzip regosft.zip 

     Please send your questions, comments, and corrections to Qingtang Jiang
     
      jiangq@umsl.edu

2.  List of functions included:

    MAIN ROUTINES:

    sobsmthest.m     Sobolev smoothness estimate for refinable vectors
                     with an arbitrary (isotropic) dilation M 
                     and the spatial dimension d<=2.
                      
    holdsmthest.m    Holder smoothness  estimate for refinable vectors 
                     with dilation M=2, M=2*I_2, or M=[1 2;-2 -1] 

    MAJOR ROUTINES:  

    sumruleorder.m   Computes the sum rule order k of the mask with 
                     an arbitrary dilation M and the spatial dimension d<=2.
                     (For the vector case, k is not the largest 
                           possible sum rule order.)
    
    stable.m         Computes the stability indicator for the stability 
                     of the refinable vector with an arbitrary dilation M 
                     and the spatial dimension d<=2.
      
    tranop.m         Matrix representation of the transition operator 
                     with an arbitrary dilation M and the dimension d<=2.
    
    smalltranop.m    Matrix representations of the transition operators for 
                     Holder smoothness
      
    restrsop.m       Matrix representations of the transition operators 
                     restricted to the common invariant subspace.   

    jointspec.m      Computes upper and lower estimates for the joint 
                     spectral radius of  matrices from restrsop.m


3. References:

       This software was developed based on the theory about the 
       smoothness of refinable functions in the following two papers 
       [JZ] (the scalar case) and [JJ] (the vector case), and 
       was explained and used in [JO]. 

       [JZ] R. Q. Jia, S. R. Zhang,  Spectral properties of the transition 
                 operator associated to a multivariate refinement equation,
                    Linear Algebra Appl. 292 (1999), 155--178. 

       [JJ] R. Q. Jia, Q. T. Jiang,  Characterization of smoothness of 
              multivariate refinable vectors, Manuscript, July 2000.

       [JO] Q. T. Jiang, P. Oswald, On the analysis of sqrt(3)-subdivision, 
                  Manuscript, March 2001.