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## Mathematisch-Naturwissenschaftliche FakultätMathematisches Institut - Lehrstuhl Kunoth

This directory contains the mask matrices of the biorthogonal generators and wavelets on the interval as constructed in

W. Dahmen, A. Kunoth, K. Urban, Biorthogonal Spline-Wavelets on the Interval - Stability and Moment
Conditions
, Appl. Comp. Harm. Anal. 6, 1999, 132-196

All the following explanations refer to this article. (Mathematical symbols are written using latex-commands.)

For each pair d, \tilde d, the decomposition and reconstruction matrices M_j and G_j are given for minimal level j=j_0 which is determined by (3.2.31). These matrices can be found in the Matlab files

• Md{\tilde d}_j.m and Gd{\tilde d}_j.m

where d, \tilde d and j are replaced by the corresponding numbers and j stands for the minimal level j_0. For instance, the matrices M24_5.m and its counterpart G24_5.m are the refinement matrices for the case d=2, \tilde d=4 for which by (3.2.31) j_0=5. The matrices defined in the Matlab files are denoted by Md{\tilde d}_j and Gd{\tilde d}_j. One can check that Gd{\tilde d}_j is the inverse of Md{\tilde d}_j. Each matrix Md{\tilde d}_j is divided as written in Proposition 2.5 into two matrices M_{j,0} and M_{j,1} which have the structure (3.5.1). Correspondingly, Gd{\tilde d}_j consists of two block matrices G_{j,0} and G_{j,1}, see Theorem 4.7 (i). Recall that these transformation matrices perform decomposition and reconstruction between two levels. For more levels, one assembles M_j and G_j for each level, noting that the corner blocks of the matrices due to the boundary adaptation always remain the same and only the interior blocks grow. The matrices in the following Matlab files are given in sparse matlab format.

For a significant reduction of the condition number of the transformation matrices for higher values of d, \tilde d, see

W. Dahmen, A. Kunoth, K. Urban, Wavelets in Numerical Analysis and their Quantitative Properties,
in: Surface Fitting and Multiresolution Methods, A. Le Mehaute, C. Rabut, and L. L. Schumaker (eds.), Vanderbilt Univ. Press, Nashville, TN, 1997, 93-130.

Since there are several cases that can be distinguished, we do not display the matrices for these cases here. The matrices can be obtained by contacting Titus Barsch, Karsten Urban or me.