Research Group of Prof. Dr. M. Griebel
Institute for Numerical Simulation
[1] F. Koster and M. Griebel. Orthogonal wavelets on the interval. Technical Report No. 576, SFB 256, Universität Bonn, Germany, 1998.
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In this paper we generalize the constructions [?], [?] and [?] of wavelets on the interval. These schemes give boundary modifications of compactly supported orthogonal wavelets ψinL2( R) with supp ψ=[-N+1,N], where N denotes the number of vanishing moments of ψ. Our new scheme overcomes this restrictive condition. Furthermore, the constructions [?], [?], [?] involve Gram matrices for explicit orthogonalization steps. These Gram matrices tend to be very ill conditioned for increasing N. It is shown that for the present scheme the condition numbers of the resulting matrices are smaller by orders of magnitude. Therefore our scheme is numerically more stable. We also point out how wavelets can be obtained satisfying homogeneous Dirichlet or Neumann conditions. In addition, we deal with the requirement of the discrete wavelet transform on the interval to find, e.g. from nodal values of u inL2([0,1]), an approximation u by a linear combination of dilated scaling functions. We present and compare two methods. One method has already been used in [?] and [?]. Our experiments show that this scheme is not suited for data compression on the interval. The other method however is designed for data compression applications and leads to cheap and very well conditioned approximation mapping