The Leveraged Waveletsand Galerkin - Wavelets

We present a scheme that leverage orthonormal or biorthogonal wavelets to a new system of biorthogonal wavelets. The leveraged biorthogonal wavelets will have some nice properties. If we start with orthonormal wavelets, the leveraged scaling functions and wavelets are compactly supported and are diierentiable. The derivatives of the leveraged wavelets are orthogonal to their translations; the derivatives of the leveraged scaling functions are nearly orthogonal to their translations; and the derivatives of the leveraged scaling functions and wavelets are orthogonal to each other. This feature may be valuable for the numerical solution of diier-ential equations. If we start with B-splines and cooperating with the lifting scheme of Sweldens, our leverage scheme can reproduce all of those biorthogonal wavelets by Cohen, Daubechies and Feauveau. There is a simple algorithm to calculate new lter coeecients from the old lter coeecients. In the end of this article we test the newly constructed biorthogonal wavelets in the framework of Galerkin-wavelet methods. 1. Motivation. The motivation of this work is the following formula (1:1) (x) = Z x ?1 (t) ? (t ? 1) dt that was suggested by Xu and Shann in their paper on Galerkin-wavelets methods 17]. Here (x) is an orthonormal scaling function. In that paper, they were studying the Galerkin basis functions derived from orthonormal wavelets that will produce well-conditioned stiiness matrices and will satisfy the Dirichlet boundary conditions. The formula (1:1) was suggested to ensure the compact supportedness of the Galerkin-wavelets basis functions. We follow this hint and discovered that it is actually the starting point of a general scheme to derive from a wavelet to another wavelet with some better properties. We call it the leverage scheme of biorthogonal wavelets. Following the spirit of the Galerkin-wavelets methods 17], the ultimate goal for the leverage scheme is the application for numerical solution of diierential equations. In Section 2, we will brieey review the fundamentals and deene the notations. In Section 3, we will deene the leverage scheme and present the main properties for the orthonormal case. As a preview, let (x) and (x) be a pair of orthonormal scaling function and wavelet. Let (x) be the leverage of (x) as deened in (1:1). Then (x) is again a scaling function and it generates a multiresolution analysis (MRA) of L 2 (R). If (x) has the approximation degree p, then (x) will have the approximation degree p + 1. The derivatives …