Bessel Wavelets and the Galerkin Analysis of the Bessel Operator

We construct a complete orthonormal system j, k4j , k for L2Ž , dx.such that each Bessel wa elet j, k has a compactly supported Hankel transform. We apply this system in the Galerkin solution of the differential equation Lu f on , where L is the Bessel operator. The associated linear system can be preconditioned using a simple diagonal preconditioning matrix to give a system which is sparse and has condition number which is bounded independent of the Galerkin subspace. Thus we obtain the same advantages for the singular operator L that have been obtained for uniformly elliptic operators using standard wavelets. The main step is a characterization of the Bessel Sobolev norm of a function in terms of its Bessel wavelet coefficients. We also present numerical work which shows that the condition numbers are small enough for this method to be practical.