A wavelet method for the first kind integral equations with kernel k(x-y)

We study the first kind integral equation integral(0)(+infinity) k(x - y)sigma(y)dy =g(x) by the wavelet method. The integral equation is discretized with respect to two different wavelet bases. We then have two different linear systems. On e of them is a Toeplitz system and the other one is a system with condition number kappa = O(1) after a diagonal scaling. By using the preconditioned conjugate gradient (PCG) method with the fast wavelet transform (FWT) and t he fast Fourier transform (FFT), we can solve the systems in O(n log n) ope rations where n is the size of the systems.