Singular integral transforms and fast numerical algorithms (Q1275441)

Fast algorithms for the evaluation of the singular integral transforms \[ T_mh(\sigma)= -{1\over\pi} \int \int_{B(0,1)} {h(\zeta)\over(\zeta- \sigma)^m} d\xi d\eta, \] \(\zeta= \xi+ i\eta\), where \(m= 1,2,\dots\), a complex-valued function \(h\) defiend on \(B(0,1)= \{z:| z|< 1\}\), are presented. They are based on some recursive relations in Fourier space and fast Fourier transform and have theoretical computational complexity of the order \(O(\log N)\) per point, where \(N^2\) is the total number of grid points.