Applications of nonuniform fast transform algorithms in numerical solutions of differential and integral equations

We review our recent efforts to apply the nonuniform fast Fourier transform (NUFFT) and related fast transform algorithms to numerical solutions of Ma xwell's equations in time and frequency domains. The NUFFT is a fast algori thm to perform the discrete Fourier transform of data sampled nonuniformly (NUDFT), Through oversampling and fast interpolation, the forward and inver se NUFFT's can be achieved with O(N log(2) N) arithmetic operations, asympt otically the same as the regular fast Fourier transform (FFT) algorithms. U sing the NUFFT scheme, me develop nonuniform fast cosine transform (NUFCT) and fast Hankel transform (NUFHT) algorithms. These algorithms provide an e fficient tool for numerical differentiation and integration, the key in the solutions to differential equations and volume integral equations. We pres ent sample applications of these nonuniform fast transform algorithms in th e numerical solution to Maxwell's equations.