Coordinate mapping can be used to dramatically enhance the efficiency of phase space sampling. A new mapped pseudospectral approach that readily exploits a fast transform algorithm is presented for both cylindrical and spherical coordinates. The combined use of a power law mapping and of a sine Fourier expansion requires only two fast transforms per Laplacian operation. The mapping algorithm can be symmetrized for the purpose of efficient diagonalization techniques. Special emphasis is given to the treatment of radial singularities as the new procedure is illustrated with the eigenvalue calculation of the harmonic oscillator, the Coulomb and the H-2(+) benchmark problems,