  The goals and contributions of this paper are twofold. It provides a new
computational tool for data driven Koopman spectral analysis by taking up the
formidable challenge to develop a numerically robust algorithm by following the
natural formulation via the Krylov decomposition with the Frobenius companion
matrix, and by using its eigenvectors explicitly -- these are defined as the
inverse of the notoriously ill-conditioned Vandermonde matrix. The key step to
curb ill-conditioning is the discrete Fourier transform of the snapshots; in
the new representation, the Vandermonde matrix is transformed into a
generalized Cauchy matrix, which then allows accurate computation by specially
tailored algorithms of numerical linear algebra. The second goal is to shed
light on the connection between the formulas for optimal reconstruction weights
when reconstructing snapshots using subsets of the computed Koopman modes. It
is shown how using a certain weaker form of generalized inverses leads to
explicit reconstruction formulas that match the abstract results from Koopman
spectral theory, in particular the Generalized Laplace Analysis.
