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An algorithm is presented to compute Zolotarev rational functions, that is, rational functions of a given degree that are as small as possible on one set E relative to their size on another set F (the third Zolotarev problem). Along the way we also approximate the sign function relative to E and F (the fourth Zolotarev problem). 

A direct solver is introduced for solving overdetermined linear systems involving nonuniform discrete Fourier transform matrices. Such matrices can be transformed into a Cauchy-like form that has hierarchical low rank structure. The rank structure of this matrix is explained, and it is shown that the ranks of the relevant submatrices grow only logarithmically with the number of columns of the matrix. A fast rank-structured hierarchical approximation method based on this analysis is developed, along with a hierarchical least-squares solver for these and related systems. This result is a direct method for inverting nonuniform discrete transforms with a complexity that is usually nearly linear with respect to the degrees of freedom in the problem. This solver is benchmarked against various iterative and direct solvers in the setting of inverting the one-dimensional type-II (or forward) transform, for a range of condition numbers and problem sizes (up to 4 × 10 by 2 × 10 ). These experiments demonstrate that this method is especially useful for large problems with multiple right-hand sides.