Solution of a system of linear equations with an incomplete circulant matrix by a discrete Fourier transformation

An algorithm is described for finding the solution with least Euclidean norm of a subdefinite system of linear equations with an incomplete circulant matrix of dimensions $m\times n$, in which the calculation of the pseudo-inverse matrix is replaced by some discrete Fourier transformations of vectors of dimension $n$ and the inverse of a positive-definite matrix of order $n-m$.