In this paper, we would like first to promote an interesting idea developed by T. Wu and Y. Xu which leads to identify a local minimizer of a nonconvex optimization problem with a global minimizer of a convex optimization one. Secondly, to give an extension of the sparse regularization model for inverting incomplete Fourier transforms introduced by T. Wu and Y. Xu. Thirdly, following the same lines as in "nverting Incomplete Fourier Transforms by a Sparse Regularization Model and Applications in Seismic Waveeld Modeling. arXiv 10 June 2022", to develop efficient, convergence guaranteed iteration algorithm for solving the resulting non-convex, non-smooth optimization problem but here using applied nonlinear analysis tools. These lead both to a simplification of the proofs and to make a connection with classical works in this filed through a startling comment.