Fourier transformation as inverse problem — An improved algorithm
M. Dobróka, Hajnalka Szegedi, Péter Vass, E. Turai · Acta Geodaetica et Geophysica Hungarica · 2012
This paper presents a new algorithm for the inversion-based 1D Fourier transformation. The continuous Fourier spectra are assumed as a series expansion with the scaled Hermite functions as square-integrable set of basis functions. The expansion coefficients are determined by solving an over-determined inverse problem. In order to define a quick and easy-to-use formula in calculating the Jacobi matrix of the problem a special feature of the Hermite functions are used. It is well-known, that the basic Hermite functions are eigenfunctions of the Fourier transformation. This feature is generalized by extending its validity for the scaled Hermite functions. Using the eigenvalues, given by this generalization, a very simple formula can be derived for the Jacobi matrix of the problem resulting in a quick and more accurate inversion-based Fourier transform algorithm. The new procedure is numerically tested by using synthetic data.