An Approximate Representation of the Fourier Spectra of Irregularly Sampled Multidimensional Functions: A Cost-Effective, Memory-Saving Algorithm

This article discusses how to estimate the Fourier spectra of irregularly sampled multidimensional functions in an approximate way, using current fast Fourier transform (FFT) algorithms. This estimate may be an alternative for more rigorous approaches when the inversion of huge matrices is prohibitively expensive. The approximation results from a Taylor expansion of the Fourier transform kernel, which is a series on the product of wavenumbers and displacements from centers of a grid where a regular, discrete Fourier transform (DFT) is defined. Convergence and efficiency, as well as some shortcuts for implementation, is indicated. The problem of finding a Fourier spectrum of a function given a finite set of irregular measurements is usually associated with the idea of regularization/interpolation. This, in turn, raises the question of representativeness of continuous functions via its discrete measurements. Although this question is central for the very Fourier spectrum estimation, the scope of this article will be restricted to the trigonometric interpolation, postponing representativeness issues. For the sake of simplicity, the proposed approximation is first derived for the one-dimensional problem where most related aspects are more clearly stated. Then, extensions to higher dimensions are straightforwardly presented.