In this paper, we introduce the novel Chebyshev Polynomials Least-Squares Fourier
Transformation (C-LSQ-FT) and its robust variant with the Iteratively Reweighted Least-Squares
technique (C-IRLS-FT). These innovative techniques for Fourier transformation are
predicated on the concept of inversion, and the C-LSQ-FT method establishes an overdetermined
inverse problem within the realm of Fourier transformation. However, given the LSQ
approach’s vulnerability to data outliers, we note the potential for considerable
errors and potentially unrepresentative model estimations. To circumvent these shortcomings,
we incorporate Steiner’s Most Frequent Value method into our framework, thereby providing
a more reliable alternative. The fusion of the Iteratively Reweighted Least-Squares
(IRLS) algorithm with Cauchy-Steiner weights enhances the robustness of our Fourier
transformation process, culminating in the C-IRLS-FT method. We use Chebyshev polynomials
as the basis functions in both methods, leading to the approximation of continuous
Fourier spectra through a finite series of Chebyshev polynomials and their corresponding
coefficients. The coefficients were obtained by solving an overdetermined non-linear
inverse problem. We validated the performance of both the traditional Discrete Fourier
Transform (DFT) and the newly developed C-IRLS-FT through numerical tests on synthetic
datasets. The results distinctly exhibited the reduced sensitivity of the C-IRLS-FT
method to outliers and dispersed noise, in comparison with the traditional DFT. We
leveraged the newly proposed (C-IRLS-FT) technique in the application of low-pass
filtering in the context of gravity data. The results corroborate the technique’s
robustness and adaptability, making it a promising method for future applications
in geophysical data processing.