A tutorial on the Fourier transform and discrete Fourier transform, with comments on the Hilbert transform, in the context of denoising spectra

In another article in this special collection, [Lizarbe et al., J. Vac. Sci. Technol. A 43, 033401 (2025)] we made the case for the Fourier denoising of X-ray photoelectron spectroscopy data. We believe that this capability will be useful when it is not possible to collect the best possible data and when the data are to be used for artificial intelligence (AI) applications, as noise tends to confuse AI algorithms. Nevertheless, while Fourier analysis appears to be the most powerful way for removing noise from data, some who might like to use it are not experts in it. This document describes the Fourier transform and discrete Fourier transform in both general and specific ways, e.g., in two dimensions, in the context of denoising spectra. It also discusses the Hilbert transform, which in generally calculated in reciprocal space via Fourier analysis. That is, after some introductory words, which motivate the topic, this article first provides enough information to the future user to be able to denoise many spectra in a reasonable fashion. Thereafter, it covers some of the mathematical theory of the Fourier transform, the discrete Fourier transform, and the Hilbert transform. Even a partial understanding of this material should help those who wish to use Fourier theory to employ it more successfully. The Fourier transform and discrete Fourier transform are powerful tools for data analysis.