Fourier Transform Spectrometers are instruments that measure the spectral distribution of an electromagnetic signal. They are based on the principle of interferometry, which leverages interference patterns to reconstruct high-resolution observations by solving an inverse problem defined on a continuous domain. The forward model is a linear integral operator, which is, in most cases, discretized by considering a uniform grid and associated Riemann sums to estimate integrals. This approximation introduces an important reconstruction error, which can be significantly reduced by using instead appropriate bases of functions to represent the signal of interest. In this paper, we show the judiciousness of the Fourier basis for this problem, and highlight through numerical experiments the gain it brings in terms of reconstruction precision, compared to the classical use of discrete samples. We finally propose a new basis obtained by enriching the Fourier basis with an affine component, which allows us to overcome the inherent difficulty of representing non-periodic functions with a finite number of sine functions. We show that this new basis leads to another significant improvement of the reconstruction error.