  Nonlinear interferometers allow for mid-infrared spectroscopy with
near-infrared detection using correlated photons. Previous implementations have
demonstrated a spectral resolution limited by spectrally selective detection.
In our work, we demonstrate mid-infrared transmission spectroscopy in a
nonlinear interferometer using single-pixel near-infrared detection and
Fourier-transform analysis. A sub-wavenumber spectral resolution allows for
rotational-line-resolving spectroscopy of gaseous samples in a spectral
bandwidth of over 700$\,$cm$^{-1}$. We use methane transmission spectra around
3.3$\,\mu$m wavelength to characterize the spectral resolution, noise
limitations and transmission accuracy of our device. The combination of
nonlinear interferometry and Fourier-transform analysis paves the way towards
performant and efficient mid-infrared spectroscopy with near-infrared
detection.
