We demonstrate compressive-sensing (CS)spectroscopy in a planar-waveguide Fourier-transform spectrometer (FTS) device.The spectrometer is implemented as an array of Mach–Zehnder interferometers (MZIs) integrated on a photonic chip. The signal from a set of MZIs is composed of an undersampled discrete Fourier interferogram, which we invert using l₁-norm minimization to retrieve a sparse input spectrum. To implement this technique, we use subwavelength-engineered spatial heterodyne FTS on a chip composed of 32 independent MZIs. We demonstrate the retrieval of three sparse input signals by collecting data from restricted sets (eight and fourteen) of MZIs and applying common CS reconstruction techniques to this data. We show that this retrieval maintains the full resolution and bandwidth of the original device, despite a sampling factor as low as one-fourth of a conventional (non-compressive) design.