Future industrial radio systems for Industry 4.0 applications will heavily rely on the management of shared spectrum to provide coexistence of different wireless communication systems and unknown wireless transmitters. A crucial component for successful coexistence management is the estimation of the spectrum of interest to identify active wireless systems and potential interference sources. Monitoring large bandwidths can be very costly in terms of the required bandwidth if classical Shannon-Nyquist signal acquisition is pursued at a sampling rate at least the bandwidth of the signal of interest. For fast and accurate spectrum estimation, we propose a novel approach called Compressive Edge Spectrum Sensing (CES) that allows for a sampling down to 5% of the Nyquist rate without losses in the detection accuracy of occupied and unoccupied spectrum regions.