A cooperative approach to the sensing task of wireless cognitive radios (CRs) is introduced based on a basis expansion model of the power spectral density (PSD) in space and frequency. Joint estimation of the model parameters enables identification of the (un)used frequency bands at arbitrary locations and thus facilitates spatial frequency reuse. The novel scheme capitalizes on the sparsity introduced by the narrowband nature of transmit-PSDs relative to the broad swaths of usable spectrum and the scarcity of position vectors where active radios are located in space. A basis pursuit scheme is developed to exploit this sparsity in the solution and reveal the unknown positions of transmitting CRs. The resultant algorithm accounts for deterministic pathloss as well as random fading propagation and can be implemented via distributed online iterations which solve quadratic programs locally (one per radio). Simulations corroborate that exploiting sparsity in CR sensing reduces spatial spectrum leakage by 15-20dB relative to least-squares (LS) alternatives.