With increasingly smaller size, more powerful sensing capabilities and higher level of autonomy, multiple unmanned aerial vehicles (UAVs) can form UAV networks to collaboratively complete missions more reliably, efficiently and economically. While UAV networks are promising for many applications, there are many outstanding issues to be resolved before large scale UAV networks are practically used. In this paper we study the application of cognitive radio technology for UAV communication networks, to provide high capacity and reliable communication with opportunistic and timely spectrum access. Compressive sensing is applied in the cognitive radio to boost the performance of spectrum sensing. However, the performance of existing compressive spectrum sensing schemes is constrained with non-strictly sparse spectrum. In addition, the reconstruction process applied in existing schemes has unnecessarily high computational complexity and low energy efficiency. We proposed a new compressive signal processing algorithm, called Iterative Compressive Filtering, to improve the UAV network communication performance. The key idea is using orthogonal projection as a bandstop filter in compressive domain. The components of primary users (PUs) in the recognized subchannels are adaptively eliminated in compressive domain, which can directly update the measurement for further detection of other active users. Experiment results showed increased efficiency of the proposed algorithm over existing compressive spectrum sensing algorithms. The proposed algorithm achieved higher detection probability in identifying the occupied subchannels under the condition of non-strictly sparse spectrum with large computational complexity reduction, which can provide strong support of reliable and timely communication for UAV networks.