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This paper investigates the issue of dynamic resource allocation (DRA) in the context of multi-user cognitive radio networks. We present a general framework adopting generalized signal expansion functions for representation of physical-layer radio resources as well as for synthesis of transmitter and receiver waveforms, which allow us to join DRA with waveform adaptation, two procedures that are currently carried out separately. Based on the signal expansion framework, we develop noncooperative games for distributed DRA, which seek to improve the spectrum utilization on a per-user basis under both transmit power and cognitive spectral mask constraints. The proposed DRA games can handle many radio platforms such as frequency, time or code division multiplexing (FDM, TDM, CDM), and even agile platforms with combinations of different types of expansion functions. To avoid the complications of having too many active expansion functions after optimization, we also propose to combine DRA with sparsity constraints. Generally, the sparsity-constrained DRA approach improves convergence of distributed games at little performance loss, since the effective resources required by a cognitive radio are in fact sparse. Finally, to acquire the channel and interference parameters needed for DRA, we develop compressed sensing techniques that capitalize on the sparse properties of the wideband signals to reduce the number of samples used for sensing and hence the sensing time.