The function of the Radio Resource Management module of a Cognitive Radio (CR) system is to evaluate the available resources and assign them to meet the Quality of Service (QoS) objectives of the Secondary User (SU), within some constraints on factors which limit the performance of the Primary User (PU). While interference mitigation to the PU spectral band from the SU's transmission has received a lot of attention in recent literature; the novelty of our work is in considering a more realistic and effective approach of dividing the PU into sub-bands, and ensuring that the interference to each of them is below a specified threshold. With this objective, and within a power budget, we execute the tasks of power allocation, bit loading and sizing the sub-carrier bandwidth for an orthogonal frequency division multiplexing (OFDM)-based SU. After extensively analyzing the solution form of the optimization problems posed for the resource allocation, we suggest iterative algorithms to meet the aforementioned objectives. The algorithm for sub-carrier bandwidth sizing is novel, and not previously presented in literature. A multiple SU scenario is also considered, which entails assigning sub-carriers to the users, besides the resource allocation. Simulation results are provided, for both single and multi-user cases, which indicate the effectiveness of the proposed algorithms in a CR environment.