The emerging concept of cognitive radios offers a way to use the limited radio-spectrum more efficiently by allowing networks and nodes to adaptively vary their parameters. An important element in the successful implementation of cognitive radios is the ability to estimate the varying state of spectrum usage in a wide-band channel quickly and at minimum cost. In this work, we utilize a powerful non-convex optimization approach to provide sparse and unbiased estimates of the spectrum from limited non-uniformly sampled data. Simulation results for a wide-band communication scenario show that the noise floor is significantly reduced compared to other commonly used approaches. This should help in reducing the miss-identification of occupied and vacant sub-bands of the spectrum, being a key requirement in spectrum sensing cognitive radios.