Kernel density estimation (KDE) is widely-used for non-parametric estimation of an underlying density from data. The performance of KDE is mainly dependent on the bandwidth parameter of the kernel. This study presents an alternative method of estimating the bandwidth by incorporating sparsity priors in the Fourier transform domain. By using cross-validation (CV) together with an l1 constraint, the proposed method significantly reduces the under-smoothing effect of traditional CV methods. A solution for all free parameters in the minimisation is proposed, such that the algorithm does not need any additional parameter tuning. Simulation results indicate that the new approach is able to outperform classical and more recent approaches over a set of distributions of interest.