A nonparametric version of the basis pursuit method is developed for field estimation. The underlying model entails known bases, weighted by generic functions to be estimated from the field's noisy samples. A novel field estimator is developed based on a regularized variational least-squares (LS) criterion that yields estimates spanned by thin-plate splines. Robustness considerations motivate well the adoption of an overcomplete set of basis functions, together with a sparsity-promoting regularization term, which endows the estimator with the ability to select a few of these bases that "better" explain the data. This parsimonious field representation becomes possible because the sparsity-aware spline-based method of this paper induces a group-Lasso estimator of the thin-plate spline basis expansion coefficients. The novel spline-based approach to basis pursuit is motivated by a spectrum cartography application, in which a set of sensing cognitive radios collaborate to estimate the distribution of RF power in space and frequency. Simulated tests corroborate that the estimated power spectrum density atlas yields the desired RF state awareness, since the maps reveal spatial locations where idle frequency bands can be reused for transmission, even when fading and shadowing effects are pronounced.