Reproducing a sampled sound field over a two-dimensional area using an array of loudspeakers is a problem with well-appreciated applications to acoustics and ultrasound treatement. Loudspeaker signal design has traditionally relied on a (possibly regularized) least-squares criterion. The fresh look advocated here, permeates benefits from advances in variable selection and compressive sampling by casting the sound field synthesis as a sparse linear regression problem that is solved by the least absolute shrinkage and selection operator (Lasso). Analysis and simulations demonstrate that the novel approach exhibits superb performance even for under-sampled sound fields, where least-squares methods yield inconsistent field reproduction. In addition, Lasso-based synthesis enables judicious placement of parsimonious loudspeaker arrays.