We present a new, discrete undersampling scheme designed to favor wavefield reconstruction by sparsity-promoting inversion with transform elements localized in the Fourier domain. The work is motivated by empirical observations in the seismic community, corroborated by results from compressive sampling, that indicate favorable (wavefield) reconstructions from random rather than regular undersampling. Indeed, random undersampling renders coherent aliases into harmless incoherent random noise, effectively turning the interpolation problem into a much simpler denoising problem. A practical requirement of wavefield reconstruction with localized sparsifying transforms is the control on the maximum gap size. Unfortunately, random undersampling does not provide such a control. Thus, we introduce a sampling scheme, termed jittered undersampling, that shares the benefits of random sampling and controls the maximum gap size. The contribution of jittered sub-Nyquist sampling is key in formu-lating a versatile wavefield sparsity-promoting recovery scheme that follows the principles of compressive sampling. After the behavior of the jittered-undersampling scheme in the Fourier domain is analyzed, its performance is studied for curvelet recovery by sparsity-promoting inversion (CRSI). The findings on synthetic and real seismic data indicate an improvement of several decibels over recovery from regularly undersampled data for the same amount of data collected.