Compressed plane waves - compactly supported multiresolution basis for the Laplace operator

This paper describes an L 1 regularized variational framework for de- veloping a spatially localized basis, compressed plane waves (CPWs), that spans the eigenspace of a differential operator, for instance, the Laplace operator. Our approach generalizes the concept of plane waves to an orthogonal real-space basis with multi-resolution capa- bilities. the non-convex optimization problem for constructing BCPWs, and a fast CPW transform and fast inverse CPW transform are developed for transforming between frequency space and real space. Numerical experiments demonstrate that CPWs can efficiently represent spatially localized functions, suggest- ing advantages over canonical basis of extended functions such as plane waves.