This paper introduces a general principle for constructing multiscale kernels on surface meshes, and presents a construction of the multiscale pre-biharmonic and multiscale biharmonic kernels. Our construction is based on an optimization problem that seeks to minimize a smoothness criterion, the Laplacian energy, subject to a sparsity inducing constraint. Namely, we use the lasso constraint, which sets an upper bound on the l1 -norm of the solution, to obtain a family of solutions parametrized by this upper-bound parameter. The interplay between sparsity and smoothness results in smooth kernels that vanish away from the diagonal. We prove that the resulting kernels have gradually changing supports, consistent behavior over partial and complete meshes, and interesting limiting behaviors (e.g. in the limit of large scales, the multiscale biharmonic kernel converges to the Green's function of the biharmonic equation); in addition, these kernels are based on intrinsic quantities and so are insensitive to isometric deformations. We show empirically that our kernels are shape-aware, are robust to noise, tessellation, and partial object, and are fast to compute. Finally, we demonstrate that the new kernels can be useful for function interpolation and shape correspondence.