  Laplacian spectral kernels and distances (e.g., biharmonic, heat diffusion,
wave kernel distances) are easily defined through a filtering of the Laplacian
eigenpairs. They play a central role in several applications, such as
dimensionality reduction with spectral embeddings, diffusion geometry, image
smoothing, geometric characterisations and embeddings of graphs. Extending the
results recently derived in the discrete
setting~\citep{PATANE-STAR2016,PATANE-CGF2017} to the continuous case, we
propose a novel definition of the Laplacian spectral kernels and distances,
whose approximation requires the solution of a set of inhomogeneous Laplace
equations. Their discrete counterparts are equivalent to a set of sparse,
symmetric, and well-conditioned linear systems, which are efficiently solved
with iterative methods. Finally, we discuss the optimality of the Laplacian
spectrum for the approximation of the spectral kernels, the relation between
the spectral and Green kernels, and the stability of the spectral distances
with respect to the evaluation of the Laplacian spectrum and to multiple
Laplacian eigenvalues.
