Invariant kernels on Riemannian symmetric spaces: a harmonic-analytic
  approach
Nathael Da Costa Cyrus Mostajeran Juan-Pablo Ortega Salem Said
This work aims to prove that the classical Gaussian kernel, when defined on a
non-Euclidean symmetric space, is never positive-definite for any choice of
parameter. To achieve this goal, the paper develops new geometric and
analytical arguments. These provide a rigorous characterization of the
positive-definiteness of the Gaussian kernel, which is complete but for a
limited number of scenarios in low dimensions that are treated by numerical
computations. Chief among these results are the L$^{\!\scriptscriptstyle
p}$-$\hspace{0.02cm}$Godement theorems (where $p = 1,2$), which provide
verifiable necessary and sufficient conditions for a kernel defined on a
symmetric space of non-compact type to be positive-definite. A celebrated
theorem, sometimes called the Bochner-Godement theorem, already gives such
conditions and is far more general in its scope, but is especially hard to
apply. Beyond the connection with the Gaussian kernel, the new results in this
work lay out a blueprint for the study of invariant kernels on symmetric
spaces, bringing forth specific harmonic analysis tools that suggest many
future applications.