  Since its introduction, the Discrete Variable Representation (DVR) basis set
has become an invaluable representation of state vectors and Hermitian
operators in non-relativistic quantum dynamics and spectroscopy calculations.
On the other hand reproducing kernel (positive definite) functions have been
widely employed for a long time to a wide variety of disciplines: detection and
estimation problems in signal processing; data analysis in statistics;
generating observational models in machine learning; solving inverse problems
in geophysics and tomography in general; and in quantum mechanics.
  In this article it was demonstrated that, starting with the axiomatic
definition of DVR provided by Littlejohn [1], it is possible to show that the
space upon which the projection operator, defined in ref [1], projects is a
Reproducing Kernel Hilbert Space (RKHS) whose associated reproducing kernel
function can be used to generate DVR points and their corresponding DVR
functions on any domain manifold (curved or not). It is illustrated how, with
this idea, one may be able to `neatly' address the long-standing challenge of
building multidimensional DVR basis functions defined on curved manifolds.
