Reproducing Kernel Functions: A general framework for Discrete Variable Representation

TL;DR

This paper establishes a connection between Discrete Variable Representation (DVR) basis sets and Reproducing Kernel Hilbert Spaces (RKHS), enabling the construction of DVR functions on curved manifolds for quantum and spectral applications.

Contribution

It demonstrates that DVR basis functions can be generated using reproducing kernel functions within RKHS, extending DVR applicability to curved manifolds.

Findings

DVR basis functions can be derived from reproducing kernels.

RKHS provides a framework for DVR on curved manifolds.

The approach addresses multidimensional DVR construction challenges.

Abstract

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.