  In this paper, we focus on barycentric weights and Lebesgue constants for
Lagrange interpolation of arbitrary node distributions on \([-1,1]\). The
following three main works are included: estimates of upper and lower bounds on
the barycentric weights are given in terms of the logarithmic potential
function; for interpolation of non-equilibrium potentials, lower bounds with
exponentially growing parts of Lebesgue constants are given; and for
interpolation consistent with equilibrium potentials, non-exponentially growing
upper bounds on their Lebesgue constants are given. Based on the work in this
paper, we can discuss the behavior of the Lebesgue constant and the existence
of exponential convergence in a unified manner in the framework of potential
theory.
