Stability analysis of boundary element methods for the wave equation: The weakly singular operator in $1d$

We discuss the construction of stable discrete pairs of trial and test spaces for the wave equation. Building on a general abstract operator-theoretic framework, we show how optimal pairings can be derived for a one-dimensional boundary element formulation of the wave equation with prescribed Dirichlet data, based on an indirect single layer potential ansatz. In this setting, the optimal test space can be computed analytically and has a physical interpretation, reflecting the non-local nature of the underlying operator.