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Although many mathematical models in wave theory lead to hyperbolic initial-boundary value problems which are inherently local due to the finite domain-of-dependence of the solution at any point on past values, there are also examples where nonlocal operators, in particular space-time integral operators, arise. A primary example is the radiation boundary condition needed to truncate an unbounded domain to a finite one to enable numerical solutions, as well as closely-related operators for unidirectional propagation. In this work we show how to leverage results from rational function approximation theory to construct spectrally-convergent local algorithms for evaluating such operators. For an important class of problems - systems equivalent to the scalar wave equation, such as acoustics or Maxwell’s equations in homogeneous, isotropic media, we will explain the construction, analysis, and implementation of our complete radiation boundary conditions, which are in a certain sense optimal. We will also discuss other applications of these approximation methods, as well as the fundamental barriers to extending the successful methods to other systems.