A Note on Approximations of Bi-continuous Cosine Families

Abstract

We investigate Trotter–Kato approximation results for the class of bi-continuous cosine families. After a concise overview of bi-continuous cosine families, we introduce uniformly bi-continuous cosine families and prove, under natural resolvent-convergence assumptions, a Trotter–Kato approximation theorem. This result both generalizes and refines existing approximation theorems for strongly continuous cosine families. To illustrate the power of our approach, we construct an explicit mollification procedure on Cb(R), yielding a practical approximation of solutions of the wave equation. The techniques developed here open new directions for rigorous numerical analysis of evolution equations that lack strong continuity and provide a framework functional analytic approaches tp partial differential equations.