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We study generalized eigenfunction expansions of multiplicity one, obtaining precise convergence estimates. We apply the theory to the expansion for the Laplacian and Hecke operators on the fundamental domain for the modular group, where the convergence estimates are shown to be optimal.doi:10.1216/rmj-2010-40-1-243 fatcat:6dxrh2qdorgundu63pvgiee6uy