Chapter 5 builds a rigorous, implementable theory for computing spectral measures of unitary operators. It mirrors Chapter 4, but there are key differences. For eigenvalue-based methods, discretisations must be unitary. We show how to enforce this via polar decomposition, and that only weak-operator convergence is needed for the resulting measures to converge. On the resolvent side, Stone-type formulas on the circle and the Carathéodory function lead to Poisson-kernel smoothing computed by solving linear systems. To reduce cost and sharpen resolution, we introduce higher-order periodic kernels—rational resolvent kernels and trigonometric-polynomial Fourier filters—that accelerate convergence. A change of variables then transfers the machinery to bounded self-adjoint operators, yielding an infinite-dimensional convergence theory for the kernel polynomial method from Chebyshev moments. The techniques are highly flexible and illustrated on challenging examples, including singular spectral measures and the unitary almost Mathieu operator.

Chapter 4 shows how spectral measures generalise diagonalisation for self-adjoint operators with continuous spectrum and provides practical, provably convergent methods for computing them. Using the spectral theorem, we introduce projection-valued and scalar measures, their Lebesgue decomposition, and analyse how discretisations produce discrete measures. Strong and strong-resolvent convergence conditions ensure weak convergence of the measures and convergence of the functional calculus, clarifying spectral invisibility and pollution. From Stone’s formula, we derive resolvent-based algorithms: spectral projections are approximated by kernel convolutions (Poisson smoothing) evaluated by solving linear systems with adaptive error control as the smoothing parameter shrinks. This yields SCI-hierarchy classifications for setwise computation, functional calculus, and weak convergence. Finally, we accelerate Stone’s formula with high-order convolution and rational kernels, reducing the computational burden, prove convergence rates in several settings, and illustrate the approach on magneto-graphene, Dirac operators, and pseudodifferential operators related to internal waves.