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We develop a comprehensive theory of reflectionless canonical systems with anarbitrary Dirichlet-regular Widom spectrum with the Direct Cauchy Theoremproperty. This generalizes, to an infinite gap setting, the constructions offinite gap quasiperiodic (algebro-geometric) solutions of stationary integrablehierarchies. Instead of theta functions on a compact Riemann surface, theconstruction is based on reproducing kernels of character-automorphic Hardyspaces in Widom domains with respect to Martin measure. We also constructunitary character-automorphic Fourier transforms which generalize thePaley-Wiener theorem. Finally, we find the correct notion of almost periodicitywhich holds for canonical system parameters in Arov gauge, and we provegenerically optimal results for almost periodicity for Potapov-de Brangesgauge, and Dirac operators.

Reflectionless canonical systems, II. Almost periodicity and character-automorphic Fourier transforms