  We study boundary values of harmonic functions in spaces of quasianalytic
functionals and spaces of ultradistributions of non-quasianalytic type. As an
application, we provide a new approach to H\"ormander's support theorem for
quasianalytic functionals. Our main technical tool is a description of
ultradifferentiable functions by almost harmonic functions, a concept that we
introduce in this article. We work in the setting of ultradifferentiable
classes defined via weight matrices. In particular, our results simultaneously
apply to the two standard classes defined via weight sequences and via weight
functions.
