  We present a method for the construction of a Krein space completion for
spaces of test functions, equipped with an indefinite inner product induced by
a kernel which is more singular than a distribution of finite order. This
generalizes a regularization method for infrared singularities in quantum field
theory, introduced by G. Morchio and F. Strocchi, to the case of singularites
of infinite order. We give conditions for the possibility of this procedure in
terms of local differential operators and the Gelfand- Shilov test function
spaces, as well as an abstract sufficient condition. As a model case we
construct a maximally positive definite state space for the Heisenberg algebra
in the presence of an infinite infrared singularity.
