  I want to point out the mathematical difficulties that arise in the study of
the relation of Wightman and Euclidean quantum field theory, i.e., the relation
between the hierarchies of Wightman and Schwinger functions. The two extreme
cases where the reconstructed Wightman functions are either tempered
distributions - the well-known Osterwalder-Schrader reconstruction - or
modified Fourier hyperfunctions are discussed in some detail. Finally, some
perpectives towards a classification of Euclidean reconstruction theorems are
outlined and preliminary steps in that direction are presented.
