Lorentz-covariant ultradistributions, hyperfunctions, and analytic functionals

We generalize the theory of Lorentz-covariant distributions to broader classes of functionals including ultradistributions, hyperfunctions, and analytic functionals with a tempered growth. We prove that Lorentz-covariant functionals with essential singularities can be decomposed into polynomial covariants and establish the possibility of the invariant decomposition of their carrier cones. We describe the properties of odd highly singular generalized functions. These results are used to investigate the vacuum expectation values of nonlocal quantum fields with an arbitrary high-energy behavior and to extend the spin--statistics theorem to nonlocal field theory.