Two classes of generalized functions used in nonlocal field theory

We elucidate the relation between the two ways of formulating causality in nonlocal quantum field theory: using analytic test functions belonging to the space S⁰ (which is the Fourier transform of the Schwartz space ) and using test functions in the Gelfand-Shilov spaces S⁰. We prove that every functional defined on S⁰ has the same carrier cones as its restrictions to the smaller spaces S⁰. As an application of this result, we derive a Paley-Wiener-Schwartz-type theorem for arbitrarily singular generalized functions of tempered growth and obtain the corresponding extension of Vladimirovs algebra of functions holomorphic in a tubular domain.__________Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 143, No. 2, pp. 195–210, May, 2005.