We elucidate the relation between the two ways of formulating causality innonlocal quantum field theory: using analytic test functions belonging to thespace $S^0$ (which is the Fourier transform of the Schwartz space $\mathcal D$)and using test functions in the Gelfand-Shilov spaces $S^0_\alpha$. We provethat every functional defined on $S^0$ has the same carrier cones as itsrestrictions to the smaller spaces $S^0_\alpha$. As an application of thisresult, we derive a Paley-Wiener-Schwartz-type theorem for arbitrarily singulargeneralized functions of tempered growth and obtain the corresponding extensionof Vladimirov's algebra of functions holomorphic on a tubular domain.
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机译：我们阐明了在非局部量子场论中提出因果关系的两种方式之间的关系：使用属于空间$ S ^ 0 $的解析测试函数（这是Schwartz空间$ \ mathcal D $的傅立叶变换），并使用Gelfand-Shilov空格$ S ^ 0_ \ alpha $。我们证明，在$ S ^ 0 $上定义的每个函数都具有与其对较小空间$ S ^ 0_ \ alpha $的限制相同的载波锥。作为此结果的应用，我们推导了任意奇异广义化的回火生长函数的Paley-Wiener-Schwartz型定理，并在管状域上获得了Vladimirov代数全同函数的代数的相应扩展。
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