A Uniqueness Theorem and Its Application to Field-Theoretical Models with a Fundamental Length

It is shown that if a distribution V of exponential growth has support in a proper convex cone and its Fourier transform is carried by a closed cone different from whole space, then V=0. The application of this result to a {\em quasi-local} quantum field theory (where the fields are localizable only in regions greater than a certain scale of nonlocality) is contemplated. In particular, we show that a number of physically important predictions of {\em local} quantum field theory also hold in a quantum field theory with a fundamental length, as indicated from string theory.