Nonpertubative Effects of Extreme Localization in Noncommutative Geometry

``Extremely'' localized wavefunctions in noncommutative geometry have disturbances that are localized to distances smaller than $\sqrt{\theta}$, where $\theta$ is the ``area'' parameter that measures noncommutativity. In particular, distributions such as the sign function or the Dirac delta function are limiting cases of extremely localized wavefunctions. It is shown that Moyal star products of extremely localized wavefunctions cannot be correctly computed perturbatively in powers of $\theta$. Nonperturbative effects as a function of $\theta$ are explicitly displayed through exact computations in several examples. In particular, for distributions, star products end up being functions of $\theta ^{-1}$ and have no expansion in positive powers of $\theta$. This result provides a warning for computations in noncommutative space that often are performed with perturbative methods. Furthermore, the result may have interesting applications that could help elucidate the role of noncommutative geometry in several areas of physics.