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Small corrections to the uncertainty relations, with effects in theultraviolet and/or infrared, have been discussed in the context of stringtheory and quantum gravity. Such corrections lead to small but finite minimaluncertainties in position and/or momentum measurements. It has been shown thatthese effects could indeed provide natural cutoffs in quantum field theory. Thecorresponding underlying quantum theoretical framework includes small`noncommutative geometric' corrections to the canonical commutation relations.In order to study the full implications on the concept of locality it iscrucial to find the physical states of then maximal localisation. These statesand their properties have been calculated for the case with minimaluncertainties in positions only. Here we extend this treatment, though still inone dimension, to the general situation with minimal uncertainties both inpositions and in momenta.Comment: Latex, 21 pages, 2 postscript figure

Maximal localization in the presence of minimal uncertainties in positions and in momenta