Abstract. We consider the families of finite Abelian groups ${\msbm Z}/p{\msbm Z}\times{\msbm Z}/p{\msbm Z}$, ${\msbm Z}/p^2{\msbm Z}$ and ${\msbm Z}/p{\msbm Z}\times{\msbm Z}/q{\msbm Z}$ for $p,q$ two distinct prime numbers. For the two first families we give a simple characterization of all functions whose support has cardinality $k$ while the size of the spectrum satisfies a minimality condition. We do it for a large number of values of $k$ in the third case. Such equality cases were previously known when $k$ divides the cardinality of the group, or for groups ${\msbm Z}/p{\msbm Z}$.

AMS Subject Classification (1991): 42A99

Keyword(s): uncertainty principle, finite Abelian groups, Fourier matrices

Received January 26, 2010, and in final form September 9, 2013. (Registered under 5/2010.)