This is a study of Arens regularity in the context of quotients of the Fourier algebra on a non-discrete locally compact abelian group (or compact group).
¶ (1) If a compact set $E$ of $G$ is of bounded synthesis and is the support of a pseudofunction, then $A(E)$ is weakly sequentially complete. (This implies that every point of $E$ is a Day point.)
¶ (2) If a compact set $E$ supports a synthesizable pseudofunction, then $A(E)$ has Day points. (The existence of a Day point implies that $A(E)$ is not Arens regular.)
¶ We use be $L^{2}$-methods of proof which do not have obvious extensions to the case of $A_{p}(E)$.
¶ Related results, context (historical and mathematical), and open questions are given.