  Let $A_p(G)$ denote the Figa-Talamanca-Herz Banach Algebra of the locally
compact group $G$, thus $A_2(G)$ is the Fourier Algebra of $G$. If $G$ is
commutative then $A_2(G)=L^1(\hat{G}){\hat{}}$. Let $A^r_p(G)=A_p\cap L^r(G)$
with norm $||u||_{A_p^r}=||u||_{A_p}+||u||_{L^r}$. We investigate a property
which insures not only existence of solutions to optimization problems but
moreover, facility in testing that an algorithm converges to such solutions
namely the RNP. Theorem(a): If $G$ is weakly amenable then $A_p^r$ is a dual
Banach space with RNP if $1\leq r\leq p'$. This does not hold if $G=SL(2,R)$,
$p=2$ and $r>2$. Theorem(b): If $G$ is weakly amenable and second countable and
$A^t_p$ has the RNP for $t=s$, then it has the RNP for all $1\leq t\leq s$,
where $s=\infty$ is allowed. In particular second countable noncompact groups
$G$, for which $A_p(G)$ has RNP, namely Fell groups, have to satisfy that
$A_p^r(G)$ has the RNP for all $1\leq r<\infty$. The results are new, even if
$G=\mathbb{Z}$, the additive integers.
