Abstract
Let $X$ be the set of all function $x: P_e(G) \\ o C$ satisfying the inequality $|\\sum_{i=1}^n \\alpha_i x(p_i)-x(p)x(q)| \\leq C(x) \\sup_{g \\in K(x)} | \\sum_{i=1}^n \\alpha_i p_i(g) -p(g)q(g)|$, for every $n,\\alpha_1,\\alpha_2,...,\\alpha_n \\in C$ and $p_1,p_2,...,p_n,p,q \\in P_e(G)$,where $C(x)>0$ and $K(x)$ is a compact subset of $G$. Then a topological group structure can be defined on $X$ so that $X$ is isomorphic to $G$. The inequality can be replaced by $|\\sum_{i=1}^n \\alpha_i x(p_i)-x(p)x(q)| \\leq \\sup_{g \\in G} |f(g)(\\sum_{i=1}^n \\alpha_ip_i (g)-p(g)q(g))|$, where $f=f_x \\in C_\\infty(G)$.
