We investigate varphi -factorable operators and Weyl-Heise- nberg frames with respect to a function-valued inner product, the so called varphi -bracket product on L^2(G) , where G is a locally compact abelian group and varphi is a topological isomorphism on G . We introduce varphi -factorable operators on L^2(G) and extend the Riesz Representation Theorems for these operators. Finally, as an application of the varphi -bracket product, we show that several well known theorems for Weyl-Heisenberg frames in L^2( mathbb{R}) remain valid in L^2(G) , and they are unified within of group theory, in connection with the varphi -bracket product.

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