Functions on noncompact lie groups with positive fourier transforms

Let G be a homogeneous group with the graded Lie algebra or a noncompact semisimple Lie group with finite center. We define the Fourier transform f of f as a family of operators, and we say that f is positive if all f(n) are positive. Then, we construct an integrable function f on G with positive f and the restriction of f to any ball centered at the origin of G is square-integrable, however, f is not square-integrable on G.