Functions on noncompact Lie groups with positive Fourier transforms

Abstract

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 ^ \hat f of f as a family of operators f ^ ( π ) = ∫ G f ( x ) π ( x ) d x ( π ∈ G ^ ) \hat f(\pi ) = {\smallint _G}f(x)\pi (x)dx(\pi \in \hat G) , and we say that f ^ \hat f is positive if all f ^ ( π ) \hat f(\pi ) are positive. Then, we construct an integrable function f on G with positive f ^ \hat 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.