Asymptotic Separation of Variables

We study generalized functions ƒ(x) that admit the asymptotic separation of variables ƒ(λx) ∼ ρ1(λ)h1(x) + ρ2(λ)h2(x) + ρ3(λ)h3(x) + •••, as λ → ∞, where {ρn(λ)} is an asymptotic sequence. Among other results, we show that when the asymptotic separation of variables holds, the terms have to be homogeneous and associated homogeneous generalized functions. The asymptotic expansion of ρ(λx), as λ → ∞, where ρ is a regularly varying function, is also considered