Abstract

We study continuous extension operators for smooth functions from [0, ∞) to R. If A is a topological vector space of smooth functions in R, let us denote by A[0, ∞) the space of restrictions of functions of A to [0, ∞). We show that when A is any of the standard test function spaces D, S, or K then there is a continuous linear operator E from A[0, ∞) to A that satisfies that E (φ) (t) = φ (t) for t ≥ 0 and that satisfies the invariant condition E {ϕ (λx) ;t} = E {ϕ (x) ; λt} , for λ ≥ 0 . However, we show that when A is E, the space of all smooth functions, then such an operator E does not exist.