More precisely, it follows from the Paley–Wiener theorem (see Theorem IX.11 in [346]) that ω f ∈ 𝒮 with ˆ f(k) = 0 for |k| ≥ R if and only if f possesses an analytic extension ¯ n f : ℂ → ℂ such that for each N = 0,1,2,... there exists a constant CN with
R |Im(ζ)| |f¯(ζ)| ≤ CN-e-------, ζ ∈ ℂn. (1+ |ζ|)N