Using the Fourier-Laplace transform for functionals, we describe the duals of some spaces of the infinitely differentiable functions given on convex compact sets or convex domains in R N and such that the growth of their derivatives is determined by weight sequences of a general form.A weight function (or a weight) is an arbitrary nondecreasing function ϕ : [0, ∞) → R that is convex from some t 0 ≥ 0 on and enjoys the following conditions:(Denote the class of all weights by W.Let K be a compact set in R N whose nonempty interior int K has the closure coinciding with K. Given a weight ϕ, define the Banach spaceHere and in the sequel we use the following notations: C ∞ (K) is the set of all infinitely differentiable functions on a compact set K; N 0 := N ∪ {0}; f (α) is the derivative of f corresponding to a multiindex α ∈ N N 0 ; and ϕ * (s) := sup{rs − ϕ(r) : r ≥ 0} is the monotone conjugate function of ϕ. Denote by W ↑ (W ↓ ) the collection of all sequences Φ = {ϕ n } ∞ n=1 of weight functions such that, for each n ∈ N, there is C n satisfyingfor every t ≥ 0. Given Φ in W ↑ (W ↓ ) and a compact set K in R N , we naturally define the space