This was later improved as follows: if f ∈ ∈ L 2 ( R d ) is such that ∫ ∫ R d ∫ ∫ R d | f ( x ) | | f ^ ^ ( ξ ξ ) | e π π | ⟨ ⟨ x , ξ ξ ⟩ ⟩ | ( 1 + | x | + | ξ ξ | ) N d x d ξ ξ < + ∞ ∞ , then f ( x ) = P ( x ) e − − π π ⟨ ⟨ A x , x ⟩ ⟩ , where P is a polynomial of degree (N − d)/2 and A is a real d × d positive definite matrix.