Appendix

On Fourier series

We consider the domain Ω = (0, L[1]) × … × (0, L[d]) and a L-periodic vector field u: Ω → ℝᵐ. The Fourier coefficients of u are defined as follows, for all k ∈ ℝᵈ

û(k) = |Ω|⁻¹ ∫ u(x) exp(-i k ⋅ x) dx.
             Ω

The following synthesis formula then holds (under non-specified, but non-restrictive, regularity conditions)

u(x) =   ∑    û(kₚ) exp(i kₚ ⋅ x),
       p ∈ ℤᵈ

where kₚ is the following wave-vector

        2π p[i]
kₚ[i] = ───────,    for all    i ∈ {1, …, d}    and    p ∈ ℤᵈ.
          L[i]