AbstractIn this chapter, we discuss the higher order differentiability properties of Σg when g lies in $$\mathcal {C}^r\cap \mathcal {D}^p\cap \mathcal {K}^{\max \{p,r\}}$$ C r ∩ D p ∩ K max { p , r } for any $$p,r\in \mathbb {N}$$ p , r ∈ ℕ . In particular, we show the fundamental fact that Σg also lies in $$\mathcal {C}^r$$ C r and that the sequence $$n \mapsto D^rf^p_n[g]$$ n ↦ D r f n p [ g ] converges uniformly on any bounded subinterval of $$\mathbb {R}_+$$ ℝ + to Dr Σg.