A basis for a set C of functions on natural numbers is a set F of functions such that C is the closure with respect to substitution of the projection functions and the functions in F. This paper introduces three new bases, comprehending only common functions, for the Grzegorczyk classes ℰ_n with n ≥ 3. Such results are then applied in order to show that ℰ_{n+1} = K_n for n ≥ 2, where {K_n}n∈ℕ is the Axt hierarchy.