Dense Markov Spaces and Unbounded Bernstein Inequalities Original Research Article

An infinite Markov system {f0, f1, ...} of C2 functions on [a, b] has dense span in C[a, b] if and only if there is an unbounded Bernstein inequality on every subinterval of [a, b]. That is if and only if, for each [α, β] ⊂ [a, b], α ≠ β and γ > 0, we can find g ∈ span{f0, f1, ...} with ∥g′∥[α,β] > γ ∥g∥[a,b]. This is proved under the assumption (f1/f0)′ does not vanish on (a, b). Extension to higher derivatives are also considered. An interesting consequence of this is that functions in the closure of the span of a non-dense C2 Markov system are always Cn on some subinterval.