Dense Markov Spaces and Unbounded Bernstein Inequalities
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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. © 1994 Academic Press Limited.
author list (cited authors)
Borwein, P., & Erdelyi, T.