Dense Markov Spaces and Unbounded Bernstein Inequalities

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] 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. The principal result of this paper will be a characterization of denseness of the span of a Markov system by whether or not it possesses an unbounded Bernstein Inequality. In order to make sense of this result we require the following definitions. Definition 1 (Chebyshev System). Let f0, . . . , fn be elements of C[a, b] the real valued continuous functions on [a, b]. Suppose that span{f0, . . . , fn} over R is an n + 1 dimensional subspace of C[0, 1]. Then {f0, . . . , fn} is called a Chebyshev system of dimension n + 1 if any element of span{f0, . . . , fn} that has n + 1 distinct zeros in [0, 1] is identically zero. If {f0, . . . , fn} is a Chebyshev system, then span{f0, . . . , fn} is called a Chebyshev space. Definition 2 (Markov System). We say that {f0, . . . , fn} is a Markov system on [a, b] if each fi ∈ C[a, b] and {f0, . . . , fm} is a Chebyshev system for every m ≥ 0. (We allow n to tend +∞ in which case we call the system an infinite Markov system.) If {f0, · · · , fn} is a Markov system then span{f0, . . . , fn} is called a Markov space. Definition 3 (Unbounded Bernstein Inequality). Let A be a subset of C[a, b]. We say that A has an everywhere unbounded Bernstein inequality if for every [α, β] ⊂ [a, b], α 6= β 1991 Mathematics Subject Classification. 41A17, 41A540.