DENSE MARKOV SPACES AND UNBOUNDED BERNSTEIN INEQUALITIES

An infinite Markov system {f(0), f(1),...} of C-2 functions on [a, b] has dense span in C[a,b] if and only if there is an unbounded Bernstei n inequality on every subinterval of [a,b]. That is if and only if, fo r each [alpha,beta]subset of[a, b], alpha not equal beta and gamma>0, we can find g epsilon span {f(0),f(1),...} with parallel to g'parallel to([alpha,beta])>gamma parallel to g parallel to([a,b]). This is prov ed under the assumption (f(1)/f(0))' does not vanish on (a,b). Extensi on to higher derivatives are also considered. An interesting consequen ce of this is that functions in the closure of the span of a non-dense C-2 Markov system are always C-n on some subinterval. (C) 1995 Academ ic Press, Inc.