Abstract
The mean ergodic theorem is equivalent to the assertion that for every function K and every epsilon, there is an n with the property that the ergodic averages A_m f are stable to within epsilon on the interval [n,K(n)]. We show that even though it is not generally possible to compute a bound on the rate of convergence of a sequence of ergodic averages, one can give explicit bounds on n in terms of K and || f || / epsilon. This tells us how far one has to search to find an n so that the ergodic averages are "locally stable" on a large interval. We use these bounds to obtain a similarly explicit version of the pointwise ergodic theorem, and show that our bounds are qualitatively different from ones that can be obtained using upcrossing inequalities due to Bishop and Ivanov. Finally, we explain how our positive results can be viewed as an application of a body of general proof-theoretic methods falling under the heading of "proof mining."
Highlights
Let T be a nonexpansive linear operator on a Hilbert space H, that is, a linear operator satisfying T f ≤ f for all f ∈ H
Even in situations where the sequence Anf does not have a computable limit, one can give explicit bounds on such n in terms of K and f /ε. This tells us how far one has to search to find an n so that the ergodic averages are “locally stable” on a large interval. We use these bounds to obtain a explicit version of the pointwise ergodic theorem, and we show that our bounds are qualitatively different from ones that can be obtained using upcrossing inequalities due to Bishop and Ivanov
We show that there are a computable Lebesgue measure preserving transformation of the unit interval [0, 1] and a computable characteristic function f = χA such that the limit of the sequence (Anf ) is not a computable element of L2([0, 1]). For this we rely on standard notions of computability for Hilbert spaces, which we review there
Summary
Let T be a nonexpansive linear operator on a Hilbert space H, that is, a linear operator satisfying T f ≤ f for all f ∈ H. If r(ε) is a function producing a witness to the existential quantifier, rather than computing an absolute rate of convergence, r(ε) provides, for each ε > 0, a value n such that the ergodic averages Amf are stable to within ε on the interval [n, K(n)]. Our extractions of bounds can be viewed as applications of a body of proof theoretic results that fall under the heading “proof mining” (see, for example, [19, 20, 21]) What makes it difficult to obtain explicit information from the usual proofs of the mean ergodic theorem is their reliance on a nonconstructive principle, namely, the assertion that any bounded increasing sequence of real numbers converges.
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