Oscillation and the mean ergodic theorem

Journal d'Analyse Mathématique, 2005

We consider generalizations of the pointwise and mean ergodic theorems to ergodic theorems averaging along different subsequences of the integers or real numbers.

Abstract and Applied Analysis, 2013

We present a simple way to produce good weights for several types of ergodic theorem including the Wiener-Wintner type multiple return time theorem and the multiple polynomial ergodic theorem. These weights are deterministic and come from orbits of certain bounded linear operators on Banach spaces. This extends the known results for nilsequences and return time sequences of the form for a measure preserving system and , avoiding in the latter case the problem of finding the full measure set of appropriate points .

Mathematical Research Letters, 2011

We create a general framework for the study of rates of decay in mean ergodic theorems. As a result, we unify and generalize results due to Assani, Cohen, Cuny, Derriennic, and Lin dealing with rates in mean ergodic theorems in a number of cases. In particular, we prove that the Cesàro means of a power-bounded operator applied to elements from the domain of its abstract one-sided ergodic Hilbert transform decay logarithmically, and this decay is best possible under natural spectral assumptions.

Proceedings of the National Academy of Sciences, 1996

Let a(x) be a real function with a regular growth as x --> infinity. [The precise technical assumption is that a(x) belongs to a Hardy field.] We establish sufficient growth conditions on a(x) so that the sequence ([a(n)])(infinity)(n=1) is a good averaging sequence in L2 for the pointwise ergodic theorem. A sequence (an) of positive integers is a good averaging sequence in L2 for the pointwise ergodic theorem if in any dynamical system (Omega, Sigma, m, T) for f [symbol, see text] in L2(Omega) the averages [equation, see text] converge for almost every omicron in. Our result implies that sequences like ([ndelta]), where delta > 1 and not an integer, ([n log n]), and ([n2/log n]) are good averaging sequences for L2. In fact, all the sequences we examine will turn out to be good averaging for Lp, p > 1; and even for L log L. We will also establish necessary and sufficient growth conditions on a(x) so that the sequence ([a(n)]) is good averaging for mean convergence. Note tha...

Proceedings of the American Mathematical Society, Series B, 2021

Convergence properties of random ergodic averages have been extensively studied in the literature. In these notes, we exploit a uniform estimate by Cohen \& Cuny who showed convergence of a series along randomly perturbed times for functions in $L^2$ with $\int \max(1,\log (1+|t|)) d\mu_f<\infty$. We prove universal pointwise convergence of a class of random averages along randomly perturbed times for $L^2$ functions with $\int \max(1,\log\log(1+|t|)) d\mu_f<\infty$. For averages with additional smoothing properties, we obtain a universal variational inequality as well as universal pointwise convergence of a series define by them for all functions in $L^2$.

2007

Let C be a bounded closed convex subset of a uniformly convex Banach space X and let T be an asymptotically nonexpansive in the intermediate mapping from C into itself. In this paper, we first provide a ergodic retraction theorem and a mean ergodic convergence theorem. Using this result, we show that the set F (T ) of fixed points of T is a sunny, nonexpansive retract of C if the norm of X is uniformly Gâteaux differentiable. Moreover, we discuss the strong convergence of the sequence {xn} defined by xn = anx+ (1− an)T (μ)xn for n = 0, 1, 2, . . . , where x ∈ C, μ is a Banach limit on l∞ and an is a real sequence in (0, 1].

Proceedings of the American Mathematical Society, 1988

We characterize those positive linear operators with positive inverse for which the dominated ergodic estimate holds. We also prove that for such operators one has mean and a.e. convergence.

SIAM Journal on Mathematical Analysis, 1990

Some Nonlinear Weak Ergodic Theorems. [SIAM Journal on Mathematical Analysis 21, 436 (1990)]. Roger D. Nussbaum. Abstract. Let $C$ be a cone with nonempty interior in a Banach space and, for $j \geqq 1$, let $f_j :\mathop ...

Illinois Journal of Mathematics, 1999

We investigate sequences of complex numbers a {at'} for which the modulated averages "t'=l at, Tt' f converge in norm for every weakly almost periodic linear operator T in a Banach space. For Dunford-Schwartz operators on probability spaces, we study also the a.e. convergence in Lp. The limit is identified in some special cases, in particular when T is a contraction in a Hilbert space, or when a {st'4,()} for some positive Dunford-Schwartz operator S on a Lebesgue space and4, L2. We also obtain necessary and sufficient conditions on a for the norm convergence of the modulated averages for every, mean ergodic power bounded T, and identify the limit.

Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 2000

Recently, Sarrión and the authors gave a sufficient condition on invertible Lamperti operators on Lp which guarantees the convergence in the Cesàro-α sense of the ergodic averages and the ergodic Hilbert transform for all f ∈ Lp with p > 1/(1 + α) and −1 < α ≤ 0. The result does not hold for the space L1/(1 + α). In this paper we give a positive result for the smaller Lorentz space L1/(1 + α),1.