Turkish Journal of Mathematics
DOI
-
Abstract
Let $(X, {\cal F}, \lambda)$ be the unit circle $\Bbb S^1 = \{z \in \Bbb C : z = 1\}$ with the usual $\sigma$-algebra ${\cal F}$ of Lebesgue measurable subsets and the normalized Lebesgue measure $\lambda$. Consider a sequence $\nu_n: \Bbb N \ra \Bbb R, \;\; \nu_n(k) \geq 0, \;\; \Sigma^{\infty}_{k=1} \nu_n(k) = 1$. For any measure-preserving $\tau : X \ra X$, this sequence induces a sequence $(T_n)^{\infty}_1$ of bounded, linear operators on $L^p(X), \;\; 1 \leq p \leq \infty$, by defining \[ T_n f = \sum^{\infty}_{k=1} \nu_n(k) \; f \circ \tau^k, \quad n = 1, 2, \ldots . \] We shall prove that under suitable conditions imposed on $\tau$ and $(\nu_n)^{\infty}_1$, there exists a large collection of measurable characteristic functions $f$ for which $\lim \sup_{n \ra \infty} T_n f - \lim \inf_{n \ra \infty} T_n f = 1$ a.e on $X$.
Keywords
Weights, weighted averages, Fourier transforms.
First Page
61
Last Page
68
Recommended Citation
Ha, M.D. (1998) "Weighted Ergodic Averages," Turkish Journal of Mathematics: Vol. 22: No. 1, Article 6. Available at: https://journals.tubitak.gov.tr/math/vol22/iss1/6
