** Authors:**
M.D. Ha

** 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.

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