Differential and ergodic transforms
Let \(\tau\) be an ergodic transformation with respect to a non-atomic measure. Operators of the form \[ T_nf(x)=\sum_{k=1}^nv_k\left((1/n_k)\sum_{j=0}^{n_k-1}f(\tau^jx)- (1/n_{k-1})\sum_{j=0}^{n_{k-1}-1}f(\tau^jx)\right) \] for a lacunary sequence \((n_k)\) and \(\ell^{\infty}\) sequence of weights \((v_k)\) are shown to converge a.e. and in \(L^p\), \(1<p<\infty\). Related results on dyadic differentiation and the dyadic martingale are found.
scientific article; zbMATH DE number 1801614