Advisor: Prof. Zhiqin Lu

Classical ergodic averages give good norm approximations, but these averages are not necessarily giving the best norm approximation among all possible averages. We consider
1) what the optimal Cesaro norm approximation can be in terms of the transformation and the function,
2) when these optimal Cesaro norm approximations are comparable to the norm of the usual ergodic average, and
3) oscillatory behavior of these norm approximations.