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In the setting of functional data analysis, we derive optimal rates ofconvergence in the supremum norm for estimating the H\"older-smooth meanfunction of a stochastic processes which is repeatedly and discretely observedat fixed, multivariate, synchronous design points and with additional errors.Similarly to the rates in $L_2$ obtained in Cai and Yuan (2011), for sparsedesign a discretization term dominates, while in the dense case the $\sqrt n$rate can be achieved as if the $n$ processes were continuously observed withouterrors. However, our analysis differs in several respects from Cai and Yuan(2011). First, we do not assume that the paths of the processes are as smoothas the mean, but still obtain the $\sqrt n$ rate of convergence withoutadditional logarithmic factors in the dense setting. Second, we show that inthe supremum norm, there is an intermediate regime between the sparse and densecases dominated by the contribution of the observation errors. Third, and incontrast to the analysis in $L_2$, interpolation estimators turn out to besub-optimal in $L_\infty$ in the dense setting, which explains their poorempirical performance. We also obtain a central limit theorem in the supremumnorm and discuss the selection of the bandwidth. Simulations and real dataapplications illustrate the results.

From dense to sparse design: Optimal rates under the supremum norm for estimating the mean function in functional data analysis