  We obtain minimax-optimal convergence rates in the supremum norm, including
infor-mation-theoretic lower bounds, for estimating the covariance kernel of a
stochastic process which is repeatedly observed at discrete, synchronous design
points. In particular, for dense design we obtain the $\sqrt n$-rate of
convergence in the supremum norm without additional logarithmic factors which
typically occur in the results in the literature. Surprisingly, in the
transition from dense to sparse design the rates do not reflect the
two-dimensional nature of the covariance kernel but correspond to those for
univariate mean function estimation. Our estimation method can make use of
higher-order smoothness of the covariance kernel away from the diagonal, and
does not require the same smoothness on the diagonal itself. Hence, as in
Mohammadi and Panaretos (2024) we can cover covariance kernels of processes
with rough sample paths. Moreover, the estimator does not use mean function
estimation to form residuals, and no smoothness assumptions on the mean have to
be imposed. In the dense case we also obtain a central limit theorem in the
supremum norm, which can be used as the basis for the construction of uniform
confidence sets. Simulations and real-data applications illustrate the
practical usefulness of the methods.
