  In this paper we consider proper orthogonal decomposition (POD) methods that
do not include difference quotients (DQs) of snapshots in the data set. The
inclusion of DQs have been shown in the literature to be a key element in
obtaining error bounds that do not degrade with the number of snapshots. More
recently, the inclusion of DQs has allowed to obtain pointwise (as opposed to
averaged) error bounds that decay with the same convergence rate (in terms of
the POD singular values) as averaged ones. In the present paper, for POD
methods not including DQs in their data set, we obtain error bounds that do not
degrade with the number of snapshots if the function from where the snapshots
are taken has certain degree of smoothness. Moreover, the rate of convergence
is as close as that of methods including DQs as the smoothness of the function
providing the snapshots allows. We do this by obtaining discrete counterparts
of Agmon and interpolation inequalities in Sobolev spaces. Numerical
experiments validating these estimates are also presented.
