In this paper the H-infinity filtering problem of a continuous-time process is considered under the assumption that the measurement is sampled by a generalized sampling device. Both cases of known and unknown initial conditions are considered. The problem is treated via the lifting transformation, which converts the problem to a pure discrete-time setting. It is shown that if a suboptimal filter exists, it can always be presented in the form of a finite-dimensional discrete-time (possibly time-varying) filter followed by a stationary generalized hold function. This extends previous results in the literature, where a suboptimal filter of this structure (with an LTI discrete-time part of the filter) was obtained for the case of known initial conditions only. Also, the existence conditions of a suboptimal filter derived in this paper are simpler than those known in the literature.