  We use analytical calculations and Monte Carlo simulations to determine the
thermal fluctuation spectrum of a membrane patch of a few tens of nanometer in
size, whose corners are located at a fixed distance $d$ above a plane rigid
surface. Our analysis shows that the surface influence on the bilayer
fluctuations can be effectively described in terms of a uniform confining
potential that grows quadratically with the height of the membrane $h$ relative
to the surface: $V=(1/2)\gamma h^2$. The strength $\gamma$ of the harmonic
confining potential vanishes when the corners of the membrane patch are placed
directly on the surface ($d=0$), and achieves its maximum value when $d$ is of
the order of a few nanometers. However, even at maximum strength the
confinement effect is quite small and has noticeable impact only on the
amplitude of the largest bending mode.
