A two-dimensional homogeneous random surface {
y
(
X
)} is generated from another such surface {
z
(
X
)} by a process of smoothing represented by
y
(
X
) = ∫
∞
d
u
w
(
u
–
X
)
z
(
u
), where
w
(
X
) is a deterministic weighting function satisfying certain conditions. The two-dimensional autocorrelation and spectral density functions of the smoothed surface {
y
(
X
)} are calculated in terms of the corresponding functions of the reference surface {
z
(
X
)} and the properties of the ‘footprint’ of the contact
w
(
X
). When the surfaces are Gaussian, the statistical properties of their peaks and summits are given by the continuous theory of surface roughness. If only sampled values of the surface height are available, there is a corresponding discrete theory. Provided that the discrete sampling interval is small enough, profile statistics calculated by the discrete theory should approach asymptotically those calculated by the continuous theory, but it is known that such asymptotic convergence may not occur in practice. For a smoothed surface {
y
(
X
)} which is generated from a reference surface {
z
(
X
)} by a ‘good’ footprint of finite area, it is shown in this paper that the expected asymptotic convergence does occur always, even if the reference surface is ideally white. For a footprint to be a good footprint,
w
(
X
) must be continuous and smooth enough that it can be differentiated twice everywhere, including at its edges. Sample calculations for three footprints, two of which are good footprints, illustrate the theory.
Global asymptotic convergence of nonlinear relaxation equations realised through a recurrent perceptron
