  We propose a fast bivariate smoothing approach for symmetric surfaces that
has a wide range of applications. We show how it can be applied to estimate the
covariance function in longitudinal data as well as multiple additive
covariances in functional data with complex correlation structures. Our
symmetric smoother can handle (possibly noisy) data sampled on a common, dense
grid as well as irregularly or sparsely sampled data. Estimation is based on
bivariate penalized spline smoothing using a mixed model representation and the
symmetry is used to reduce computation time compared to the usual non-symmetric
smoothers. We outline the application of our approach in functional principal
component analysis and demonstrate its practical value in two applications. The
approach is evaluated in extensive simulations. We provide documented open
source software implementing our fast symmetric bivariate smoother building on
established algorithms for additive models.
