  Smoothing of noisy sample covariances is an important component in functional
data analysis. We propose a novel covariance smoothing method based on
penalized splines and associated software. The proposed method is a bivariate
spline smoother that is designed for covariance smoothing and can be used for
sparse functional or longitudinal data. We propose a fast algorithm for
covariance smoothing using leave-one-subject-out cross validation. Our
simulations show that the proposed method compares favorably against several
commonly used methods. The method is applied to a study of child growth led by
one of coauthors and to a public dataset of longitudinal CD4 counts.
