  Despite increasing accessibility to function data, effective methods for
flexibly estimating underlying functional trend are still scarce. We thereby
develop functional version of trend filtering for estimating trend of
functional data indexed by time or on general graph by extending the
conventional trend filtering, a powerful nonparametric trend estimation
technique, for scalar data. We formulate the new trend filtering by introducing
penalty terms based on $L_2$-norm of the differences of adjacent trend
functions. We develop an efficient iteration algorithm for optimizing the
objective function obtained by orthonormal basis expansion. Furthermore, we
introduce additional penalty terms to eliminate redundant basis functions,
which leads to automatic adaptation of the number of basis functions. The
tuning parameter in the proposed method is selected via cross validation. We
demonstrate the proposed method through simulation studies and applications to
real world datasets.
