  Despite increasing accessibility to function data, effective methods for
flexibly estimating underlying trend structures are still scarce. We thereby
develop locally adaptive smoothing methods for both functional time series and
spatial data by extending trend filtering that is a powerful nonparametric
trend estimation technique for scalar data. We formulate the functional version
of trend filtering by introducing $L_2$-norm of the difference of adjacent
trend functions. Using orthonormal basis expansion, we simplify the objective
function to squared loss for coefficient vectors with grouped fused lasso
penalty, and develop an efficient iteration algorithm for optimization. The
tuning parameter in the proposed method is selected via cross validation. We
also consider an extension of the proposed algorithm to spatial functional
data. The proposed methods are demonstrated by simulation studies and an
application to two real world datasets.
