  This research focuses on the estimation of a non-parametric regression
function designed for data with simultaneous time and space dependencies. In
such a context, we study the Trend Filtering, a nonparametric estimator
introduced by \cite{mammen1997locally} and \cite{rudin1992nonlinear}. For
univariate settings, the signals we consider are assumed to have a kth weak
derivative with bounded total variation, allowing for a general degree of
smoothness. In the multivariate scenario, we study a $K$-Nearest Neighbor fused
lasso estimator as in \cite{padilla2018adaptive}, employing an ADMM algorithm,
suitable for signals with bounded variation that adhere to a piecewise
Lipschitz continuity criterion. By aligning with lower bounds, the minimax
optimality of our estimators is validated. A unique phase transition
phenomenon, previously uncharted in Trend Filtering studies, emerges through
our analysis. Both Simulation studies and real data applications underscore the
superior performance of our method when compared with established techniques in
the existing literature.
