We derive the complete duality theory underlying the Hodrick--Prescott filter, whose rank-deficient second-difference penalty admits infinitely many equivalent trend representations via generalized inverses. Constructing two canonical choices---the Moore--Penrose-based \emph{B-representation} and an alternative \emph{A-representation}---we prove that the extracted trend is invariant across representations, obtain a closed-form Bregman-type divergence quantifying their disagreement under a shared coefficient vector, and show this divergence vanishes as $λ\to\infty$. We further mollify the non-smooth $\ell_1$ trend filter with a compactly supported biweight kernel to obtain a closed-form $C^3$ \emph{Convolutional LASSO} that restores Newton-type quadratic convergence without sacrificing the kink- setecting character of $\ell_1$ regularization. Theoretical results are verified numerically and benchmarked against ADMM/IRLS on NVIDIA's 2013--2018 daily closing prices, where the sparse filter isolates genuine growth-regime breaks---chiefly the November 2016 post-earnings acceleration---cleanly separated from the smooth $L_2$ trend.

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