We decompose a time series into two parts, one is the trend, and the other is the seasonality.
$$ y_t=g_t+c_t $$
, where \(g_t\) is the trend, and \(c_t\) represents seasonality. Or, one can understand those two components as a low-frequent part, and a high-frequent part.
The filer tells that,
$$\min_{g} \sum_i^N (y_i-g_i)^2+\lambda \sum_i^{N-1} (g_i^2-2g_{i+1}+g_{i+2}^2 )^2$$
$$\min_{g} \sum_i^N (y_i-g_i)^2+\lambda \sum_i^{N-1} [(g_i-g_{i+1})-(g_{i+1}-g_{i+2}]^2$$
,which can be also written as,
$$\min_{g} || y-g||^2+\lambda||\nabla^2 g||$$
We can see the first term represents how far the trend term \(g\) is away from the original series \(y\), and the second term means to smooth the trend term \(g\).
$$ g=argmin_g || y-g||^2+\lambda||\nabla^2 g||^2$$
We replace \( \nabla^2 g \) by \(Dg\).
$$ || y-g||^2+\lambda||D g||$$
$$ (y-g)^T (y-g) +\lambda (Dg)^T(Dg)$$
We take the first gradient (f.o.c.) to solve for the trend term \(g\).
$$ -(y-g)+\lambda D^TDg =0$$
Therefore,
$$ y=(I+\lambda D^T D)g $$
and,
$$ g=(I+\lambda D^T D)^{-1}y $$