Quadratic smoothing spline for interval averages (${\sum }_i(Y_i-L_{i}g{)}^2+\lambda {\int }_a^b(g^{\prime }{)}^2$)

Let $L_{i}g$ be the average of $g$ on successive knot intervals. Minimizing the displayed loss over $C^1[a,b]$ gives a quadratic spline with constant tails outside the first and last knots. To prove the reduction, choose the unique constant-tail quadratic spline $q$ with the same interval averages as a candidate $g$, and put $v=g-q$. Since $q^{\prime \prime }$ is constant on every knot interval, $q^{\prime }$ is continuous and vanishes on both tails, integration by parts gives $\int q^{\prime }{v}^{\prime }=-{\sum }_i{q}^{\prime \prime }{|}_{I_i}{\int }_{I_i}v=0$. Consequently the data loss is unchanged and $\int (g^{\prime }{)}^2=\int (q^{\prime }{)}^2+\int (v^{\prime }{)}^2$. The penalty decreases strictly unless $v$ is constant; its zero interval averages then force $v=0$. The finite-dimensional quadratic optimization on this spline space also gives a unique minimizer because the interval-average map is injective.