Minimum Dirichlet energy of a linear interpolant (${\int }_a^b|u^{\prime }{|}^2={\int }_a^b|s^{\prime }{|}^2+{\int }_a^b|u^{\prime }-s^{\prime }{|}^2$)

For nodes $a=x_0<\cdots <x_n=b$ and prescribed real node values, let $s$ be the linear spline joining them. Among all absolutely continuous functions $u$ with the same node values and square-integrable derivative, $s$ uniquely minimizes the Dirichlet energy ${\int }_a^b|u^{\prime }{|}^2$.

On each node interval, $s^{\prime }$ is constant and $u-s$ vanishes at both endpoints. The fundamental theorem of calculus therefore gives $\int s^{\prime }(u^{\prime }-s^{\prime })=s^{\prime }[(u-s)(x_{i+1})-(u-s)(x_i)]=0$. Expanding the square and summing proves the displayed energy decomposition. Its final term is nonnegative; equality forces $u^{\prime }=s^{\prime }$ almost everywhere and the endpoint values fix the additive constant, proving uniqueness. In particular, the minimum is ${\sum }_i(u(x_{i+1})-u(x_i){)}^2/(x_{i+1}-x_i)$. If all interior node values are zero and there is at least one interior node, only the two endpoint intervals contribute to this minimum.