Convergence and Gibbs' phenomenon in the cubic spline interpolation of discontinuous functions (Q1379703)

Let \(s_n(t)\) be the complete cubic spline which interpolates the Heaviside function \(f\), defined by \(f(t)=0\) on \([-1,0)\), \(f(0)= {1\over 2}\) and \(f(t)=1\) on \((0,1]\), on a grid \(-1=t_0 < t_1< \dots <t_n=1\) which contains the point \(t_m=0\). It is shown that \(s_n\) converges to \(f\) in the \(L^p\) norm at a rate \(O(h^{1/p})\) for quasi-uniform meshes with maximum grid size \(h\) for \(1\leq p <\infty\). The rate \(O(h^{1/p})\) is optimal. There is no convergence for \(p=\infty\). The spline oscillates near the discontinuity. Although the oscillation decays exponentially away from the discontinuity, the maximum overshoot approaches for uniform meshes the value \(0.039\dots\) as \(h\rightarrow 0\). This ``Gibbs' phenomenon'' is apparently related to the boundary conditions. For cubic B-spline interpolation of \(f\) no oscillation occur.