Convergence and Gibbsʹ phenomenon in cubic spline interpolation of discontinuous functions

Convergence of cubic spline interpolation for discontinuous functions are investigated. It is shown that the complete cubic spline interpolation of the Heaviside step function converges in the Lp-norm at rate O(h1p) for quasi-uniform meshes when 1⩽p<∞, and diverges in the L∞-norm when the uniform meshes are used. No matter how small the uniform mesh size is, the complete cubic spline interpolation always oscillates near the discontinuity. Although this oscillation decays exponentially away from the discontinuous point, the maximum overshoot is not decreasing. Especially, we obtain the asymptotic maximum overshoot when the uniform mesh size goes to zero. The knowledge on the Heaviside function is utilized to discuss convergence properties of cubic spline interpolation for functions with isolated discontinuous points.