On the Convergence of Cubic Interpolating Splines

Given n ≥ 2, a partition Δn = {0 = xo < x1 < ... < xn =1} of [0,1], and a function f ∈ C[0,1] = {g ∈ C[0,1]: g(0) = g(1)}, let Pnf be the periodic cubic spline interpolating f at {xi} o n . (For a precise definition of Pnf, see §2). The following question has received considerable attention recently (see e.g., [1 – 4, 6 – 9, 11 – 12] and references therein): Given a sequence ‹Δn› of partitions of [0, 1] with $$ \left\| {{\Delta _n}} \right\| = \mathop {\max }\limits_{1 \leqslant i \leqslant n} ({x_i} - {x_{i - 1}}) \to 0\quad as\quad n \to \infty $$ (1.1) , what further conditions on ‹Δn› are needed to guarantee that the sequence of spline interpolants ‹Pnf› converges uniformly to f as n → ∞? The study of this question was stimulated by the discovery by Nord [9] of an example of a sequence ‹Δn› satisfying (1.1) and a function f ∈ C[0,1] such that ‖f - Pnf‖ ↛ 0 as n → ∞.