Abstract:
Let the sequence of nets Δn be such that limn→∞maxih(n)i=0, where h(n)i are the lengths of the segments of a net. The bound max|i−j|=1h(n)ih(n)j1−α⩽R<∞ is necessary in order that interpolating parabolic and cubic splines converge for any function in C(α=0) or Cα(0<α<1), where Cα is the class of functions satisfying a Lipschitz condition of order α. It is also shown that this bound cannot essentially be weakened.
Citation:
N. L. Zmatrakov, “A necessary condition for convergence of interpolating parabolic and cubic splines”, Mat. Zametki, 19:2 (1976), 165–178; Math. Notes, 19:2 (1976), 100–107
This publication is cited in the following 3 articles:
I. A. Blatov, A. I. Zadorin, E. V. Kitaeva, “Cubic spline interpolation of functions with high gradients in boundary layers”, Comput. Math. Math. Phys., 57:1 (2017), 7–25
I. A. Blatov, A. I. Zadorin, E. V. Kitaeva, “Parabolic spline interpolation for functions with large gradient in the boundary layer”, Siberian Math. J., 58:4 (2017), 578–590
Yu. S. Volkov, Yu. N. Subbotin, “50 years to Schoenberg's problem on the convergence of spline interpolation”, Proc. Steklov Inst. Math. (Suppl.), 288, suppl. 1 (2015), 222–237