Yu.S. Volkov. Example of parabolic spline interpolation with bounded Lebesgue constant ... P. 85-91

We consider an example of a sequence of geometric data grids for which the Lebesgue constant of interpolation by the classical parabolic splines (Subbotin’s scheme) with periodic boundary conditions is unbounded; i.e., the interpolation process may diverge. We propose an alternative scheme for choosing the knots of a parabolic spline. In Subbotin’s scheme, knots of a spline are chosen as the midpoints of intervals of the data grid, whereas the location of a knot in the alternative scheme is defined proportionally to the lengths of the adjacent intervals (we consider two variants). In the case of interpolation by the alternative scheme in the example under consideration, the process converges for any continuous function; i.e., the Lebesgue constant is bounded. The sequence of grids studied in the paper is the “worst” from the viewpoint of the convergence of the interpolation process in the classical case.

Keywords: parabolic splines, interpolation, convergence, Lebesgue constant

Received September 01, 2018

Revised October 08, 2018

Accepted October 15, 2018

Funding Agency: This work was supported by Program 0314-2016-0013 for Fundamental Research of the Siberian Branch of the Russian Academy of Sciences.

Yuriy Stepanovich Volkov, Dr. Phys.-Math. Sci, Prof., Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk, 630090 Russia; Novosibirsk State University, Novosibirsk, 630090 Russia, e-mail: volkov@math.nsc.ru


1.   Stechkin S.B., Subbotin Yu.N. Splainy v vychislitel’noi matematike [Splines in numerical mathematics]. Moscow: Nauka Publ., 1976, 248 p.

2.   Volkov Yu.S. Divergence of interpolational splines of odd degree. Vychisl. Sist., 1984, vol. 106, pp. 41–56.

3.   Zmatrakov N.L. Convergence of an interpolation process for parabolic and cubic splines. Proc. Steklov Inst. Math., 1977, vol. 138, pp. 75–99.

4.   Volkov Yu.S. The general problem of polynomial spline interpolation. Proc. Steklov Inst. Math., 2018, vol. 300, suppl. 1, pp. 187–198. doi: 10.1134/S0081543818020190 .

5.   Schoenberg I.J., Whitney A. On P$\acute{\mathrm{o}}$lya frequency functions. III. The positivity of translation determinants with an application to the interpolation problem by spline curves. Trans. Amer. Math. Soc., 1953, vol. 74, no. 2, pp. 246–259. doi: 10.2307/1990881

6.   Richards F. B. Best bounds for the uniform periodic spline interpolation operator. J. Approxim. Theory., 1973, vol. 7, no. 3, pp. 302–317.

7.   Volkov Yu.S. A new method for constructing cubic interpolating splines. Comput. Math. Math. Phys., 2004, vol. 44, no. 2, pp. 215–224.

8.   Volkov Yu.S. Inverses of cyclic band matrices and the convergence of interpolation processes for derivatives of periodic interpolation splines. Num. Anal. Appl., 2010, vol. 3, no. 3, pp. 199–207. doi: 10.1134/S1995423910030018

9.   Volkov Yu.S., Miroshnichenko V.L. Norm estimates for the inverses of matrices of monotone type and totally positive matrices. Siberian Math. J., 2009, vol. 50, no. 6, pp. 982–987. doi: 10.1007/s11202-009-0108-2