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DOI: https://doi.org/10.33048/mattrudy.2019.22.102

Abstract: We consider the problem on shape-preserving interpolation by classical cubic splines. Namely, we consider conditions guaranteeing that, for a positive function (or a function whose $k$th derivative is positive), the cubic spline (respectively, its $k$th derivative) is positive. We present a survey of known results, completely describe the cases in which boundary conditions are formulated in terms of the first derivative, and obtain similar results for the second derivative. We discuss in detail mathematical methods for obtaining sufficient conditions for shape-preserving interpolation. We also develop such methods, which allows us to obtain general conditions for a spline and its derivative to be positive. We prove that, for a strictly positive function (or a function whose derivative is positive), it is possible to find an interpolant of the same sign as the initial function (respectively, its derivative) by thickening the mesh.

Key words: cubic spline, shape-preserving interpolation, monotonicity, convexity.

Funding agency	Grant number
Russian Academy of Sciences - Federal Agency for Scientific Organizations	0314-2016-0013
The study was carried out within the framework of the state contract of the Sobolev Institute of Mathematics, SB RAS (project 0314-2016-0013)

Received: 13.09.2018
Revised: 04.02.2019
Accepted: 27.02.2019

English version:
Siberian Advances in Mathematics, 2019, Volume 29, Issue 4, Pages 231–262
DOI: https://doi.org/10.3103/S1055134419040011

Bibliographic databases:

Citation: V. V. Bogdanov, Yu. S. Volkov, “Shape-preservation conditions for cubic spline interpolation”, Mat. Tr., 22:1 (2019), 19–67; Siberian Adv. Math., 29:4 (2019), 231–262

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This publication is cited in the following articles:

1. Yu. S. Volkov, V. V. Bogdanov, “On error estimates of local approximation by splines”, Siberian Math. J., 61:5 (2020), 795–802          
2. D. Igaz, K. Sinka, P. Varga, G. Vrbicanova, E. Aydin, A. Tarnik, “The evaluation of the accuracy of interpolation methods in crafting maps of physical and hydro-physical soil properties”, Water, 13:2 (2021), 212      
3. Yu. S. Volkov, “Shape Preserving Conditions for Integro Quadratic Spline Interpolation in the Mean”, Proc. Steklov Inst. Math. (Suppl.), 319, suppl. 1 (2022), S291–S297          

Citing articles in Google Scholar: Russian citations, English citations
Related articles in Google Scholar: Russian articles, English articles