Yu.S. Volkov. Shape preserving conditions for integro quadratic spline interpolation in the mean ... P. 71-77

Earlier, Yu. N. Subbotin considered the problem of interpolation in the mean, where the interpolated values of the function are replaced by averaged values on an interval. In his paper, the grid was uniform, but the space grid step could differ from the size of the averaging intervals. Subbotin investigated the existence of such splines and their convergence in different metrics. In the literature, splines of this type are also called integro or histosplines. The present paper considers such an interpolating in the mean quadratic spline on an arbitrary nonuniform grid of a closed interval, where the averaging intervals are the grid intervals. Sufficient conditions are obtained for the inheritance by an integro spline of certain properties of the approximated function such as nonnegativity, monotonicity, and convexity.

Keywords: integro spline, interpolation in the mean, shape preserving, quadratic splines

Received August 14, 2022

Revised September 5, 2022

Accepted September 12, 2022

Funding Agency: This work was carried out under a state contract of the Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences (project no. FWNF–2022–0015).

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


1.   Subbotin Yu.N. Extremal problems of functional interpolation and interpolation-in-the-mean splines. Proc. Steklov Inst. Math., 1977, vol. 138, pp. 127–185.

2.   Schoenberg I.J. Splines and histograms. In: Spline Functions and Approximation Theory, Meir A., Sharma A. (eds). ISNM: Internat. Ser. Numer. Math., vol. 21, Basel: Birkhäuser, 1973, pp. 277–327. doi: 10.1007/978-3-0348-5979-0_13 

3.   Wu J., Zhang X. Integro quadratic spline interpolation. Appl. Math. Model., 2015, vol. 39, no. 10-11, pp. 2973–2980. doi: 10.1016/j.apm.2014.11.015 

4.   Lang F.-G., Xu X.-P. On the superconvergence of some quadratic integro-splines at the mid-knots of a uniform partition. Appl. Math. Comput., 2018, vol. 338, pp. 507–514. doi: 10.1016/j.amc.2018.06.046 

5.   Zhanlav T. Shape preserving properties of some $C^2$ cubic spline approximations. Sci. Trans. NUM, 2000, vol. 7, pp. 21–35.

6.   Zhanlav T., Mijiddorj R. Convexity and monotonicity properties of the local integro cubic spline. Appl. Math. Comput., 2017, vol. 293, pp. 131–137. doi: 10.1016/j.amc.2016.08.017 

7.   Zhanlav T., Mijiddorj R.-O. Construction of a family of $C^1$ convex integro cubic splines. Comm. Math. Appl., 2020, vol. 11, no. 4, pp. 527–538. doi: 10.26713/cma.v11i4.1386 

8.   Volkov Yu.S., Bogdanov V.V., Miroshnichenko V.L., Shevaldin V.T. Shape-preserving interpolation by cubic splines. Math. Notes, 2010, vol. 88, no. 5-6, pp. 798–805. doi: 10.1134/S0001434610110209 

9.   Volkov Yu.S., Shevaldin V.T. Shape preserving conditions for quadratic spline interpolation in the sense of Subbotin and Marsden. Tr. Inst. Mat. Mekh. UrO RAN., 2012, vol. 18, no. 4, pp. 145–152 (in Russian).

10.   Bogdanov V.V., Volkov Yu.S. Shape-preservation conditions for cubic spline interpolation. Siberian Adv. Math., 2019, vol. 29, no. 4, pp. 231–262. doi: 10.3103/S1055134419040011 

11.   Miroshnichenko V.L. Convex and monotone spline interpolation. In: “Constructive theory of functions”: Proc. Int. Conf., Varna, 1984, B. Sendov, P. Petrushev, R. Maleev, S. Tashev (eds), Sofia: Publ. House Bulgar. Acad. Sci., 1984, pp. 610–620.

12.   Zavyalov Yu.S., Kvasov B.I., Miroshnichenko V.L. Metody splain-funkcii [Methods of spline functions]. Moscow: Nauka Publ., 1980, 352 p.

13.   Bogdanov V.V., Volkov Yu.S. Selection of parameters of generalized cubic splines with convexity preserving interpolation. Sib. Zh. Vychisl. Mat., 2006, vol. 9, no. 1, pp. 5–22 (in Russian).

14.   Bogdanov V.V., Volkov Yu.S. Near-optimal tension parameters in convexity preserving interpolation by generalized cubic splines. Numer. Algorithms, 2021, vol. 86, no. 2, pp. 833–861. doi: 10.1007/s11075-020-00914-9 

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

Cite this article as: Yu.S. Volkov. Shape preserving conditions for integro quadratic spline interpolation in the mean. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2022, vol. 28, no. 4, pp. 71–77; Proceedings of the Steklov Institute of Mathematics (Suppl.), 2022, Vol. 319, Suppl. 1, pp. S291–S197.