On the relationship between the second divided difference and the second derivative in the problem of extremal interpolation in the mean
https://doi.org/10.22405/2226-8383-2025-26-1-131-141
Abstract
In the paper, the general problem of extremal functional interpolation in the mean for real functions that have derivative of order 𝑛 almost everywhere is formulated on an arbitrary partition Δ = {𝑥𝑘}∞𝑘
=−∞ of the real axis. It is required to find the smallest value of the 𝐿∞-
norm of the 𝑛-derivative among functions that interpolate in the mean (with averaging intervals of length 2ℎ) any sequence of real numbers 𝑦 = {𝑦𝑘}∞𝑘 =−∞ from a class 𝑌 of sequences whose divided differences of order 𝑛 are bounded from above on such a grid. In this paper, the problem is considered in the case of 𝑛 = 2. We give the above and below estimates for the 𝐿∞-norm of the second derivative in terms of grid steps ℎ𝑘 = 𝑥𝑘+1−𝑥𝑘 provided that 2ℎ ⩽ ℎ = inf𝑘 ℎ𝑘. The
obtained results are developments is research of Yu. N. Subbotin, the author and S. I. Novikov in the well-known Yanenko—Stechkin problem of extremal functional interpolation. This problem was put in the early 60-s years of the last century for the case of the uniform grid.
About the Author
Valerii Trifonovich Shevaldin
Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences
Russian Federation
Russian Federation
doctor of physical and mathematical sciences
References
1. Subbotin Yu. N. “On the connection between finite differences and corresponding derivatives”, Proc. Steklov Inst. Math., 1965, vol. 78, pp. 24–42 (in Russian).
2. Subbotin Yu. N. “Functional interpolation in the mean with smallest 𝑛-derivative”, Proc. Steklov Inst. Math., 1967, vol. 88, pp. 31–63.
3. Subbotin Yu. N. “Extremal problem of functional interpolation, and mean interpolation splines”, Proc. Steklov Inst. Math., 1977, vol. 138, pp. 127–185.
4. Subbotin Yu. N. “Extremal functional interpolation in the mean with least value of thenth derivative for large averaging intervals”, Math. Notes, 1996, vol. 59, no. 1, pp. 83–96. doi: 10.1007/BF02312469 .
5. Subbotin Yu. N. “Extremal 𝐿𝑝 interpolation in the mean with intersecting averaging intervals”, Izv. Math., 1997, vol. 61, no. 1, pp. 183–205. doi: 10.1070/im1997v061n01ABEH000110 .
6. Subbotin Yu. N., Novikov S. I., Shevaldin V. T. “Extremal function interpolation and splines”, Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2018, vol. 24, no. 3, pp. 200–225. doi: 10.21538/0134-4889-2018-24-3-200-225 (in Russian).
7. Favard J. “Sur interpolation”, J. Math. Pures Appl., 1940, vol. 19, no. 9, pp. 281–306. 8. de Boor C. “How small can one make the derivatives of an interpolating function?” J. Approx. Theory, 1975, vol. 13, no. 2, pp. 105–116.
8. de Boor C. “A smooth and local interpolant with small 𝑘-th derivative”, Numerical solutions of boundary value problems for ordinary differential equations. N.-Y: Acad. Press, 1975, pp. 177–197.
9. Kunkle Th. “Favard’s interpolation problem in one or more variables”, Constructive Approx., 2002, vol. 18, no. 4, pp. 467–478. doi: 10.1007/s00365-001-0015-7 .
10. Novikov S. I., Shevaldin V. T. “On the connection between the second divided difference and the second derivative”, Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2020, vol. 26, no. 2, pp. 216–224. doi: 10.21538/0134-4889-2020-26-2-216-224 (in Russian).
11. Novikov S. I., Shevaldin V. T. “Extremal interpolation on the semiaxis with the smallest norm of the third derivative”, Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2020, vol. 26, no. 4, pp. 210–223. doi: 10.21538/0134-4889-2020-26-4-210-223 (in Russian).
12. Shevaldin V. T. “Extremal interpolation with the least value of the norm of the second derivative in 𝐿𝑝(R)”, Izvestiya: Mathematics, 2022, vol. 86, no. 1, pp. 203–219. doi: 10.1070/IM9125 .
13. Volkov Yu. S. “A remark on the connection between the second divided difference and the second derivative”, Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2021, vol. 27, no. 1, pp. 19–21. doi: 10.21538/0134-4889-2021-27-1-19-21 (in Russian).
14. Shevaldin V. T. “Some problems of extremal interpolation in the mean for linear differential operators”, Proc. Steklov Inst. Math., 1985, vol. 164, pp. 233–273.
15. Shevaldin V. T. “Extremal interpolation in the mean with overlapping averaging intervals and ℒ-splines”, Izv. Math., 1998, vol. 62, no. 4, pp. 201–224. doi: 10.4213/im193 .
16. Shevaldin V. T. “Extremal interpolation in the mean with overlapping averaging intervals and the smallest norm of a linear differential operator”, Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2023, vol. 29, no. 1, pp. 219–232. doi: 10.21538/0134-4889-2023-29-1-219-232 (in Russian).
Review
For citations:
Shevaldin V.T. On the relationship between the second divided difference and the second derivative in the problem of extremal interpolation in the mean. Chebyshevskii Sbornik. 2025;26(1):131-141. (In Russ.) https://doi.org/10.22405/2226-8383-2025-26-1-131-141
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