A solution is given to Stechkin's problem on the best approximation on the real axis of differentiation operators of fractional (more precisely, real) order $k$ in the space $L_2$ by bounded linear operators from the space $L$ to the space $L_2$ on the class of functions whose fractional derivative of order $n$, $0\le k<n,$ is bounded in the space $L_2$. An upper estimate is obtained for the best constant in the corresponding Kolmogorov inequality. It is shown that the well-known Stechkin lower estimate for the value of the problem of approximating the differentiation operator via the best constant in the Kolmogorov inequality is strict in this case; in other words, Stechkin's problem and the Kolmogorov inequality are not consistent.
Keywords: fractional order differentiation operator, Stechkin's problem, Kolmogorov inequality, Carlson inequality
Received June 19, 2024
Revised September 17, 2024
Accepted September 23, 2024
Vitalii Vladimirovich Arestov, Dr. Phys.-Math. Sci., Prof., Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia; Ural Federal University, Yekaterinburg, 620000 Russia, e-mail: vitalii.arestov@urfu.ru
REFERENCES
1. Stein E.M., Weiss G. Introduction to Fourier analysis on euclidean spaces, Princeton, Princeton Univ. Press, 1971, 312 p. ISBN: 9780691080789 . Translated to Russian under the title Vvedenie v garmonicheskii analiz na evklidovykh prostranstvakh, Moscow, Mir Publ., 1974, 333 p.
2. Shilov G.E. Matematicheskii analiz. Vtoroi spetsial’nyi kurs [Mathematical analysis. The second special course]. Moscow, Nauka Publ., 1965, 328 p.
3. Buslaev A.P., Magaril-Il’yaev G.G., Tikhomirov V.M. Existence of extremal functions in inequalities for derivatives. Math. Notes, 1982, vol. 32, pp. 898–904. doi: 10.1007/BF01145874
4. Szőkefalvi-Nagy B. Über integralunglechungen zwischen einer funktion und ihrer ableitung. Acta Sci. Math., 1941, vol. 10, pp. 64–74.
5. Levin V.I. Exact constants in inequalities of the Carlson type. Doklady Akad. Nauk SSSR, 1948, vol. 59, no. 4, pp. 635–638 (in Russian).
6. Arestov V.V. Approximation of linear operators, and related extremal problems. Proc. Steklov Inst. Math., 1977, vol. 138, pp. 31–44.
7. Osipenko K.Yu. Sharp Carlson type inequalities with many weights. Proc. Steklov Inst. Math., 2023, vol. 323, suppl. 1, pp. S211–S221. doi: 10.1134/S0081543823060184
8. Hardy G.H., Littlewood J.E. Contribution to the arithmetic theory of series. Proc. London Math. Soc. 2, 1912, vol. 11, no. 1, p. 411–478.
9. Arestov V.V., Gabushin V.N. Best approximation of unbounded operators by bounded operators. Russian Math. (Iz. VUZ), 1995, vol. 39, no. 11, pp. 38–63.
10. Arestov V.V. Approximation of unbounded operators by bounded operators and related extremal problems. Russian Math. Surveys, 1996, vol. 51, iss. 6, pp. 1093–1126. doi: 10.1070/RM1996v051n06ABEH003001
11. Babenko V.F., Korneichuk N.P., Kofanov V.A., Pichugov S.A. Neravenstva dlya proizvodnykh i ikh prilozheniya [Inequalities for derivatives and their applications]. Kiev, Naukova Dumka, 2003, 590 p.
12. Tikhomirov V.M., Magaril-Il’yaev G.G. Neravenstva dlya proizvodnykh [Inequalities for derivatives]. In: Izbrannye Trudy. Matematika i Mekhanika [Selected Works. Mathematics and Mechanics]. Moscow, Nauka Publ., 1985, pp. 387–390.
13. Kolmogorov A.N. O neravenstvakh mezhdu verkhnimi granyami posledovatel’nykh proizvodnykh proizvol’noy funktsii na beskonechnom intervale [On inequalities between upper bounds of successive derivatives of an arbitrary function on an infinite interval]. In: Izbrannyye tr. Matematika, mekhanika [Selected proceedings. Mathematics, mechanics]. Moscow, Nauka Publ., 1985, pp. 252–263.
14. Arestov V.V. Inequalities for fractional derivatives on the half-line. Approx. Theory, Warsawa, Banach Cent. Publ., 1979, vol. 4, pp. 19–34. doi: 10.4064/-4-1-19-34
15. Babenko V.F., Parfinovich N.V., Pichugov S.A. Kolmogorov-type inequalities for norms of Riesz derivatives of functions of several variables with Laplacian bounded in $L_\infty$ and related problems. Math. Notes, 2014, vol. 95, no. 1, pp. 3–14. doi: 10.1134/S0001434614010015
16. Babenko V., Kovalenko O., Parfinovych N. On approximation of hypersingular integral operators by bounded ones. J. Math. Anal. Appl., 2022, vol. 513, iss. 2, art. 126215. doi: 10.1016/j.jmaa.2022.126215
17. Stechkin S.B. Best approximation of linear operators. Math. Notes, 1967, vol. 1, iss. 2, pp. 91–99. doi: 10.1007/BF01268056
18. Arestov V.V. Approximation of operators invariant with respect to a shift. Proc. Steklov Inst. Math., 1977, vol. 138, pp. 45–74.
19. Hörmander L. Estimates for translation invariant operators in $L_p$ spaces. Acta Math., 1960, vol. 104, nos. 1–2, pp. 93–140. doi: 10.1007/BF02547187
20. Arestov V.V., Akopyan R.R. Stechkin’s problem on the best approximation of an unbounded operator by bounded ones and related problems. Trudy Inst. Mat. i Mekh. UrO RAN, 2020, vol. 26, no. 4, pp. 7–31 (in Russian). doi: 10.21538/0134-4889-2020-26-4-7-31
21. Arestov V.V., Filatova M.A. Best approximation of the differentiation operator in the space $L_2$ on the semiaxis. J. Approx. Theory, 2014, vol. 187, pp. 65–81. doi: 10.1016/j.jat.2014.08.001
22. Arestov V.V. Approximation of differentiation operators by bounded linear operators in Lebesgue spaces on the axis and related problems in the spaces of (p,q)-multipliers and their predual spaces. Ural Math. J., 2023, vol. 9, no. 2, pp. 4–27. doi: 10.15826/umj.2023.2.001
23. Arestov V.V. Best approximation of translation invariant unbounded operators by bounded linear operators. Proc. Steklov. Inst. Math., 1994, vol. 198, pp. 1–16.
24. Buslaev A.P. Approximation of a differentiation operator. Math. Notes, 1981, vol. 29, no. 5, pp. 372–378. doi: 10.1007/BF01158361
25. Timoshin O.A. Best approximation of the operator of second mixed derivative in the metrics of L and C on the plane. Math. Notes, 1984, vol. 36, iss. 3, pp. 683–686. doi: 10.1007/BF01141940
26. Timoshin O.A. Sharp inequalities between norms of partial derivatives of second and third order. Dokl. Akad. Nauk, 1995, vol. 344, no. 1, pp. 20–22.
27. Timofeev V.G. Landau inequality for function of several variables. Math. Notes, 1985, vol. 37, iss. 5, pp. 369–377. doi: 10.1007/BF01157968
28. Magaril-Il’yaev G.G., Osipenko K.Yu., Sivkova E.O. Optimal recovery of pipe temperature from inaccurate measurements. Proc. Steklov Inst. Math., 2021, vol. 312, iss. 1, pp. 207–214. doi: 10.1134/S0081543821010120
29. Subbotin Yu.N., Taikov L.V. Best approximation of a differentiation operator in $L_2$-space. Math. Notes, 1968, vol. 3, no. 2, pp. 100–105. doi: 10.1007/BF01094328
Cite this article as: V.V. Arestov. A variant of Stechkin’s problem on the best approximation of a fractional order differentiation operator on the axis. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2024, vol. 30, no. 4, pp. 37–54.
