A solution is given to Stechkin's problem on the best approximation in the uniform norm on the real axis of differentiation operators of fractional (more precisely, real) order $k$ by bounded linear operators from the space $L^2$ to the space $C$ on the class of functions $\mathcal{Q}^n$ whose Fourier transform of the $n$th-order fractional derivative, $0\le k<n,$ is integrable. The corresponding exact Kolmogorov inequality is given. A solution is obtained to the problem of optimal recovery of the differentiation operator offractional order $k$ on functions of the class $\mathcal{Q}^n$ defined with a known error in the space $L^2.$
Keywords: fractional-order differentiation operator, Stechkin's problem, Kolmogorov inequality, optimal differеntiation.
Received March 13, 2025
Revised April 3, 2025
Accepted April 7, 2025
Published online May 30, 2025
Funding Agency:
This work was supported by Russian Science Foundation, project 25-21-00118,
https://rscf.ru/project/25-21-00118/ .
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. Stechkin S.B. Best approximation of linear operators. Math. Notes, 1967, vol. 1, no. 2, pp. 91–99. https://doi.org/10.1007/BF01268056
2. 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.
3. Arestov V.V. Approximation of unbounded operators by bounded operators and related extremal problems. Russian Math. Surv., 1996, vol. 51, no. 6, pp. 1093–1126. https://doi.org/10.1070/RM1996v051n06ABEH003001
4. 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. https://doi.org/10.1016/j.jat.2014.08.001
5. 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. Mekh. UrO RAN, 2020, vol. 26, no. 4, pp. 7–31 (in Russian). https://doi.org/10.21538/0134-4889-2020-26-4-7-31
6. Babenko V., Kovalenko O., Parfinovych N. On approximation of hypersingular integral operators by bounded ones. J. Math. Anal. Appl., 2022, vol. 513, no. 2, art. no. 126215. https://doi.org/10.1016/j.jmaa.2022.126215
7. 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. https://doi.org/10.15826/umj.2023.2.001
8. 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, 591 p.
9. 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.
10. Arestov V.V. Uniform regularization of the problem of calculating the values of an operator. Math. Notes, 1977, vol. 22, no. 2, pp. 618–626. https://doi.org/10.1007/BF01780971
11. Ivanov V.K., Vasin V.V., Tanana V.P. Theory of linear ill-posed problems and its applications. In: Inverse and ill-posed problems series, vol. 36. Utrecht etc., VSP, 2002, 281 p. https://doi.org/10.1515/9783110944822 . Original Russian text published in Ivanov V.K., Vasin V.V., Tanana V.P. Teoriya lineinykh nekorrektnykh zadach i ee prilozheniya, Moscow, Nauka Publ., 1978, 206 p.
12. Osipenko K.Yu. Vvedeniye v teoriyu optimal’nogo vosstanovleniya [Introduction to the theory of optimal recovery]. St. Petersburg, Lan’, 2022, 388 p. ISBN: 978-5-507-44358-1 .
13. 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.
14. Shilov G.E. Matematicheskii analiz. Vtoroi spetsial’nyi kurs [Mathematical analysis. The second special course]. Moscow, Nauka Publ., 1965, 328 p.
15. Gabushin V.N. Inequalities for the norms of a function and its derivatives in metric $L_p$. Math. Notes, 1967, vol. 1, no. 3, pp. 194–198. https://doi.org/10.1007/BF01098882
16. Szőkefalvi-Nagy B. Über integralunglechungen zwischen einer funktion und ihrer ableitung. Acta Sci. Math., 1941, vol. 10, pp. 64–74.
17. Gabushin V.N. Exact constants in inequalities between norms of derivatives of functions. Math. Notes, 1968, vol. 4, no. 2, pp. 624–630. https://doi.org/10.1007/BF01094963
Cite this article as: V.V. Arestov. Best approximation of a fractional-order differentiation operator in the uniform norm on the axis on the class of functions with integrable Fourier transform of the highest derivative. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2025, vol. 31, no. 3, pp. 47–63
