Skip to main content
Log in

Best Uniform Approximation of the Differentiation Operator by Operators Bounded in the Space L2

  • Published:
Proceedings of the Steklov Institute of Mathematics Aims and scope Submit manuscript

Abstract

We give a solution of the problem on the best uniform approximation on the number axis of the first-order differentiation operator on the class of functions with bounded second derivative by linear operators bounded in the space L2. This is one of the few cases of the exact solution of the problem on the approximation of the differentiation operator in some space with the use of approximating operators that are bounded in another space. We obtain a related exact inequality between the uniform norm of the derivative of a function, the variation of the Fourier transform of the function, and the L-norm of its second derivative. This inequality can be regarded as a nonclassical variant of the Hadamard-Kolmogorov inequality.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. S. B. Stechkin, “Best approximation of linear operators,” Math. Notes 1 (2), 91–99 (1967).

    Article  MathSciNet  MATH  Google Scholar 

  2. V. V. Arestov, “Approximation of unbounded operators by bounded operators and related extremal problems,” Russ. Math. Surv. 51 (6), 1093–1126 (1996).

    Article  MATH  Google Scholar 

  3. V. F. Babenko, N. P. Korneichuk, V. A. Kofanov, and S. A. Pichugov, Inequalities for Derivatives and Their Applications (Naukova Dumka, Kiev, 2003) [in Russian].

    Google Scholar 

  4. V. K. Ivanov, V. V. Vasin, and V. P. Tanana, Theory of Linear Ill-Posed Problems and Its Applications (Nauka, Moscow, 1978; VSP, Utrecht, 2002).

    MATH  Google Scholar 

  5. V. N. Gabushin, “Best approximations of functionals on certain sets,” Math. Notes Acad. Sci. USSR 8 (5), 780–785 (1970).

    MathSciNet  Google Scholar 

  6. Yu. Babenko and D. Skorokhodov, “Stechkin’s problem for differential operators and functionals of first and second orders,” J. Approx. Theory 167, 173–200 (2013). doi 10.1016/j.jat.2012.12.003

    Article  MathSciNet  MATH  Google Scholar 

  7. V. F. Babenko, N. V. Parfinovich, and S. A. Pichugov, “Kolmogorov-type inequalities for norms of Riesz derivatives of functions of several variables with Laplacian bounded in Loo and related problems,” Math. Notes 95 (1), 3–14 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  8. E. Berdysheva and M. Filatova, “On the best approximation of the infinitesimal generator of a contraction semigroup in a Hilbert space,” Ural Math. J. 3 (2), 40–45 (2017). doi 10.15826/umj.2017.2.006

    Article  MathSciNet  Google Scholar 

  9. R. R. Akopyan, “Approximation of the differentiation operator on the class of functions analytic in an annulus,” Ural Math. J. 2017, 3 (2), 6–13. doi 10.15826/umj.2017.2.002

    Article  MathSciNet  Google Scholar 

  10. R. R. Akopyan, “Optimal recovery of a derivative of an analytic function from values of the function given with an error on a part of the boundary,” Anal. Math. 44 (1), 3–19 (2018). doi 10.1007/sl0476-018-0102-7

    Article  MathSciNet  MATH  Google Scholar 

  11. V. V. Arestov, “On the best approximation of the differentiation operator,” Ural Math. J. 1 (1), 20–29 (2015). doi 10.15826/umj.2015.1.002

    Article  MATH  Google Scholar 

  12. V. V. Arestov and M. A. Filatova, “Best approximation of the differentiation operator in the space L2 on the semiaxis,” J. Approx. Theory 187, 65–81 (2014). doi 10.1016/j.jat.2014.08.001

    Article  MathSciNet  MATH  Google Scholar 

  13. A. P. Buslaev, G. G. Magaril-IPyaev, and V. M. Tikhomirov, “Existence of extremal functions in inequalities for derivatives,” Math. Notes 32 (6), 898–904 (1982).

    Article  MathSciNet  MATH  Google Scholar 

  14. E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces (Princeton Univ. Press, Princeton, 1971; Mir, Moscow, 1974).

    MATH  Google Scholar 

  15. E. Landau, “Einige Ungleichungen für zweimal differentierbare Funktionen,” Proc. London Math. Soc. (2) 13, 43–49 (1913). doi 10.1112/plms/s2-13.1.43

    MATH  Google Scholar 

  16. A. N. Kolmogorov, “On inequalities between upper bounds of sequential derivatives of an arbitrary function on an infinite interval,” in Selected Works: Mathematics and Mechanics (Nauka, Moscow, 1985), pp. 252–263 [in Russian].

    Google Scholar 

  17. J. Hadamard, “Sur le module maximum d’une fonction et de ses dérivées,” S. R. des Séances Soc. Math. France 41, 68–72 (1914).

    Google Scholar 

  18. Yu. G. Bosse (G. E. Shilov), “On inequalities between derivatives,” in Collection of Works of Student Research Societies of Moscow State University (Izd. MGU, Moscow, 1937), Vol. 1, pp. 68–72 [in Russian].

    Google Scholar 

  19. V. N. Gabushin, “The best approximation of the differentiation operator in the metric of Lp,” Math. Notes 12 (5), 756–760 (1972).

    Article  MathSciNet  Google Scholar 

  20. V. M. Tikhomirov and G. G. Magaril-Il’yaev, “Inequalities for derivatives,” in A. N. Kolmogorov’s Selected Works (Nauka, Moscow, 1985), pp. 387–390 [in Russian].

    Google Scholar 

  21. V. V. Arestov, “Approximation of operators invariant with respect to a shift,” Proc. Steklov Inst. Math. 138, 45–74 (1975).

    MATH  Google Scholar 

  22. V. V. Arestov, “Approximation of operators of convolution type by bounded linear operators,” Proc. Steklov Inst. Math. 145, 1–18 (1981).

    MATH  Google Scholar 

  23. V. V. Arestov, “Approximation of invariant operators,” Math. Notes 34 (1), 489–499 (1983).

    Article  MathSciNet  MATH  Google Scholar 

  24. V. V. Arestov, “On the best approximation of the differentiation operator,” in Approximation of Functions by Polynomials and Splines (IMM AN SSSR, Sverdlovsk, 1985), pp. 3–14 [in Russian].

    Google Scholar 

  25. V.V. Arestov “Best approximation of translation invariant unbounded operators by bounded linear operators” Proc. Steklov Inst. Math.11–16 1994

    Google Scholar 

  26. L. Hormander, “Estimates for translation invariant operators in Lp spaces,” Acta Math. 104 (1–2), 93–140 (1960).

    Article  MathSciNet  MATH  Google Scholar 

  27. R. Larsen, An Introduction to the Theory of Multipliers (Springer, Berlin, 1971).

    Book  MATH  Google Scholar 

  28. S. B. Stechkin, “Inequalities between the norms of derivatives of an arbitrary function,” Acta Sci. Math. 26 (3–4), 225–230 (1965).

    MathSciNet  Google Scholar 

  29. V. V. Arestov, “On the best approximation of differentiation operators,” Math. Notes 1 (2), 100–103 (1967).

    Article  MathSciNet  MATH  Google Scholar 

  30. A. P. Buslaev, “Approximation of a differentiation operator,” Math. Notes 29 (5), 372–378 (1981).

    Article  MATH  Google Scholar 

  31. Y. Domar, “An extremal problem related to Kolmogoroff’s inequality for bounded functions,” Arkiv for Mat. 7 (5), 433–441 (1968).

    Article  MathSciNet  MATH  Google Scholar 

  32. Yu. N. Subbotin and L. V. Taikov, “Best approximation of a differentiation operator in L2-space,” Math. Notes 3 (2), 100–105 (1968).

    Article  Google Scholar 

  33. G. H. Hardy, J. E. Littlewood, and G. Polya, Inequalities (Cambridge Univ. Press, Cambridge, 1934; Inostrannaya Lit., Moscow, (1948).

    MATH  Google Scholar 

  34. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Moscow, Fizmatgiz, 1963; Academic, New York, 1980).

    MATH  Google Scholar 

  35. G. Polya and G. Szego, Problems and Theorems in Analysis (Springer, Berlin, 1972; Nauka, Moscow, 1978), Vol. 2.

    Book  MATH  Google Scholar 

  36. A. I. Markushevich, Theory of Analytic Functions (Nauka, Moscow, 1967), Vol. 1 [in Russian].

    Google Scholar 

  37. G. M. Fikhtengol’ts, A Course in Differential and Integral Calculus (Lan, St. Petersburg, 1997) [in Russian].

    Google Scholar 

  38. N. Dunford and J. Schwartz, Linear Operators: General Theory (Interscience, New York, 1958; Moscow, Editorial URSS, 2004).

    MATH  Google Scholar 

Download references

Funding

This work was supported by the Russian Foundation for Basic Research (project no. 18- 01-00336) and by the Russian Academic Excellence Project (agreement no. 02.A03.21.0006 of August 27, 2013, between the Ministry of Education and Science of the Russian Federation and Ural Federal University).

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to V. V. Arestov.

Additional information

Russian Text © The Author(s), 2018, published in Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2018, Vol. 24, No. 4, pp. 34–56.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Arestov, V.V. Best Uniform Approximation of the Differentiation Operator by Operators Bounded in the Space L2. Proc. Steklov Inst. Math. 308 (Suppl 1), 9–30 (2020). https://doi.org/10.1134/S0081543820020029

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0081543820020029

Keywords

Navigation