Skip to main content
Log in

Sharp inequalities for approximations of classes of periodic convolutions by odd-dimensional subspaces of shifts

  • Published:
Mathematical Notes Aims and scope Submit manuscript

Abstract

Sharp Akhiezer-Krein-Favard-type inequalities for classes of periodic convolutions with kernels that do not increase oscillation are obtained. A large class of approximating odd-dimensional subspaces constructed from uniform shifts of one function with extremal widths is specified. As a corollary, sharp Jackson-type inequalities for the second-ordermodulus of continuity are derived.

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. J. Favard, “Sur les meilleurs procédés d’approximation de certaines classes des fonctions par des polynômes trigonométriques,” Bull. Sci. Math. (2) 61, 209–224, 243–256 (1937). !! “polynomes” instead of “polynômes” in the Russian original !!

    Google Scholar 

  2. N. Achyeser and M. Krein, “Sur la meilleure approximation des fonctions periodiques derivables au moyen de sommes trigonometrqiues,” C. R. (Dokl.) Acad. Sci. URSS, n. Ser. 15, 107–111 (1937).

    Google Scholar 

  3. S. M. Nikol’skii, “Approximation of functions in the mean by trigonometric polynomials,” Izv. Akad. Nauk SSSR. Ser.Mat. 10, 207–256 (1946).

    Google Scholar 

  4. V. M. Tikhomirov, “Best methods of approximation and interpolation of differentiable functions in the space C [−1,1],” Mat. Sb. 80(2), 290–304 (1969.).

    MathSciNet  Google Scholar 

  5. N. P. Korneichuk, “Exact error bound of approximation by interpolating splines on L-metric on the classes W r p (1 ≤ p < ∞) of periodic functions,” Anal. Math. 3(2), 109–117 (1977).

    Article  MathSciNet  Google Scholar 

  6. N. P. Korneichuk, Exact Constants in Approximation Theory (Nauka, Moscow, 1987) [in Russian].

    Google Scholar 

  7. N. P. Korneichuk, Splines in Approximation Theory (Nauka, Moscow, 1984) [in Russian].

    Google Scholar 

  8. A. A. Ligun, “Inequalities for upper bounds of functionals,” Anal.Math. 2(1), 11–40 (1976).

    Article  MathSciNet  Google Scholar 

  9. A. Pinkus, n-Widths in Approximation Theory, in Ergeb. Math. Grenzgeb. (3) (Springer-Verlag, Berlin, 1985), Vol. 7.

    Google Scholar 

  10. V. M. Tikhomirov, “On extremal subspaces for classes of functions defined by kernels that do not increase oscillation,” VestnikMoskov. Univ. Ser. I Mat., Mekh., No. 4, 16–19 (1997) [Moscow Univ. Math. Bull. 52, no. 4, 17–20 (1997)].

  11. V. F. Babenko, “Extremal problems in approximation theory and inequalities for rearrangements,” Dokl. Akad. Nauk SSSR 290(5), 1033–1036 (1986).

    MathSciNet  Google Scholar 

  12. V. F. Babenko, “Approximation of convolution classes,” Sib. Mat. Zh. 28(5), 6–21 (1987).

    MathSciNet  Google Scholar 

  13. V. M. Tikhomirov, “Diameters of sets in functional spaces and the theory of best approximations,” Uspekhi Mat. Nauk 15(3), 81–120 (1960) [RussianMath. Surveys 15 (3), 75–111 (1960)].

    Google Scholar 

  14. O. L. Vinogradov, “Analog of the Akhiezer-Krein-Favard sums for periodic splines of minimal defect,” Probl. Mat. Anal. 25, 29–56 (2003) [J. Math. Sci. (N. Y.) 114 (5), 1608–1627 (2003)].

    MATH  Google Scholar 

  15. V. L. Velikin, “On limiting constraints between approximations of periodic functions by splines and trigonometric polynomials,” Dokl. Akad. Nauk SSSR 258(3), 525–529 (1981).

    MathSciNet  Google Scholar 

  16. Nguyĭen Thi Thiěu Hoa, “The operator D(D 2 + 12) ... (D 2 + n 2) and trigonometric interpolation,” Anal. Math. 15(4), 291–306 (1989).

    MathSciNet  Google Scholar 

  17. S. Karlin, Total Positivity, (Stanford Univ. Press, Stanford, Calif., 1968), Vol. 1.

    MATH  Google Scholar 

  18. A. F. Timan, Theory of Approximation of Functions of a Real Variable (Fizmatgiz, Moscow, 1960; Oxford, Pergamon, 1963).

    Google Scholar 

  19. A. Zygmund, Trigonometric Series, 2nd ed. (Cambridge Univ. Press, New York, 1959; Mir, Moscow, 1965), Vol. 2.

    MATH  Google Scholar 

  20. A. K. Kushpel’, “Sharp estimates for the widths of convolution classes,” Izv. Akad. Nauk SSSR Ser. Mat. 52(6), 1305–1322 (1988) Math. USSR-Izv. 33 (3), 631–649 (1989).

    Google Scholar 

  21. V. V. Zhuk, “On sharp inequalities between best approximations and moduli of continuity,” Vestnik Leningrad. Univ. Mat. Mekh. Astronom., No. 1, 21–26 (1974).

  22. V. V. Shalaev, “To the approximation of continuous periodic functions by trigonometric polynomials,” in Research on Modern Problems of Summation and Approximation of Functions and Their Applications, (Dnepropetrovsk. Gos. Univ., Dnepropetrovsk, 1977), No. 8, pp. 39–43 [in Russian].

    Google Scholar 

  23. V. V. Zhuk, Approximation of Periodic Functions (Izd. Leningrad. Gos. Univ., Leningrad, 1982) [in Russian].

    MATH  Google Scholar 

  24. O. L. Vinogradov and V.V. Zhuk, “Sharp estimates for deviations of linear approximationmethods for periodic functions by linear combinations of moduli of continuity of different order,” Probl. Mat. Anal. 25, 57–97 (2003) [J. Math. Sci. (N. Y.) 114 (5), 1628–1656 (2003)].

    MATH  Google Scholar 

  25. O. L. Vinogradov, “A sharp Jackson-type inequality for spline approximation and the second-ordermodulus of continuity,” in Modern Problems of Function Theory and Their Applications (Saratov Winter School), Saratov, Russia, 2004, (Izd. Gos. Uchebno-Nauchn. Tsentra “Kolledzh”, Saratov, 2004), pp. 45–46.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to O. L. Vinogradov.

Additional information

Original Russian Text © O. L. Vinogradov, 2009, published in Matematicheskie Zametki, 2009, Vol. 85, No. 4, pp. 569–584.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Vinogradov, O.L. Sharp inequalities for approximations of classes of periodic convolutions by odd-dimensional subspaces of shifts. Math Notes 85, 544–557 (2009). https://doi.org/10.1134/S0001434609030250

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0001434609030250

Key words

Navigation