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On exact Bernstein-type inequalities for splines

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Abstract

We establish new exact Bernstein-type and Kolmogorov-type inequalities. The main result of this work is the following exact inequality for periodic splines s of order r and defect 1 with nodes at the points iπ/n, iZ, nN:

$$\left\| {s^{(k)} } \right\|_q \leqslant n^{k + 1/p - 1/q} \frac{{\left\| {\varphi _{r - k} } \right\|_q }}{{\left\| {\varphi _r } \right\|_p }}\left\| s \right\|_p ,$$

where k, rN, k < r, p = 1 or p = 2, q > p, and ϕr is the perfect Euler spline of order r.

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References

  1. V. M. Tikhomirov, “Widths of sets in functional spaces and theory of best approximations,” Usp. Mat. Nauk, 15, No. 3, 81–120 (1960).

    Google Scholar 

  2. Yu. N. Subbotin, “On piecewise-polynomial interpolation,” Mat. Zametki, 1, No. 1, 24–29 (1967).

    MathSciNet  Google Scholar 

  3. N. P. Korneichuk, V. F. Babenko, and A. A. Ligun, Extremal Properties of Polynomials and Splines [in Russian], Naukova Dumka, Kiev (1992).

    Google Scholar 

  4. A. A. Ligun, “Exact inequalities for spline functions and best quadrature formulas for some classes of functions,” Mat. Zametki, 19, No. 6, 913–926 (1976).

    MATH  MathSciNet  Google Scholar 

  5. A. A. Ligun, “On inequalities for the norms of derivatives of periodic functions,” Mat. Zametki, 33, No. 3, 385–391 (1983).

    MathSciNet  Google Scholar 

  6. V. F. Babenko, V. A. Kofanov, and S. A. Pichugov, “Inequalities for norms of intermediate derivatives of periodic functions and their applications,” East J. Approxim., 3, No. 3, 351–376 (1997).

    MATH  MathSciNet  Google Scholar 

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

    Google Scholar 

  8. A. Pinkus and O. Shisha, “Variations on the Chebyshev and L p -theories of best approximation,” J. Approxim. Theory, 148–168 (1982).

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

    Google Scholar 

  10. V. F. Babenko, V. A. Kofanov, and S. A. Pichugov, “Inequalities of Kolmogorov type and some their applications in approximation theory,” Rend. Circolo Mat. Palermo, Ser. II, Suppl., 52, 223–237 (1998).

    MathSciNet  Google Scholar 

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

    Google Scholar 

  12. V. F. Babenko, V. A. Kofanov, and S. A. Pichugov, “Approximation of sine-shaped functions by constants in the spaces L p , p < 1,” Ukr. Mat. Zh., 56, No. 6, 745–762 (2004).

    Article  MATH  MathSciNet  Google Scholar 

  13. V. A. Kofanov, “Some exact inequalities of Kolmogorov type,” Mat. Fiz., Analiz, Geom., 9, No. 3, 412–419 (2002).

    MATH  MathSciNet  Google Scholar 

  14. V. A. Kofanov, “On Kolmogorov-type inequalities taking into account the number of changes in the sign of derivatives,” Ukr. Mat. Zh., 55, No. 4, 456–469 (2003).

    MATH  MathSciNet  Google Scholar 

  15. V. A. Kofanov, “On exact Kolmogorov-type and Bernstein-type inequalities,” in: Proceedings of the Ukrainian Mathematical Congress, Approximation Theory and Harmonic Analysis [in Ukrainian], Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (2002), pp. 84–99.

    Google Scholar 

  16. N. P. Korneichuk, A. A. Ligun, and V. G. Doronin, Approximation with Restrictions [in Russian], Naukova Dumka, Kiev (1982).

    Google Scholar 

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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 10, pp. 1357–1367, October, 2006.

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Kofanov, V.A. On exact Bernstein-type inequalities for splines. Ukr Math J 58, 1538–1551 (2006). https://doi.org/10.1007/s11253-006-0152-z

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