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, i ∈ Z, n ∈ N:
where k, r ∈ N, k < r, p = 1 or p = 2, q > p, and ϕr is the perfect Euler spline of order r.
Similar content being viewed by others
References
V. M. Tikhomirov, “Widths of sets in functional spaces and theory of best approximations,” Usp. Mat. Nauk, 15, No. 3, 81–120 (1960).
Yu. N. Subbotin, “On piecewise-polynomial interpolation,” Mat. Zametki, 1, No. 1, 24–29 (1967).
N. P. Korneichuk, V. F. Babenko, and A. A. Ligun, Extremal Properties of Polynomials and Splines [in Russian], Naukova Dumka, Kiev (1992).
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).
A. A. Ligun, “On inequalities for the norms of derivatives of periodic functions,” Mat. Zametki, 33, No. 3, 385–391 (1983).
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).
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).
A. Pinkus and O. Shisha, “Variations on the Chebyshev and L p -theories of best approximation,” J. Approxim. Theory, 148–168 (1982).
N. P. Korneichuk, Exact Constants in Approximation Theory [in Russian], Nauka, Moscow (1987).
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).
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.
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).
V. A. Kofanov, “Some exact inequalities of Kolmogorov type,” Mat. Fiz., Analiz, Geom., 9, No. 3, 412–419 (2002).
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).
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.
N. P. Korneichuk, A. A. Ligun, and V. G. Doronin, Approximation with Restrictions [in Russian], Naukova Dumka, Kiev (1982).
Author information
Authors and Affiliations
Additional information
__________
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 10, pp. 1357–1367, October, 2006.
Rights and permissions
About this article
Cite this article
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
Received:
Issue Date:
DOI: https://doi.org/10.1007/s11253-006-0152-z