We prove the following sharp inequality of different metrics:
for 2π -periodic functions x ∈ L r∞ satisfying the condition
where
and φ r is the Euler spline of order r. As a special case, we establish the Nikol’skii-type sharp inequalities for polynomials and polynomial splines satisfying the condition (A).
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V. F. Babenko, V. A. Kofanov, and S. A. Pichugov, “Comparison of rearrangements and Kolmogorov–Nagy-type inequalities for periodic functions,” in: B. Bojanov (editor), Approximation Theory: A Volume Dedicated to B. Sendov, Darba, Sofia (2002), pp. 24–53.
V. A. Kofanov, “Comparison of rearrangements and inequalities of various metrics,” East. J. Approxim., 8, No. 3, 311–325 (2002).
A. Pinkus and O. Shisha, “Variations on the Chebyshev and L q theories of best approximation,” J. Approxim. Theory, 35, No. 2, 148–168 (1982).
V. A. Kofanov, “Sharp inequalities of Bernstein and Kolmogorov type,” East. J. Approxim., 11, No. 2, 131–145 (2005).
V. A. Kofanov, “On exact Bernstein-type inequalities for splines,” Ukr. Mat. Zh., 58, No. 10, 1357–1367 (2006); English translation: Ukr. Math. J., 58, No. 10, 1538–1551 (2006).
V. A. Kofanov and V. E. Mitropolskiy, “On the best constants in inequalities of Kolmogorov type,” East. J. Approxim., 13, No. 4, 455–466 (2007).
V. A. Kofanov, “On some extremal problems of different metrics for differentiable functions on the axis,” Ukr. Mat. Zh., 61, No. 6,765–776 (2009); English translation: Ukr. Math. J., 61, No. 6, 908–922 (2009).
V. F. Babenko, V. A. Kofanov, and S. A. Pichugov, “Inequalities of Kolmogorov type and some their applications in approximation theory,” Rend. Circ. Mat. Palermo. Ser. II. Suppl., 52, 223–237 (1998).
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); English translation: Ukr. Math. J., 56, No. 6, 882–903 (2004).
A. N. Kolmogorov, “On the inequalities between the upper bounds of successive derivatives on the infinite interval,” in: Selected Works, Mathematics and Mechanics [in Russian], Nauka, Moscow (1985), pp. 252–263.
N. P. Korneichuk, V. F. Babenko, and A. A. Ligun, Extremal Properties of Polynomials and Splines [in Russian], Naukova Dumka, Kiev (1992).
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).
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 67, No. 2, pp. 202–212, February, 2015.
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Kofanov, V.A. Inequalities of Different Metrics for Differentiable Periodic Functions. Ukr Math J 67, 230–242 (2015). https://doi.org/10.1007/s11253-015-1076-2
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DOI: https://doi.org/10.1007/s11253-015-1076-2