The Landau-Kolmogorov problem on a finite interval in the Taikov case. (English) Zbl 1508.41004
Let \(n\in\mathbb{N}\), \(\sigma>0\) and let \(\|\cdot\|_2\) denote the \(L^2\)-norm on \([-1,1]\). The author find an exact bound for \(|f^{(k)}(t)|\), \(k < n\), under constraints \(\|f\|_2\leq 1\) and \(\|f^{(n)}\|_2\leq \sigma\), where \(t\in [-1,1]\) is fixed. Then, for \(n=1\) and \(n=2\), the Landau-Kolmogorov problem on the interval \([-1,1]\) in the Taikov case is solved by proving the Karlin-type conjecture. Furthermore, the smallest possible constant \(A>0\) and the smallest possible constant \(B=B(A)\) in the inequality \(\|f^{(k)}\|_{\infty}\leq A\|f\|_2 + B\|f^{(n)}\|_2\) for \(k\in \{n - 2,n - 1\}\) are found.
Reviewer: Yuri A. Farkov (Moskva)
MSC:
| 41A17 | Inequalities in approximation (Bernstein, Jackson, Nikol’skiĭ-type inequalities) |
| 26D10 | Inequalities involving derivatives and differential and integral operators |
| 34B24 | Sturm-Liouville theory |
| 41A35 | Approximation by operators (in particular, by integral operators) |
Keywords:
Landau-Kolmogorov problem; Kolmogorov-type inequalities; Stechkin problem; higher-order Sturm-Liouville problemReferences:
| [1] | Arestov, V., Approximation of unbounded operators by bounded operators and related extremal problems, Russ. Math. Surv.. Russ. Math. Surv., Usp. Mat. Nauk, 51, 6, 89-124-1126 (1996), translation from · Zbl 0947.41019 |
| [2] | Arestov, V., Uniform approximation of differentiation operators by bounded linear operators in the space \(L_r\), Anal. Math., 46, 425-445 (2020) · Zbl 1474.41035 |
| [3] | Babenko, Y., Pointwise inequalities of Landau-Kolmogorov type for functions defined on a finite segment, Ukr. Math. J., 53, 2, 270-275 (2001) · Zbl 0984.26004 |
| [4] | Babenko, V.; Babenko, Y.; Kriachko, N.; Skorokhodov, D., On the Hardy-Littlewood-Pólya and Taikov type inequalities for multiple operators in Hilbert spaces, Anal. Math., 47, 709-745 (2021) · Zbl 1495.47035 |
| [5] | Babenko, V.; Kofanov, V.; Pichugov, S., On additive inequalities for intermediate derivatives of functions given on a finite interval, Ukr. Math. J., 49, 5, 685-696 (1997) · Zbl 0935.26004 |
| [6] | Babenko, V.; Korneichuk, N.; Kofanov, V.; Pichugov, S., Inequalities for Derivatives and their Applications, 591 (2003), Naukova Dumka, Kyiv, (in Russian) |
| [7] | Bojanov, B.; Naidenov, N., Examples of Landau-Kolmogorov inequality in integral norms on a finite interval, J. Approx. Theory, 117, 1, 55-73 (2002) · Zbl 1019.41009 |
| [8] | Borwein, P.; Erdelýi, T., (Polynomials and Polynomial Inequalities. Polynomials and Polynomial Inequalities, Graduate Texts in Mathematics, vol. 161 (1995), Springer), 490 · Zbl 0840.26002 |
| [9] | Burenkov, V., Exact constants in inequalities for norms of intermediate derivatives on a finite interval, Proc. Steklov Inst. Math.. Proc. Steklov Inst. Math., Tr. Mat. Inst. Steklova, 156, 22-29-30 (1980), translation from · Zbl 0455.46023 |
| [10] | Burenkov, V., Exact constants in inequalities for norms of intermediate derivatives on a finite interval. II, Proc. Steklov Inst. Math.. Proc. Steklov Inst. Math., Tr. Mat. Inst. Steklova, 173, 38-49-50 (1986), translation from · Zbl 0643.46023 |
| [11] | Burenkov, V.; Gusakov, V., On sharp constants in inequalities for the modulus of a derivative, Proc. Steklov Inst. Math.. Proc. Steklov Inst. Math., Tr. Mat. Inst. Steklova, 243, 104-126-119 (2003), translation from · Zbl 1076.26012 |
| [12] | Chui, C.; Smith, P., A note on Landau’s problem for bounded intervals, Amer. Math. Monthly, 82, 9, 927-929 (1975) · Zbl 0321.26005 |
| [13] | Dunford, N.; Schwartz, J., Linear operators. Part II: Spectral theory, (Courant, R.; Bers, L.; Stoker, J., Pure and Applied Mathematics, vol. VII (1963), Interscience Publishers), 490 · Zbl 0128.34803 |
| [14] | Eriksson, B., Some best constants in the Landau inequality on a finite interval, J. Approx. Theory, 94, 3, 420-454 (1998) · Zbl 0911.41006 |
| [15] | Gabushin, V., On the best approximation of the differentiation operator on the half-line, Math. Notes. Math. Notes, Mat. Zametki, 6, 5, 573-582-810 (1969), translation from · Zbl 0216.13603 |
| [16] | Ganzburg, M., Sharp constants in V. A. Markov-Bernstein type inequalities of different metrics, J. Approx. Theory, 215, C, 92-105 (2017) · Zbl 1358.41008 |
| [17] | Kalyabin, G., Sharp constants in inequalities for intermediate derivatives (the Gabushin Case), Funct. Anal. Appl., 38, 3, 184-191 (2004) · Zbl 1079.41009 |
| [18] | Karlin, S., Oscillatory perfect splines and related extremal problems, (Karlin, S.; Micchelli, C. A.; Pinkus, A.; Schoenberg, I. J., Studies in Spline Functions and Approximation Theory (1976), Academic Press: Academic Press New York), 371-460 · Zbl 0348.41015 |
| [19] | Kwong, M.; Zettl, A., (Norm Inequalities for Derivatives and Differences. Norm Inequalities for Derivatives and Differences, Lecture Notes in Mathematics, vol. 1536 (1992), Springer Berlin Heidelberg), 160 · Zbl 0925.26011 |
| [20] | Labelle, G., Concerning polynomials on the unit interval, Proc. Amer. Math. Soc., 20, 2, 321-326 (1969) · Zbl 0176.01303 |
| [21] | Landau, E., Einige ungleichungen für zweimal differenzierbare funktionen, Proc. Lond. Math. Soc., 13, 43-49 (1914) · JFM 44.0463.02 |
| [22] | Lunev, A.; Oridoroga, L., Exact constants in generalized inequalities for intermediate derivatives, Math. Notes, 85, 5, 703-711 (2009) · Zbl 1180.41009 |
| [23] | Markov, V., On functions which deviate least from zero in a given interval, Math. Ann., 77, 213-258 (1916), (in Russian), German translation: · JFM 46.0415.01 |
| [24] | Milovanović, G.; Mitrinović, D.; Rassias, T., Topics in Polynomials: Extremal Problems, Inequalities, Zeros (1994), World Scientific: World Scientific Singapore · Zbl 0848.26001 |
| [25] | Mitrinović, D.; Pecaric, J.; Fink, A., (Inequalities Involving Functions and their Integrals and Derivatives. Inequalities Involving Functions and their Integrals and Derivatives, Mathematics and its Applications, vol. 53 (1991), Springer Netherlands), 587 · Zbl 0744.26011 |
| [26] | Naidenov, N., Landau-Type extremal problem for the triple \(\| f \|_\infty, \| f^\prime \|_p, \| f^{\prime \prime} \|_\infty\) on a finite interval, J. Approx. Theory, 123, 2, 147-161 (2003) · Zbl 1035.41008 |
| [27] | Naidenov, N., On an extremal problem of Kolmogorov type for functions from \(W_\infty^4 ( [ a , b ] )\), East J. Approx., 9, 1, 117-135 (2003) · Zbl 1111.41002 |
| [28] | Pinkus, A., Some extremal properties of perfect splines and the pointwise Landau problem on the finite interval, J. Approx. Theory, 23, 1, 37-64 (1978) · Zbl 0395.41006 |
| [29] | Sato, M., The Landau inequality for bounded intervals with \(f^{( 3 )}\) finite, J. Approx. Theory, 34, 2, 159-166 (1982) · Zbl 0494.41009 |
| [30] | Shadrin, A. Y., Inequalities of Kolmogorov type and estimates of spline interpolation on periodic classes \(W^m_2\), Math. Notes. Math. Notes, Mat. Zametki, 48, 4, 132-139-1063 (1990), translation from · Zbl 0753.42003 |
| [31] | A. Shadrin, To the Landau-Kolmogorov problem on a finite interval, in: Proceedings of the International Conference “Open Problems in Approximation Theory”, Vonesta Voda, 1993, pp. 192-204. |
| [32] | Shadrin, A., A note on the least constant in Landau inequality on a finite interval, (Milovanović, G. V., Recent Progress in Inequalities. Mathematics and its Applications, Vol. 430 (1998), Springer, Dordecht), 489-491 · Zbl 0898.41008 |
| [33] | Shadrin, A., Twelve proofs of the Markov inequality, (Nikolov, G.; etal., Approximation Theory: A Volume Dedicated To Borislav Bojanov (2004), Prof. M. Drinov Acad. Publ. House, Sofia), 233-298 |
| [34] | Shadrin, A., The Landau-Kolmogorov inequality revisited, Discrete Contin. Dyn. Syst., 34, 3, 1183-1210 (2014) · Zbl 1280.41012 |
| [35] | Skorokhodov, D., On the Landau-Kolmogorov inequality between \(\| f^\prime \|_\infty, \| f \|_\infty\) and \(\| f^{\prime \prime \prime} \|_1\), Res. Math., 27, 1, 55-66 (2019) · Zbl 1489.41005 |
| [36] | Stechkin, S., Inequalities between the norms of the derivatives of an arbitrary function, Acta Sci. Math., 26, 225-230 (1965) · Zbl 0139.07004 |
| [37] | Stechkin, S., Best approximation of linear operators, Math. Notes, 1, 2, 91-99 (1967) · Zbl 0168.12201 |
| [38] | Taikov, L., Kolmogorov-Type inequalities and the best formulas for numerical differentiation, Math. Notes, 4, 2, 631-634 (1968) · Zbl 0206.34901 |
| [39] | Zvyagintsev, A., Kolmogorov’s Inequalities for \(n = 4\), Latv. Mat. Ezhegodnik, 26, 165-175 (1982), (in Russian) · Zbl 0514.26004 |
| [40] | Zvyagintsev, A.; Lepin, A., Kolmogorov’s Inequalities between the upper bounds of derivatives of functions for \(n = 3\), Latv. Mat. Ezhegodnik, 26, 176-181 (1982), (in Russian) · Zbl 0514.26005 |
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