March  2014, 34(3): 1183-1210. doi: 10.3934/dcds.2014.34.1183

The Landau--Kolmogorov inequality revisited

1. 

DAMTP, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom

Received  January 2013 Revised  May 2013 Published  August 2013

We consider the Landau--Kolmogorov problem on a finite interval which is to find an exact bound for $\|f^{(k)}\|$, for $0 < k < n$, given bounds $\|f\| \le 1$ and $\|f^{(n)}\| \le \sigma$, with $\|\cdot\|$ being the max-norm on $[-1,1]$. In 1972, Karlin conjectured that this bound is attained at the end-point of the interval by a certain Zolotarev polynomial or spline, and that was proved for a number of particular values of $n$ or $\sigma$. Here, we provide a complete proof of this conjecture in the polynomial case, i.e. for $0 \le \sigma \le \sigma_n := \|T_n^{(n)}\|$, where $T_n$ is the Chebyshev polynomial of degree $n$. In addition, we prove a certain Schur-type estimate which is of independent interest.
Citation: Alexei Shadrin. The Landau--Kolmogorov inequality revisited. Discrete and Continuous Dynamical Systems, 2014, 34 (3) : 1183-1210. doi: 10.3934/dcds.2014.34.1183
References:
[1]

C. K. Chui and P. W. Smith, A note on Landau's problem for bounded intervals, Amer. Math. Monthly, 82 (1975), 927-929. doi: 10.2307/2318501.

[2]

P. Erdös and G. Szegö, On a problem of I. Schur, Ann. of Math. (2), 43 (1942), 451-470. doi: 10.2307/1968803.

[3]

B.-O. Eriksson, Some best constants in the Landau inequality on a finite interval, J. Approx. Theory, 94 (1998), 420-454. doi: 10.1006/jath.1998.3203.

[4]

M. Floater, Error formulas for divided difference expansions and numerical differentiation, J. Approx. Theory, 122 (2003), 1-9. doi: 10.1016/S0021-9045(03)00025-X.

[5]

S. Karlin, Oscillatory perfect splines and related extremal problems, in "Studies in Spline Functions and Approximation Theory", 371-460. Academic Press, New York, (1976).

[6]

E. Landau, Einige Ungleichungen für zweimal differenzierbare Funktionen, Proc. London Math. Soc., 13 (1914), 43-39. doi: 10.1112/plms/s2-13.1.43.

[7]

A. P. Matorin, On inequalities between the maxima of the absolute values of a function and its derivatives on a half-line, Ukrain. Mat. Zh., 7 (1955), 262-266 = Amer. Math. Soc. Transl. (2), 8 (1958), 13-17.

[8]

N. Naidenov, On an extremal problem of Kolmogorov type for functions from $W^4_\infty$([a,b]), East J. Approx., 9 (2003), 117-135.

[9]

A. Pinkus, Some extremal properties of perfect splines and the pointwise Landau problem on the finite interval, J. Approx. Theory, 23 (1978), 37-64. doi: 10.1016/0021-9045(78)90077-1.

[10]

M. Sato, The Landau inequality for bounded intervals with $f^{(3)}$ finite, J. Approx. Theory, 34 (1982), 159-166. doi: 10.1016/0021-9045(82)90089-2.

[11]

I. Schur, Über das Maximum des absoluten Betrages eines Polynoms in einem gegebenen Intervall, Math. Z., 4 (1919), 271-287. doi: 10.1007/BF01203015.

[12]

A. Shadrin, Interpolation with Lagrange polynomials. A simple proof of Markov inequality and some of its generalizations, Approx. Theory Appl., 8 (1992), 51-61.

[13]

A. Shadrin, To the Landau-Kolmogorov problem on a finite interval, in "Open Problems in Approximation Theory" (Ed. B. Bojanov), 192-204, SCT Publishing, Singapore, (1994).

[14]

A. Shadrin, Error bounds for Lagrange interpolation, J. Approx. Theory, 80 (1995), 25-49. doi: 10.1006/jath.1995.1003.

[15]

A. Shadrin, Twelve proofs of the Markov inequality, in "Approximation Theory: A Volume Dedicated to Borislav Bojanov" (Eds. G. Nikolov et al), 233-298, Prof. M. Drinov Acad. Publ. House, Sofia, (2004).

[16]

A. I. Zvyagintsev, Kolmogorov's inequalities for $n=4$, Latv. Mat. Ezhegodnik, 26 (1982), 165-175 (in Russian).

[17]

A. I. Zvyagintsev and A. Ya. Lepin, Kolmogorov's inequalities between the upper bounds of derivatives of functions for $n=3$, Latv. Mat. Ezhegodnik, 26 (1982), 176-181 (in Russian).

show all references

References:
[1]

C. K. Chui and P. W. Smith, A note on Landau's problem for bounded intervals, Amer. Math. Monthly, 82 (1975), 927-929. doi: 10.2307/2318501.

[2]

P. Erdös and G. Szegö, On a problem of I. Schur, Ann. of Math. (2), 43 (1942), 451-470. doi: 10.2307/1968803.

[3]

B.-O. Eriksson, Some best constants in the Landau inequality on a finite interval, J. Approx. Theory, 94 (1998), 420-454. doi: 10.1006/jath.1998.3203.

[4]

M. Floater, Error formulas for divided difference expansions and numerical differentiation, J. Approx. Theory, 122 (2003), 1-9. doi: 10.1016/S0021-9045(03)00025-X.

[5]

S. Karlin, Oscillatory perfect splines and related extremal problems, in "Studies in Spline Functions and Approximation Theory", 371-460. Academic Press, New York, (1976).

[6]

E. Landau, Einige Ungleichungen für zweimal differenzierbare Funktionen, Proc. London Math. Soc., 13 (1914), 43-39. doi: 10.1112/plms/s2-13.1.43.

[7]

A. P. Matorin, On inequalities between the maxima of the absolute values of a function and its derivatives on a half-line, Ukrain. Mat. Zh., 7 (1955), 262-266 = Amer. Math. Soc. Transl. (2), 8 (1958), 13-17.

[8]

N. Naidenov, On an extremal problem of Kolmogorov type for functions from $W^4_\infty$([a,b]), East J. Approx., 9 (2003), 117-135.

[9]

A. Pinkus, Some extremal properties of perfect splines and the pointwise Landau problem on the finite interval, J. Approx. Theory, 23 (1978), 37-64. doi: 10.1016/0021-9045(78)90077-1.

[10]

M. Sato, The Landau inequality for bounded intervals with $f^{(3)}$ finite, J. Approx. Theory, 34 (1982), 159-166. doi: 10.1016/0021-9045(82)90089-2.

[11]

I. Schur, Über das Maximum des absoluten Betrages eines Polynoms in einem gegebenen Intervall, Math. Z., 4 (1919), 271-287. doi: 10.1007/BF01203015.

[12]

A. Shadrin, Interpolation with Lagrange polynomials. A simple proof of Markov inequality and some of its generalizations, Approx. Theory Appl., 8 (1992), 51-61.

[13]

A. Shadrin, To the Landau-Kolmogorov problem on a finite interval, in "Open Problems in Approximation Theory" (Ed. B. Bojanov), 192-204, SCT Publishing, Singapore, (1994).

[14]

A. Shadrin, Error bounds for Lagrange interpolation, J. Approx. Theory, 80 (1995), 25-49. doi: 10.1006/jath.1995.1003.

[15]

A. Shadrin, Twelve proofs of the Markov inequality, in "Approximation Theory: A Volume Dedicated to Borislav Bojanov" (Eds. G. Nikolov et al), 233-298, Prof. M. Drinov Acad. Publ. House, Sofia, (2004).

[16]

A. I. Zvyagintsev, Kolmogorov's inequalities for $n=4$, Latv. Mat. Ezhegodnik, 26 (1982), 165-175 (in Russian).

[17]

A. I. Zvyagintsev and A. Ya. Lepin, Kolmogorov's inequalities between the upper bounds of derivatives of functions for $n=3$, Latv. Mat. Ezhegodnik, 26 (1982), 176-181 (in Russian).

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