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A Note on the Least Constant in Landau Inequality on a Finite Interval

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Recent Progress in Inequalities

Part of the book series: Mathematics and Its Applications ((MAIA,volume 430))

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Abstract

In the Landau inequality on the unit interval

$$\parallel f^{(k)}\parallel_{q}\leq \alpha \parallel f\parallel _{p}+\beta \parallel f^{(n)}\parallel _{r}$$

with ∥ · ∥ s := ∥ · ∥ L s[0,1], 1 ≥ p, q, r ≥ ∞, 0 ≥ k < n, we find the least value A 0 of the first constant α.

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References

  1. V. I. Burenkov, On sharp constants in inequalities between norms of intermediate derivatives on a finite interval, Trudy Mat. Inst. AN SSSR (Proc. Steklov Math. Inst.) 156 (1980), 22–29. (Russian)

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  2. V. N. Gabushin, Inequalities for the norms of a function and its derivatives in metric L p, Mat. Zametki 1 (1967), no. 3, 291–298 (Russian) [Engl. Trans.: Math. Notes 1 (1967), 194-198].

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  3. H. Kallioniemi The Landau problem on compact intervals and optimal numerical differentiation, J. Approx. Theory 63 (1990), 72–91.

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

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  5. S. B. Stechkin, Best approximation of linear operators, Mat. Zametki 1 (1967), 137–148 (Russian) [Engl. Trans.: Math. Notes 1 (1967), 91-100].

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Author information

Authors and Affiliations

  1. Computing Center, Siberian Branch, Russian Academy of Sciences, 630090, Novosibirsk, Russia

    A. Yu. Shadrin

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  1. A. Yu. Shadrin
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Editor information

Editors and Affiliations

  1. University of Niš, Niš, Yugoslavia

    G. V. Milovanović (Faculty of Electronic Engineering) (Faculty of Electronic Engineering)

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© 1998 Springer Science+Business Media New York

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Shadrin, A.Y. (1998). A Note on the Least Constant in Landau Inequality on a Finite Interval. In: Milovanović, G.V. (eds) Recent Progress in Inequalities. Mathematics and Its Applications, vol 430. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-9086-0_32

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  • DOI: https://doi.org/10.1007/978-94-015-9086-0_32

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-90-481-4945-2

  • Online ISBN: 978-94-015-9086-0

  • eBook Packages: Springer Book Archive

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