Landau–Kolmogorov inequality

Last updated April 16, 2019

In mathematics, the Landau–Kolmogorov inequality, named after Edmund Landau and Andrey Kolmogorov, is the following family of interpolation inequalities between different derivatives of a function f defined on a subset T of the real numbers:^[1]

Mathematics Field of study concerning quantity, patterns and change

Mathematics includes the study of such topics as quantity, structure, space, and change.

Edmund Georg Hermann Landau was a German mathematician who worked in the fields of number theory and complex analysis.

Andrey Nikolaevich Kolmogorov was a Soviet mathematician who made significant contributions to the mathematics of probability theory, topology, intuitionistic logic, turbulence, classical mechanics, algorithmic information theory and computational complexity.

On the real line

For k = 1, n = 2, T=R the inequality was first proved by Edmund Landau^[2] with the sharp constant C(2, 1, R) = 2. Following contributions by Jacques Hadamard and Georgiy Shilov, Andrey Kolmogorov found the sharp constants and arbitrary n, k:^[3]

Jacques Salomon Hadamard ForMemRS was a French mathematician who made major contributions in number theory, complex function theory, differential geometry and partial differential equations.

Georgi Evgen'evich Shilov was a Soviet mathematician and expert in the field of functional analysis, who contributed to the theory of normed rings and generalized functions.

C(n,k,\mathbb {R} )=a_{n-k}a_{n}^{-1+k/n}~,

where a_n are the Favard constants.

On the half-line

Following work by Matorin and others, the extremising functions were found by Isaac Jacob Schoenberg,^[4] explicit forms for the sharp constants are however still unknown.

Isaac Jacob Schoenberg American mathematician

Isaac Jacob Schoenberg was a Romanian-American mathematician, known for his discovery of splines.

Generalisations

There are many generalisations, which are of the form

\|f^{(k)}\|_{L_{q}(T)}\leq K\cdot {\|f\|_{L_{p}(T)}^{\alpha }}\cdot {\|f^{(n)}\|_{L_{r}(T)}^{1-\alpha }}{\text{ for }}1\leq k<n.

Here all three norms can be different from each other (from L₁ to L_∞, with p=q=r=∞ in the classical case) and T may be the real axis, semiaxis or a closed segment.

The Kallman–Rota inequality generalizes the Landau–Kolmogorov inequalities from the derivative operator to more general contractions on Banach spaces.^[5]

In mathematics, the Kallman–Rota inequality, introduced by Kallman & Rota (1970), is a generalization of the Landau–Kolmogorov inequality to Banach spaces. It states that if A is the infinitesimal generator of a one-parameter contraction semigroup then

In operator theory, a discipline within mathematics, a bounded operator T: X → Y between normed vector spaces X and Y is said to be a contraction if its operator norm ||T|| ≤ 1. Every bounded operator becomes a contraction after suitable scaling. The analysis of contractions provides insight into the structure of operators, or a family of operators. The theory of contractions on Hilbert space is largely due to Béla Szőkefalvi-Nagy and Ciprian Foias.

In mathematics, more specifically in functional analysis, a Banach space is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and is complete in the sense that a Cauchy sequence of vectors always converges to a well defined limit that is within the space.

Notes

↑ Weisstein, E.W. "Landau-Kolmogorov Constants". MathWorld--A Wolfram Web Resource.
↑ Landau, E. (1913). "Ungleichungen für zweimal differenzierbare Funktionen". Proc. London Math. Soc. 13: 43–49. doi:10.1112/plms/s2-13.1.43.
↑ Kolmogorov, A. (1949). "On Inequalities Between the Upper Bounds of the Successive Derivatives of an Arbitrary Function on an Infinite Interval". Amer. Math. Soc. Transl. 1–2: 233–243.
↑ Schoenberg, I.J. (1973). "The Elementary Case of Landau's Problem of Inequalities Between Derivatives". Amer. Math. Monthly. 80: 121–158. doi:10.2307/2318373.
↑ Kallman, Robert R.; Rota, Gian-Carlo (1970), "On the inequality $\Vert f^{\prime }\Vert ^{2}\leqq 4\Vert f\Vert \cdot \Vert f''\Vert$ ", Inequalities, II (Proc. Second Sympos., U.S. Air Force Acad., Colo., 1967), New York: Academic Press, pp. 187–192, MR 0278059 .

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[1] Weisstein, E.W. "Landau-Kolmogorov Constants". MathWorld--A Wolfram Web Resource.

[2] Landau, E. (1913). "Ungleichungen für zweimal differenzierbare Funktionen". Proc. London Math. Soc. 13: 43–49. doi:10.1112/plms/s2-13.1.43.

[3] Kolmogorov, A. (1949). "On Inequalities Between the Upper Bounds of the Successive Derivatives of an Arbitrary Function on an Infinite Interval". Amer. Math. Soc. Transl. 1–2: 233–243.

[4] Schoenberg, I.J. (1973). "The Elementary Case of Landau's Problem of Inequalities Between Derivatives". Amer. Math. Monthly. 80: 121–158. doi:10.2307/2318373.

[5] Kallman, Robert R.; Rota, Gian-Carlo (1970), "On the inequality $\Vert f^{\prime }\Vert ^{2}\leqq 4\Vert f\Vert \cdot \Vert f''\Vert$ ", Inequalities, II (Proc. Second Sympos., U.S. Air Force Acad., Colo., 1967), New York: Academic Press, pp. 187–192, MR 0278059 .