# Kolmogorov inequality

Kolmogorov's inequality in approximation theory is a multiplicative inequality between the norms in the spaces $L _ {s} ( J)$ of functions and their derivatives on the real axis (or half-line):

$$\| x ^ {(k)} \| _ {L _ {q} } \leq \ C \| x \| _ {L _ {r} } ^ \nu \cdot \| x ^ {(n)} \| _ {L _ {p} } ^ {1 - \nu } ,$$

where

$$0 \leq k < n,\ \ \nu = \ \frac{n - k - p ^ {-1} + q ^ {-1} }{n - p ^ {-1} + r ^ {-1} } ,$$

and $C$ does not depend on $x$. Such inequalities were first studied by G.H. Hardy (1912), J.E. Littlewood (1912), E. Landau (1913), and J. Hadamard (1914). A.N. Kolmogorov  obtained the best possible constant $C$ for the most important case $J = ( - \infty , + \infty )$, $p = q = r = \infty$ and arbitrary $k$, $n$.

Kolmogorov's inequality is connected with problems of best numerical differentiation and the stable calculation of the (unbounded) operator $D ^ {k}$ of $k$- fold differentiation. In fact, the modulus of continuity

$$\omega ( \delta ) = \ \sup \ \{ {\| x ^ {( k)} \| _ {L _ {q} } } : { \| x \| _ {L _ {r} } \leq \delta ,\ \| x ^ {( n)} \| _ {L _ {p} } \leq 1 } \}$$

of the operator $D ^ {k}$ on the class $\{ {x \in L _ {r} } : {\| x ^ {( n)} \| _ {L _ {p} } \leq 1 } \}$ is expressed by the formula $\omega ( \delta ) = \omega ( 1) \delta$, that is, Kolmogorov's inequality holds with constant $C = \omega ( 1)$.

Kolmogorov's inequality is a special case of inequalities relating to the imbedding of classes of differentiable functions (see Imbedding theorems).

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How to Cite This Entry:
Kolmogorov inequality. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Kolmogorov_inequality&oldid=51245
This article was adapted from an original article by Yu.N. Subbotin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article