Bernstein Inequality (Analysis)

The Bernstein inequalities - named after the Russian mathematician Sergei Natanowitsch Bernstein - specify an upper bound for the derivation of polynomials in a closed interval. They are used in the area of approximation theory .

Trigonometric polynomials

Bernstein considered trigonometric polynomials of degree n of the form in 1912

${\displaystyle S(\Theta )=\sum _{k=0}^{n}(a_{k}\cos(k\Theta )+b_{k}\sin(k\Theta ))}$

with and ${\displaystyle a_{n}\neq 0}$ ${\displaystyle b_{n}\neq 0}$

For these he proved the following inequality - but only for pure cosine polynomials.

Let be a trigonometric polynomial of degree less than or equal to and the first derivative, then applies ${\displaystyle S}$ ${\displaystyle n}$ ${\displaystyle S'}$ ${\displaystyle \max _{-\pi \leq \Theta \leq \pi }(|S'(\Theta )|)\leq n\cdot \max _{-\pi \leq \Theta \leq \pi }(|S(\Theta )|)}$

The represented shape with sine and cosine was described by the Hungarian mathematician Leopold Fejér in 1914, and Edmund Landau later mentioned it in a letter to Bernstein. Alternative proofs of the inequality are shown by Marcel Riesz in 1914 and George Pólya and Gábor Szegő in 1925.

This Bernstein inequality is helpful in proving a Markov inequality .

General polynomials

The Bernstein inequality can be generalized to general polynomials in the complex plane .

Let be a polynomial of degree on the complex numbers and the first derivative. Then applies on the unit circle : ${\displaystyle P}$ ${\displaystyle n}$ ${\displaystyle P'}$ ${\displaystyle |z|\leq 1}$ ${\displaystyle \max _{|z|\leq 1}(|P'(z)|)\leq n\cdot \max _{|z|\leq 1}(|P(z)|)}$

This inequality, in turn, can be generalized to higher derivatives.

Let be a polynomial of degree on the complex numbers, the -th derivative. Then applies on the unit circle : ${\displaystyle P}$ ${\displaystyle n}$ ${\displaystyle P^{(k)}}$ ${\displaystyle k}$ ${\displaystyle |z|\leq 1}$ ${\displaystyle \max _{|z|\leq 1}(|P^{(k)}(z)|)\leq {\frac {n!}{(n-k)!}}\cdot \max _{|z|\leq 1}(|P(z)|)}$

See also

• Bernstein Inequality (Stochastics)
• Markov Inequality (Analysis)

Individual evidence

1. ↑ Sergei Natanowitsch Bernstein: Sur l'ordre de la meilleure approximation des fonctions continues par des polynomes. Académie Royale de Belgique, Classe des Sciences, Mémores Collection in 4th, ser. II, Vol. 4 (1922). = Russian translation in Communications of the Kharkov Mathematical Society (CKMS) Vol. 13 (1912), 49-194.
2. ^ Leopold Fejér: About conjugated trigonometric series. Journal for pure and applied mathematics , Vol. 144 (1914), pp. 48–56 Online (accessed on May 13, 2014)
3. ↑ Sergei Natanowitsch Bernstein: Leçons sur les Propriétés Extrémales et la Meilleure Approximation des Fonctions Analytiques d'une Variable Réelle. Gauthier-Villars, Paris 1926.
4. ↑ Marcel Riesz: A trigonometric interpolation formula and some inequalities for polynomials. German Mathematicians Association , Annual Report, Vol. 23 (1914), pp. 354–368. Online (accessed May 13, 2014)
5. ↑ Marcel Riesz: Formule d'interpolation pour la dérivée d'un polynome trigonométrique. Comptes Rendus Hebdomaries, Séances de l'Académie des Sciences, Paris, Vol. 158 (1914), pp. 1152-1154. Online (accessed May 13, 2014)
6. ^ George Pólya, Gábor Szegő: Exercises and theorems from analysis. Springer, Berlin 1925.

literature

• Elliot Ward Cheney: Introduction to Approximation Theory. McGraw-Hill Book Company, 1966, Library of Congress Catalog Card Number 65-25916, ISBN 0-07-010757-2 , pp. 90-91 and 228
• Clement Frappier: Note on Bernstein's inequality for the third derivative of a polynomial. Journal of Inequalities in Pure and Applied Mathematics, Volume 5, Issue 1, Article 7, 2004 Online (accessed May 12, 2014)