Markov Inequality (Analysis)

The Markov inequalities - named after the Russian mathematicians Andrei and Wladimir Markow - indicate an upper bound for the derivation of polynomials in the closed real interval [−1, + 1]. They are used in approximation theory . Occasionally these inequalities are also referred to as the Markov brothers' inequalities .

Basic form

Andrei Markow published the following inequality in 1889:

Let be a polynomial of degree at most and its first derivative, then holds ${\displaystyle P}$ ${\displaystyle n}$ ${\displaystyle P'}$ ${\displaystyle \max _{-1\leq x\leq 1}(|P'(x)|)\leq n^{2}\cdot \max _{-1\leq x\leq 1}(|P(x)|)}$

It can be proven with the help of the Bernstein inequality from analysis .

The constant is the best possible. If you choose for the -th Chebyshev polynomial , then equality applies: ${\displaystyle n^{2}}$ ${\displaystyle P}$ ${\displaystyle n}$

${\displaystyle \max _{-1\leq x\leq 1}(|P'(x)|)=n^{2}\cdot \max _{-1\leq x\leq 1}(|P(x)|)}$

generalization

In 1892 Andrei's brother Vladimir Markov generalized this inequality for higher derivatives:

Let be a polynomial of degree less than or equal and the -th derivative, then applies ${\displaystyle P}$ ${\displaystyle n}$ ${\displaystyle P^{(k)}}$ ${\displaystyle k}$ ${\displaystyle \max _{-1\leq x\leq 1}|P^{(k)}(x)|\leq {\frac {n^{2}(n^{2}-1^{2})(n^{2}-2^{2})\cdots (n^{2}-(k-1)^{2})}{1\cdot 3\cdot 5\cdots (2k-1)}}\max _{-1\leq x\leq 1}|P(x)|}$

The first inequality is obtained for the special case . Werner Wolfgang Rogosinski found a simpler proof in 1955. ${\displaystyle k=1}$

In the 1940s and 1950s mathematicians found further generalizations and sharpening of these inequalities. Richard Duffin and Albert C. Schaeffer tightened the basic form in 1961

${\displaystyle \max _{-1\leq x\leq 1}(|P'(x)|)\leq n^{2}\cdot \max(|P(x_{i})|)}$

where the extreme values ​​of the Chebyshev polynomials are nth degree . ${\displaystyle x_{i}}$ ${\displaystyle T_{n}}$

literature

• Elliot Ward Cheney: Introduction to Approximation Theory. McGraw-Hill Book Company, 1966, ISBN 0-07-010757-2 , pp. 90-91 and 228.

Individual evidence

1. ^ Andrei Markow: Sur une question posée par Mendeleieff. Izvestia Akademii Nauk SSSR Vol. 62 (1889), pp. 1-24.
2. ↑ Elliot Ward Cheney: Introduction to Approximation Theory. 1966, pp. 90-91.
3. ↑ Elliot Ward Cheney: Introduction to Approximation Theory. 1966, p. 94, problem 8.
4. ↑ Vladimir Markov: On functions deviating the least from zero on a given interval. St. Petersburg 1892, - About polynomials that deviate as little as possible from zero in a given interval. In: Mathematical Annals . Vol. 77 (1916), pp. 213-258. ( PDF )
5. ^ Werner Wolfgang Rogosinski: Some elementary inequalities for polynomials. In: Mathematical Gazette. Vol. 39 (1955), pp. 7-12.
6. ↑ Elliot Ward Cheney: Introduction to Approximation Theory. 1966, p. 228.
7. ^ Richard Duffin, Albert C. Schaeffer: A refinement of an inequality of the brothers Markoff. In: American Mathematical Society Transactions. (TAMS), Vol. 50 (1941), pp. 517-528 ( PDF )