DFT matrix

In mathematical analysis, Bernstein's inequality is named after Sergei Natanovich Bernstein. The inequality states that on the complex plane, within the disk of radius 1, the degree of a polynomial times the maximum value of a polynomial is an upper bound for the similar maximum of its derivative.

Theorem

Let P be a polynomial of degree $n$ on complex numbers with derivative P′. Then

${\max }_{\left|z\right|\le 1}(|P^{\prime }(z)|)\le n\cdot {\max }_{\left|z\right|\le 1}(|P(z)|)$

The inequality finds uses in the field of approximation theory.

Using the Bernstein's inequality we have for the k:th derivative,

${\max }_{\left|z\right|\le 1}(|P^{(k)}(z)|)\le \frac{n!}{(n-k)!}\cdot {\max }_{\left|z\right|\le 1}(|P(z)|).$

See also

• Markov brothers' inequality
• Remez inequality

References

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