grandes-ecoles 2021 Q4.28

grandes-ecoles · France · x-ens-maths__pc Complex numbers 2 Inequalities and Estimates for Complex Expressions
Let $I = [a,b]$ be a segment of $\mathbb{R}$, and let $n$ and $m$ be two non-zero natural integers. We recall: $$C_{n,m}^I = \sup\left\{\left.\frac{\|Q\|_I \|R\|_I}{\|QR\|_I}\right\rvert\, Q \in \mathbb{C}_n[X]\backslash\{0\}, R \in \mathbb{C}_m[X]\backslash\{0\}\right\} \in \mathbb{R} \cup \{+\infty\}$$ Deduce from question 4.27 that the quantity $C_{n,m}^I$ does not depend on the segment $I$.
Let $I = [a,b]$ be a segment of $\mathbb{R}$, and let $n$ and $m$ be two non-zero natural integers. We recall:
$$C_{n,m}^I = \sup\left\{\left.\frac{\|Q\|_I \|R\|_I}{\|QR\|_I}\right\rvert\, Q \in \mathbb{C}_n[X]\backslash\{0\}, R \in \mathbb{C}_m[X]\backslash\{0\}\right\} \in \mathbb{R} \cup \{+\infty\}$$
Deduce from question 4.27 that the quantity $C_{n,m}^I$ does not depend on the segment $I$.