grandes-ecoles 2022 Q10

grandes-ecoles · France · centrale-maths2__official Sequences and Series Proof of Inequalities Involving Series or Sequence Terms
Show that there exists a constant $C > 0$ for which the interpolation inequality $$\forall f \in \mathcal{C}^K([0,1]), \quad \max_{0 \leqslant k \leqslant K-1} \left\|f^{(k)}\right\|_\infty \leqslant \left\|f^{(K)}\right\|_\infty + C \sum_{\ell=1}^K \left|f\left(x_\ell\right)\right|$$ is satisfied. 
Show that there exists a constant $C > 0$ for which the interpolation inequality
$$\forall f \in \mathcal{C}^K([0,1]), \quad \max_{0 \leqslant k \leqslant K-1} \left\|f^{(k)}\right\|_\infty \leqslant \left\|f^{(K)}\right\|_\infty + C \sum_{\ell=1}^K \left|f\left(x_\ell\right)\right|$$
is satisfied.
Paper Questions
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