grandes-ecoles 2022 Q10

grandes-ecoles · France · centrale-maths2__mp 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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