grandes-ecoles 2010 QIII.D.1

grandes-ecoles · France · centrale-maths1__pc Sequences and Series Asymptotic Equivalents and Growth Estimates for Sequences/Series

We denote by $C([-1,1])$ the vector space of continuous functions on $[-1,1]$ with real values, equipped with the infinite norm $\|f\|_\infty = \sup_{x \in [-1,1]} |f(x)|$. For every integer $n \in \mathbb{N}$, $V_n$ denotes the set of restrictions to $[-1,1]$ of polynomial functions of degree at most $n$, and $d(f, V_n) = \inf_{p \in V_n} \|f - p\|_\infty$.
Let $f \in C([-1,1])$. We assume that the sequence $(d(f, V_n))_{n \in \mathbb{N}}$ has rapid decay.
Show that we can construct a sequence $(p_n)_{n \in \mathbb{N}}$ of polynomial functions such that:

for every integer $n$, $\deg(p_n) \leqslant n$;
$(\|f - p_n\|_\infty)_{n \in \mathbb{N}}$ has rapid decay.

We denote by $C([-1,1])$ the vector space of continuous functions on $[-1,1]$ with real values, equipped with the infinite norm $\|f\|_\infty = \sup_{x \in [-1,1]} |f(x)|$. For every integer $n \in \mathbb{N}$, $V_n$ denotes the set of restrictions to $[-1,1]$ of polynomial functions of degree at most $n$, and $d(f, V_n) = \inf_{p \in V_n} \|f - p\|_\infty$.

Let $f \in C([-1,1])$. We assume that the sequence $(d(f, V_n))_{n \in \mathbb{N}}$ has rapid decay.

Show that we can construct a sequence $(p_n)_{n \in \mathbb{N}}$ of polynomial functions such that:
\begin{itemize}
  \item for every integer $n$, $\deg(p_n) \leqslant n$;
  \item $(\|f - p_n\|_\infty)_{n \in \mathbb{N}}$ has rapid decay.
\end{itemize}

Paper Questions

QI.A.1 QI.A.2 QI.A.3 QI.A.4 QI.A.5 QI.B.1 QI.B.2 QI.B.3 QI.C.1 QI.C.2 QI.C.3 QI.C.4 QII.A.1 QII.A.2 QII.A.3 QII.A.4 QII.A.5 QII.B.1 QII.B.2 QII.B.3 QII.C QIII.A.1 QIII.A.2 QIII.B.1 QIII.B.2 QIII.B.3 QIII.C.1 QIII.C.2 QIII.C.3 QIII.D.1 QIII.D.2