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## Agregation 2013 – Analysis – Part 1

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L’Agregation is the French contest for students finishing University in order to qualify for the jobs as teachers in highschools and universities. It usually consists of a big problem divided into a few parts with many helping points. Although at the begining it might seem easy, it gets harder as you close to the end. This is just the introduction with notations and the first part. The next parts will come in future posts.

1. Introduction

The object of this problem is the study of some properties of the Muntz spaces (as Banach spaces) and of the functions belonging to these spaces.

The first part establishes some preliminary results to be used from the third part on. The second part is independent of the first one. The third part uses the first two parts. The fourth part is independent of parts two and three with exception of the question 3.a. The fifth part uses only the definitions used in the first part. In the sixth part only the last two questions use the results of the previous parts.

In all the problem we consider vector spaces over ${\Bbb{C}}$. The space ${C([0,1])}$ is the space of continuous function defined on ${[0,1]}$ with complex values endowed with the norm ${\|f\|_\infty = \sup\limits_{t\in [0,1]} |f(t)|}$.

We will denote ${C(\Bbb{T})}$ the space of continuous functions on ${R}$ with complex values which are ${2\pi}$-periodic endowed with the sup norm.

For ${k \in \Bbb{Z}}$ we define the function ${e_k}$ on ${\Bbb{R}}$ by ${e_k(t)=e^{ikt}}$. We denote ${\mathcal{T}}$ the vector subspace of ${C(\Bbb{T})}$ containint the trigonometric polynomials (i.e. the subspace spanned by ${e_k}$ with ${k \in \Bbb{Z}}$.

The subspace of ${C(\Bbb{T})}$ containing the even functions will be denoted by ${C_P(\Bbb{T})}$ and the subspace generated by the functions ${ t \mapsto \cos(kt)}$ where ${k \in \{0,1,..,N\}}$ will be denoted ${\Gamma_N}$. These last two subspaces are also endowed with the sup norm.

For ${f \in C(\Bbb{T})}$ and ${k \in \Bbb{Z}}$ we denote ${\widehat f(k)}$ the ${k}$-th Fourier coefficient of ${f}$:

$\displaystyle \widehat f(k) =\frac{1}{2 \pi} \int_0^{2\pi}e_k(-t)f(t)dt.$

In all the problem ${\Lambda = (\lambda_n)_{n \in \Bbb{N}}}$ will be a strictly increasing sequence of positive integers. Fpr ${\lambda \in \Bbb{N}}$ we define the polynomial function ${v_\lambda(t)=t^\lambda}$ restricted to ${[0,1]}$. We denote ${M_\Lambda}$ the subspace spanned by the sequence of functions ${(v_{\lambda_j})_{j \in \Bbb{N}}}$ and by ${\overline{M_\Lambda}}$ the closure of ${M_\Lambda}$ in ${C([0,1])}$ (endowed with the uniform norm).

For a sequence ${(z_n)_{n \in \Bbb{N}}}$ of complex numbers if the limit ${\displaystyle\lim_{n \rightarrow \infty} \prod_{j=1}^N z_j}$ exists, we denote it by ${\displaystyle \prod_{j=1}^\infty z_j}$. By default, the product over the empty set is equal to 1.

We recall the theorem of Banach-Steinhaus: let ${(T_i)_{i \in I}}$ be a family of linear continuous applications defined on a Banach space ${X}$ with values in another Banach space ${Y}$ such that for every ${x \in X}$ we have ${\displaystyle \sup_{i \in I} \|T_i(x)\|_Y<\infty}$. Then we have ${\displaystyle \sup_{i \in I} \|T_i\|<\infty}$.

The notation ${X^*}$ will denote the topological duap of a normed vector space ${X}$ (the space of linear continuous forms on ${X}$). Recall the Hahn-Banach theorem: if ${X}$ is a subspace of a normed vector space ${Y}$ then for every ${\xi \in X^*}$ there exists ${\tilde \xi \in Y^*}$ such chat ${\tilde \xi | X =\xi}$ and ${\|\tilde \xi\|=\|\xi\|}$.

We denote by ${c}$ the space of complex sequences which are convergent and by ${c_0}$ the subspace of the sequences which converge to ${0}$ endowed with the sup norm. Recall that the topological dual of ${c_0}$ is isomorphic to ${\ell^1}$ and that ${\ell^1}$ is separable.

2. Part I: Preliminaries

Questions 1,2,3 are independent

1. (a) Prove that ${\displaystyle \lim_{A \rightarrow \infty} \int_1^A \frac{\cos x}{x} dx}$ exists.
For ${N \in \Bbb{N}}$ we define ${D_n = \sum_{k=-N}^N e_k}$.

(b) Let ${t \in (0,2\pi)}$. Prove that ${\displaystyle D_N(t)=\frac{\sin((2N+1)/2)}{\sin t/2}}$.
In the sequel we denote ${\mathcal{L}_N =\frac{1}{2\pi} \int_0^{2\pi} |D_N(t)|dt}$.

(c) Prove that

$\displaystyle \mathcal{L}_N \geq \frac{1}{\pi} \int_0^{2\pi}\frac{|\sin((2N+1)x/2)|}{x} dx \geq \frac{1}{\pi} \int_0^{\pi(2N+1)} \frac{|sin(x)|}{x}dx \geq \frac{1}{2\pi} \int_0^{\pi(2N+1)} \frac{1-\cos(2x)}{x}dx.$

(d) Deduce that there exists ${\gamma>0}$ such that for every ${N \in \Bbb{N}}$ we have ${\mathcal{L}_N \geq \gamma \ln(N+1)}$.

2. Consider a sequence of real numbers ${u=(u_n)_{n \in \Bbb{N}}}$ which is nonincreasing, convergent, with ${u_0=1}$.

(a) Prove that the sequence ${(nu_n)_{n \in \Bbb{N}}}$ converges to ${0}$.
For ${s}$ a positive integer we denote ${E_s =\displaystyle \{n \in \Bbb{N} | u_n \geq \frac{1}{s}\}}$.

(b) Prove that ${E_s}$ is finite, its cardinal ${K_s}$ tends to infinity and that ${E_s= \{0,..,K_s-1\}}$.

(c) Establish that ${\displaystyle \frac{K_s-1}{2s} \leq \frac{1}{K_s} \sum_{n \in E_s} nu_n}$.

(d) Conclude that ${\displaystyle \lim_{s \rightarrow \infty} \frac{K_s}{s}=0}$.

3. Consider the application ${T: c \rightarrow c_0}$ defined by

$\displaystyle Tu = (l,u_0-l,...,u_k-l,...)$

where ${l}$ is the limit of ${u}$.

(a) Prove that ${T}$ is well defined, linear and bijective.

(b) Prove that ${T}$ and ${T^{-1}}$ are continuous.

Hints: 1. (a) Integrate by parts

(b) Use the identity ${2\cos a\sin b=\sin(a+b)-\sin(a-b)}$ and create a telescopic sum.

(c) First inequality: ${\sin x \leq x}$. Second inequality: change of variables. Third inequality: ${|\sin x| \geq \sin^2 x}$ and ${\cos(2x)=1-2\sin^2 x}$.

(d) Use (c), split the integral in two intervals separated by ${1}$. The second integral will be a logarithm plus something bounded given by (a).

2. (a) We have

$\displaystyle (n+p)u_{n+p}\leq nu_n+u_{n+1}+...+u_{n+p}$

which combined with the convergence of the series proves that ${(nu_n)}$ is convergent (it is bounded and every two limit points are close enough). If the limit is not zero, then the series is not convergent (comparison with the harmonic series).

(b) Obvious.

(c) ${\displaystyle \sum_{n \in E_s}nu_n \geq \sum_{i=0}^{K_s-1}\frac{n}{s}}$

(d) From (c) deduce that ${\displaystyle \frac{K_s}{s}\leq \frac{K_s}{K_s-1} \max_{n \in E_s} nu_n}$ which tends to zero by (a) (or use Stolz-Cesaro to prove that the limit of the arithmetic mean of the first ${n}$ terms of a convergent sequence has the same limit as the sequence itself).

3. (a) Obvious

(b) ${|Tu|_\infty\leq \sup|u_n|+|l|\leq 3\sup |u_n|}$.

In the same way: ${T^{-1}u=(u_0,u_1+u_0,...,u_k+u_0,...)}$ and obviously ${|Tu|_\infty\leq 2|u|_\infty}$.