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Papers (191)
2025
centrale-maths1__official 40 centrale-maths2__official 42 mines-ponts-maths1__mp 20 mines-ponts-maths1__pc 21 mines-ponts-maths1__psi 21 mines-ponts-maths2__mp 28 mines-ponts-maths2__pc 24 mines-ponts-maths2__psi 26 polytechnique-maths-a__mp 27 polytechnique-maths__fui 16 polytechnique-maths__pc 27 x-ens-maths-a__mp 18 x-ens-maths-c__mp 9 x-ens-maths-d__mp 38 x-ens-maths__pc 27 x-ens-maths__psi 38
2024
centrale-maths1__official 28 centrale-maths2__official 29 geipi-polytech__maths 9 mines-ponts-maths1__mp 25 mines-ponts-maths1__pc 20 mines-ponts-maths1__psi 19 mines-ponts-maths2__mp 23 mines-ponts-maths2__pc 21 mines-ponts-maths2__psi 21 polytechnique-maths-a__mp 44 polytechnique-maths-b__mp 37 x-ens-maths-a__mp 43 x-ens-maths-b__mp 35 x-ens-maths-c__mp 22 x-ens-maths-d__mp 45 x-ens-maths__pc 24 x-ens-maths__psi 26
2023
centrale-maths1__official 44 centrale-maths2__official 33 e3a-polytech-maths__mp 4 mines-ponts-maths1__mp 15 mines-ponts-maths1__pc 23 mines-ponts-maths1__psi 23 mines-ponts-maths2__mp 22 mines-ponts-maths2__pc 18 mines-ponts-maths2__psi 22 polytechnique-maths__fui 23 x-ens-maths-a__mp 25 x-ens-maths-b__mp 24 x-ens-maths-c__mp 20 x-ens-maths-d__mp 20 x-ens-maths__pc 18 x-ens-maths__psi 15
2022
centrale-maths1__mp 48 centrale-maths1__official 48 centrale-maths1__pc 37 centrale-maths1__psi 43 centrale-maths2__mp 32 centrale-maths2__official 32 centrale-maths2__pc 39 centrale-maths2__psi 45 mines-ponts-maths1__mp 25 mines-ponts-maths1__pc 24 mines-ponts-maths1__psi 24 mines-ponts-maths2__mp 24 mines-ponts-maths2__pc 19 mines-ponts-maths2__psi 20 x-ens-maths-a__mp 13 x-ens-maths-b__mp 40 x-ens-maths-c__mp 27 x-ens-maths-d__mp 46 x-ens-maths1__mp 13 x-ens-maths2__mp 40 x-ens-maths__pc 15 x-ens-maths__pc_cpge 15 x-ens-maths__psi 22 x-ens-maths__psi_cpge 23
2021
centrale-maths1__mp 40 centrale-maths1__official 40 centrale-maths1__pc 36 centrale-maths1__psi 29 centrale-maths2__mp 30 centrale-maths2__official 29 centrale-maths2__pc 38 centrale-maths2__psi 37 x-ens-maths2__mp 39 x-ens-maths__pc 44
2020
centrale-maths1__mp 42 centrale-maths1__official 42 centrale-maths1__pc 36 centrale-maths1__psi 40 centrale-maths2__mp 38 centrale-maths2__official 38 centrale-maths2__pc 40 centrale-maths2__psi 39 mines-ponts-maths1__mp_cpge 24 mines-ponts-maths2__mp_cpge 21 x-ens-maths-a__mp_cpge 18 x-ens-maths-b__mp_cpge 20 x-ens-maths-d__mp 14 x-ens-maths1__mp 18 x-ens-maths2__mp 20 x-ens-maths__pc 18
2019
centrale-maths1__mp 37 centrale-maths1__official 37 centrale-maths1__pc 40 centrale-maths1__psi 39 centrale-maths2__mp 37 centrale-maths2__official 37 centrale-maths2__pc 39 centrale-maths2__psi 49 x-ens-maths1__mp 24 x-ens-maths__pc 18 x-ens-maths__psi 26
2018
centrale-maths1__mp 47 centrale-maths1__official 47 centrale-maths1__pc 41 centrale-maths1__psi 44 centrale-maths2__mp 44 centrale-maths2__official 44 centrale-maths2__pc 35 centrale-maths2__psi 38 x-ens-maths1__mp 19 x-ens-maths2__mp 17 x-ens-maths__pc 22 x-ens-maths__psi 24
2017
centrale-maths1__mp 45 centrale-maths1__official 45 centrale-maths1__pc 22 centrale-maths1__psi 17 centrale-maths2__mp 30 centrale-maths2__official 30 centrale-maths2__pc 28 centrale-maths2__psi 44 x-ens-maths1__mp 26 x-ens-maths2__mp 16 x-ens-maths__pc 18 x-ens-maths__psi 26
2016
centrale-maths1__mp 42 centrale-maths1__pc 31 centrale-maths1__psi 33 centrale-maths2__mp 25 centrale-maths2__pc 47 centrale-maths2__psi 27 x-ens-maths1__mp 18 x-ens-maths2__mp 46 x-ens-maths__pc 15 x-ens-maths__psi 20
2015
centrale-maths1__mp 42 centrale-maths1__pc 18 centrale-maths1__psi 42 centrale-maths2__mp 44 centrale-maths2__pc 18 centrale-maths2__psi 33 x-ens-maths1__mp 16 x-ens-maths2__mp 31 x-ens-maths__pc 30 x-ens-maths__psi 22
2014
centrale-maths1__mp 28 centrale-maths1__pc 26 centrale-maths1__psi 27 centrale-maths2__mp 24 centrale-maths2__pc 26 centrale-maths2__psi 27 x-ens-maths1__mp 9 x-ens-maths2__mp 16 x-ens-maths__pc 4 x-ens-maths__psi 24
2013
centrale-maths1__mp 22 centrale-maths1__pc 45 centrale-maths1__psi 29 centrale-maths2__mp 31 centrale-maths2__pc 52 centrale-maths2__psi 32 x-ens-maths1__mp 24 x-ens-maths2__mp 35 x-ens-maths__pc 22 x-ens-maths__psi 9
2012
centrale-maths1__mp 36 centrale-maths1__pc 28 centrale-maths1__psi 33 centrale-maths2__mp 27 centrale-maths2__psi 18
2011
centrale-maths1__mp 27 centrale-maths1__pc 17 centrale-maths1__psi 24 centrale-maths2__mp 29 centrale-maths2__pc 17 centrale-maths2__psi 10
2010
centrale-maths1__mp 19 centrale-maths1__pc 30 centrale-maths1__psi 13 centrale-maths2__mp 32 centrale-maths2__pc 37 centrale-maths2__psi 27
2010 centrale-maths1__pc

30 maths questions

QI.A.1 Addition & Double Angle Formulae Trigonometric Identity Proof or Derivation View
For every integer $n \in \mathbb{N}$, we set $F_n(x) = \cos(n \arccos x)$.
a) Show that the functions $F_n$ are defined on the same domain $D$ which should be specified.
b) Calculate $F_1(x), F_2(x)$ and $F_3(x)$ for all $x \in D$.
c) Calculate $F_n(1), F_n(0)$ and $F_n(-1)$ for all $n \in \mathbb{N}$.
d) Specify the parity properties of $F_n$ as a function of $n$.
QI.A.2 Addition & Double Angle Formulae Trigonometric Identity Proof or Derivation View
For every integer $n \in \mathbb{N}$, we set $F_n(x) = \cos(n \arccos x)$.
Calculate $F_{n+1}(x) + F_{n-1}(x)$ for all $n \in \mathbb{N}^*$ and all $x \in D$.
QI.A.3 Taylor series Formal power series manipulation (Cauchy product, algebraic identities) View
For every integer $n \in \mathbb{N}$, we set $F_n(x) = \cos(n \arccos x)$.
Deduce from the above that $F_n$ extends to $\mathbb{R}$ as a unique polynomial function, whose degree and leading coefficient should be specified.
QI.A.5 Taylor series Recursive or implicit derivative computation for series coefficients View
Throughout the rest of this problem, we set $T_0(x) = 1$. For $n \in \mathbb{N}^*$, we denote by $T_n$ the polynomial function satisfying $T_n(x) = 2^{1-n} F_n(x)$ for all $x \in \mathbb{R}$.
Determine two real numbers $a$ and $b$ such that $$\forall x \in \mathbb{R}, \forall n \in \mathbb{N}^*, T_{n+2}(x) = a x T_{n+1}(x) + b T_n(x)$$
QI.B.1 Differentiating Transcendental Functions Regularity and smoothness of transcendental functions View
Let $n \in \mathbb{N}$.
a) Show that the function $F_n$ is of class $C^\infty$ on $\mathbb{R}$.
b) For $x \in ]-1,1[$, give a simple expression for $F_n'(x)$. Justify the calculation carefully.
QI.B.2 Differentiating Transcendental Functions Evaluate derivative at a point or find tangent slope View
Let $n \in \mathbb{N}^*$.
a) Show that $\arccos(x) \sim \sqrt{2(1-x)}$ as $x \rightarrow 1$.
b) Deduce the calculation of $F_n'(1)$ and $F_n'(-1)$.
QI.B.3 Differentiating Transcendental Functions Higher-order or nth derivative computation View
Throughout the rest of this problem, we set $T_0(x) = 1$. For $n \in \mathbb{N}^*$, we denote by $T_n$ the polynomial function satisfying $T_n(x) = 2^{1-n} F_n(x)$ for all $x \in \mathbb{R}$.
Show that, for all $n \in \mathbb{N}^*$ and all real $x$, the following relation holds: $$\left(1 - x^2\right) T_n''(x) - x T_n'(x) + n^2 T_n(x) = 0.$$
QI.C.1 Indefinite & Definite Integrals Convergence and Evaluation of Improper Integrals View
We denote by $E$ the vector space of polynomial functions on $\mathbb{R}$ and, for all $n \in \mathbb{N}$, we denote by $E_n$ the vector subspace of $E$ formed by polynomial functions of degree at most $n$.
Show that, for every pair $(f, g)$ of $E \times E$, the function: $$x \mapsto f(x) g(x) \frac{1}{\sqrt{1 - x^2}}$$ is integrable on the interval $]-1,1[$.
QI.C.2 Sequences and Series Inner Product Spaces, Orthogonality, and Hilbert Space Structure on Sequence/Function Spaces View
We denote by $E$ the vector space of polynomial functions on $\mathbb{R}$ and, for all $n \in \mathbb{N}$, we denote by $E_n$ the vector subspace of $E$ formed by polynomial functions of degree at most $n$.
The previous question shows that the following application $\varphi$ is well defined: $$\varphi : \left\{ \begin{aligned} E \times E & \rightarrow \mathbb{R} \\ (f, g) & \mapsto \int_{-1}^{1} \frac{1}{\sqrt{1 - x^2}} f(x) g(x)\, dx \end{aligned} \right.$$
Show that $\varphi$ defines an inner product on $E$.
QI.C.3 Sequences and Series Inner Product Spaces, Orthogonality, and Hilbert Space Structure on Sequence/Function Spaces View
We denote by $E$ the vector space of polynomial functions on $\mathbb{R}$, for all $n \in \mathbb{N}$, $E_n$ the vector subspace of $E$ formed by polynomial functions of degree at most $n$, and the space $E$ is equipped with the inner product $(\cdot|\cdot)$ defined by $\varphi(f,g) = \int_{-1}^{1} \frac{1}{\sqrt{1-x^2}} f(x) g(x)\, dx$.
a) Show that there exists a sequence of polynomial functions $(p_n)_{n \in \mathbb{N}}$ such that, for all $n \in \mathbb{N}$, $p_n$ has degree $n$ and leading coefficient 1, and that, for all $n \in \mathbb{N}^*$, $p_n$ is orthogonal to all elements of $E_{n-1}$.
b) Show that there exists a unique family $(Q_n)_{n \in \mathbb{N}}$ of polynomial functions satisfying the following conditions:
  • [i)] the family $(Q_n)_{n \in \mathbb{N}}$ is orthogonal for the inner product $(\cdot|\cdot)$;
  • [ii)] for all $n \in \mathbb{N}$, $Q_n$ has degree $n$ and leading coefficient 1.
QI.C.4 Reduction Formulae Prove Orthogonality or Algebraic Relations Between Integral-Defined Objects View
We denote by $E$ the vector space of polynomial functions on $\mathbb{R}$, for all $n \in \mathbb{N}$, $E_n$ the vector subspace of $E$ formed by polynomial functions of degree at most $n$. For $n \in \mathbb{N}^*$, $T_n$ denotes the polynomial function satisfying $T_n(x) = 2^{1-n} F_n(x)$ and $T_0(x) = 1$. The space $E$ is equipped with the inner product $(\cdot|\cdot)$ defined by $(f|g) = \int_{-1}^{1} \frac{1}{\sqrt{1-x^2}} f(x) g(x)\, dx$.
Calculate $(T_m | T_n)$ for all $(m, n) \in \mathbb{N} \times \mathbb{N}$. What can we deduce from this?
QII.A.1 Proof Direct Proof of an Inequality View
We seek to show that the inequality $|\sin(n\theta)| \leqslant n \sin(\theta)$ is satisfied for all $n \in \mathbb{N}^*$ and all $\theta \in \left[0, \frac{\pi}{2}\right]$.
a) Show that $\sin(n\theta) \leqslant n \sin(\theta)$ for all $n \in \mathbb{N}^*$ and all $\theta \in \left[0, \frac{\pi}{2n}\right]$.
b) Show that, for all $\theta \in \left[0, \frac{\pi}{2}\right]$, we have $\sin(\theta) \geqslant \frac{2}{\pi} \theta$.
c) Deduce that: $$\forall \theta \in \left[\frac{\pi}{2n}, \frac{\pi}{2}\right], \quad 1 \leqslant n \sin(\theta)$$
d) Conclude.
e) For which values of $\theta \in \left[0, \frac{\pi}{2}\right]$ do we have $|\sin(n\theta)| = n \sin(\theta)$?
QII.A.2 Differentiating Transcendental Functions Higher-order or nth derivative computation View
For $n \in \mathbb{N}^*$, $T_n$ denotes the polynomial function satisfying $T_n(x) = 2^{1-n} F_n(x)$ where $F_n(x) = \cos(n \arccos x)$.
Show that for all $n \in \mathbb{N}^*$, $$\sup_{x \in [-1,1]} \left| T_n'(x) \right| = 2^{1-n} n^2$$
Is this supremum attained? If so, specify for which values of $x$.
QII.A.3 Roots of polynomials Location and bounds on roots View
For $n \in \mathbb{N}^*$, $T_n$ denotes the polynomial function satisfying $T_n(x) = 2^{1-n} F_n(x)$ where $F_n(x) = \cos(n \arccos x)$.
Let $n \in \mathbb{N}^*$. Show that the polynomial function $T_n$ has exactly $n$ distinct zeros all belonging to $]-1,1[$. For $j \in \{1, 2, \ldots, n\}$, we denote by $x_{n,j}$ the $j$-th zero of $T_n$ in increasing order. Give the value of $x_{n,j}$.
QII.A.4 Roots of polynomials Proof of polynomial identity or inequality involving roots View
For $n \in \mathbb{N}^*$, $T_n$ denotes the polynomial function satisfying $T_n(x) = 2^{1-n} F_n(x)$ where $F_n(x) = \cos(n \arccos x)$, and $x_{n,j}$ denotes the $j$-th zero of $T_n$ in increasing order.
Let $n \in \mathbb{N}^*$ and $x \in \mathbb{R} \setminus \{x_{n,j},\, 1 \leqslant j \leqslant n\}$. Show that: $$\frac{T_n'(x)}{T_n(x)} = \sum_{j=1}^{n} \frac{1}{x - x_{n,j}}$$
QII.A.5 Roots of polynomials Polynomial evaluation, interpolation, and remainder View
For $n \in \mathbb{N}^*$, $T_n$ denotes the polynomial function satisfying $T_n(x) = 2^{1-n} F_n(x)$ where $F_n(x) = \cos(n \arccos x)$, and $x_{n,j}$ denotes the $j$-th zero of $T_n$ in increasing order. We denote by $E_{n-1}$ the vector subspace of polynomial functions of degree at most $n-1$.
Let $n \in \mathbb{N}^*$, $x \in \mathbb{R} \setminus \{x_{n,j},\, 1 \leqslant j \leqslant n\}$ and $P \in E_{n-1}$.
a) Show that: $$P(x) = \sum_{j=1}^{n} \frac{P(x_{n,j})}{T_n'(x_{n,j})} \frac{T_n(x)}{x - x_{n,j}}$$
b) Deduce that: $$P(x) = \frac{2^{n-1}}{n} \sum_{j=1}^{n} (-1)^{n-j} \sqrt{1 - x_{n,j}^2}\, P(x_{n,j}) \frac{T_n(x)}{x - x_{n,j}}$$
QII.B.1 Proof Direct Proof of an Inequality View
For $n \in \mathbb{N}^*$, $T_n$ denotes the polynomial function satisfying $T_n(x) = 2^{1-n} F_n(x)$ where $F_n(x) = \cos(n \arccos x)$, and $x_{n,j}$ denotes the $j$-th zero of $T_n$ in increasing order.
Show that, for all $x \in [x_{n,1}, x_{n,n}]$, we have $$\sqrt{1 - x^2} \geq \frac{1}{n}$$
QII.B.2 Proof Bounding or Estimation Proof View
For $n \in \mathbb{N}^*$, $T_n$ denotes the polynomial function satisfying $T_n(x) = 2^{1-n} F_n(x)$ where $F_n(x) = \cos(n \arccos x)$, and $x_{n,j}$ denotes the $j$-th zero of $T_n$ in increasing order. We denote by $E_{n-1}$ the vector subspace of polynomial functions of degree at most $n-1$.
a) Let $n \in \mathbb{N}^*$ and $P \in E_{n-1}$ such that $\sup_{x \in [-1,1]} \sqrt{1 - x^2}\, |P(x)| \leq 1$.
Show that $$\sup_{x \in [-1,1]} |P(x)| \leq n$$ (Distinguish three cases according to whether $x$ belongs to one of the intervals $[-1, x_{n,1}[$, $[x_{n,1}, x_{n,n}]$ or $]x_{n,n}, 1]$.)
b) Deduce that for all $n \in \mathbb{N}^*$ and for all $P \in E_{n-1}$, we have: $$\sup_{x \in [-1,1]} |P(x)| \leqslant n \sup_{x \in [-1,1]} \sqrt{1 - x^2}\, |P(x)|.$$
QII.B.3 Proof Deduction or Consequence from Prior Results View
Let $T$ be a trigonometric polynomial of the form $$T(\theta) = a_0 + \sum_{k=1}^{n} \left[ a_k \cos(k\theta) + b_k \sin(k\theta) \right]$$ where $a_0, a_1, b_1, \ldots, a_n, b_n \in \mathbb{R}$.
a) Let $k \in \mathbb{N}^*$. Show that there exists a polynomial function $B_k$ of degree $(k-1)$ such that: $$\forall \theta \in \mathbb{R}, \quad \sin(k\theta) = B_k(\cos(\theta)) \sin(\theta).$$
b) Let $\theta_0 \in \mathbb{R}$. Show that there exists a polynomial function $P \in E_{n-1}$ such that, for all $\theta \in \mathbb{R}$, we have: $$T(\theta_0 + \theta) - T(\theta_0 - \theta) = 2 P(\cos\theta) \sin\theta$$
c) Deduce that: $$\sup_{x \in [-1,1]} |P(x)| \leqslant n \sup_{\theta \in \mathbb{R}} |T(\theta)|.$$
d) Show that: $$\sup_{\theta \in \mathbb{R}} \left| T'(\theta) \right| \leq n \sup_{\theta \in \mathbb{R}} |T(\theta)|.$$
QII.C Proof Direct Proof of an Inequality View
Let $P \in E_n$. Show that: $$\sup_{x \in [-1,1]} \left| P'(x) \right| \leq n^2 \sup_{x \in [-1,1]} |P(x)|$$ (One may use the trigonometric polynomial $T(\theta) = P(\cos(\theta))$.)
QIII.A.1 Sequences and Series Convergence/Divergence Determination of Numerical Series View
We denote by $\mathcal{S}$ the set of sequences of real numbers with rapid decay (i.e., sequences $(\alpha_n)_{n \in \mathbb{N}}$ such that for every integer $k \in \mathbb{N}$, the sequence $(n^k \alpha_n)_{n \in \mathbb{N}}$ is bounded).
Let $(\alpha_n)_{n \in \mathbb{N}}$ be a sequence of $\mathcal{S}$ and $j \in \mathbb{N}$.
Show that the numerical series $\sum_{n \in \mathbb{N}} n^j \alpha_n$ is convergent.
QIII.A.2 Sequences and Series Asymptotic Equivalents and Growth Estimates for Sequences/Series View
We denote by $\mathcal{S}$ the set of sequences of real numbers with rapid decay (i.e., sequences $(\alpha_n)_{n \in \mathbb{N}}$ such that for every integer $k \in \mathbb{N}$, the sequence $(n^k \alpha_n)_{n \in \mathbb{N}}$ is bounded).
Let $(\alpha_n)_{n \in \mathbb{N}}$ be a sequence of $\mathcal{S}$ and $j \in \mathbb{N}$. We set, for $n \in \mathbb{N}$, $$R_n(j) = \sum_{p=n+1}^{+\infty} p^j \alpha_p$$
Show that the sequence $(R_n(j))_{n \in \mathbb{N}}$ has rapid decay.
QIII.B.1 Sequences and Series Uniform or Pointwise Convergence of Function Series/Sequences View
We denote by $\mathcal{S}$ the set of sequences of real numbers with rapid decay. Let $(\alpha_n)_{n \in \mathbb{N}}$ be a sequence of $\mathcal{S}$. For every integer $n \in \mathbb{N}$, $F_n(x) = \cos(n \arccos x)$ extended as a polynomial to $\mathbb{R}$.
Show that, for all $x \in [-1,1]$, the series $$\sum_{n \geqslant 0} \alpha_n F_n(x)$$ is convergent.
QIII.B.2 Sequences and Series Uniform or Pointwise Convergence of Function Series/Sequences View
We denote by $\mathcal{S}$ the set of sequences of real numbers with rapid decay. Let $(\alpha_n)_{n \in \mathbb{N}}$ be a sequence of $\mathcal{S}$. For every integer $n \in \mathbb{N}$, $F_n(x) = \cos(n \arccos x)$ extended as a polynomial to $\mathbb{R}$. We define the function $f$ on the segment $[-1,1]$ by: $$\forall x \in [-1,1], \quad f(x) = \sum_{n=0}^{+\infty} \alpha_n F_n(x).$$
Show that $f$ is of class $C^\infty$ on $[-1,1]$.
QIII.B.3 Sequences and Series Asymptotic Equivalents and Growth Estimates for Sequences/Series View
We denote by $\mathcal{S}$ the set of sequences of real numbers with rapid decay. Let $(\alpha_n)_{n \in \mathbb{N}}$ be a sequence of $\mathcal{S}$. For every integer $n \in \mathbb{N}$, $F_n(x) = \cos(n \arccos x)$ extended as a polynomial to $\mathbb{R}$. We define the function $f$ on the segment $[-1,1]$ by: $$\forall x \in [-1,1], \quad f(x) = \sum_{n=0}^{+\infty} \alpha_n F_n(x).$$ 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 $d(f, V_n) = \inf_{p \in V_n} \|f - p\|_\infty$.
Show that the sequence $(d(f, V_n))_{n \in \mathbb{N}}$ has rapid decay.
QIII.C.1 Sequences and Series Asymptotic Equivalents and Growth Estimates for Sequences/Series View
For a function $h \in C([-1,1])$, we denote by $\widetilde{h}$ the following $2\pi$-periodic function: $$\tilde{h} : \begin{cases} \mathbb{R} \rightarrow \mathbb{R} \\ \theta \mapsto h(\cos(\theta)) \end{cases}$$ The Fourier coefficients of $\widetilde{h}$ are given by: $$a_0(\widetilde{h}) = \frac{1}{2\pi} \int_{-\pi}^{\pi} \widetilde{h}(t)\, dt, \quad a_n(\widetilde{h}) = \frac{1}{\pi} \int_{-\pi}^{\pi} \widetilde{h}(t) \cos(nt)\, dt, \quad b_n(\widetilde{h}) = \frac{1}{\pi} \int_{-\pi}^{\pi} \widetilde{h}(t) \sin(nt)\, dt.$$
Let $f \in C^\infty([-1,1])$.
Show that the sequence $(a_n(\widetilde{f}))_{n \in \mathbb{N}}$ has rapid decay. What is the value of $b_n(\widetilde{f})$?
QIII.C.2 Sequences and Series Uniform or Pointwise Convergence of Function Series/Sequences View
For a function $h \in C([-1,1])$, we denote by $\widetilde{h}$ the following $2\pi$-periodic function: $$\tilde{h} : \begin{cases} \mathbb{R} \rightarrow \mathbb{R} \\ \theta \mapsto h(\cos(\theta)) \end{cases}$$
Let $f \in C^\infty([-1,1])$.
Show that the Fourier series of $\widetilde{f}$ converges normally to $\widetilde{f}$.
QIII.C.3 Sequences and Series Functional Equations and Identities via Series View
For a function $h \in C([-1,1])$, we denote by $\widetilde{h}$ the following $2\pi$-periodic function: $$\tilde{h} : \begin{cases} \mathbb{R} \rightarrow \mathbb{R} \\ \theta \mapsto h(\cos(\theta)) \end{cases}$$ For every integer $n \in \mathbb{N}$, $F_n(x) = \cos(n \arccos x)$ extended as a polynomial to $\mathbb{R}$.
Let $f \in C^\infty([-1,1])$.
Show that there exists a sequence $(\alpha_n(f))_{n \in \mathbb{N}}$ with rapid decay such that $$f(x) = \sum_{n=0}^{+\infty} \alpha_n(f) F_n(x)$$ for all $x \in [-1,1]$. Give an expression for $\alpha_n(f)$ in terms of $f$ and $n$.
QIII.D.1 Sequences and Series Asymptotic Equivalents and Growth Estimates for Sequences/Series View
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.
QIII.D.2 Sequences and Series Uniform or Pointwise Convergence of Function Series/Sequences View
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 $d(f, V_n) = \inf_{p \in V_n} \|f - p\|_\infty$. For a function $h \in C([-1,1])$, $\widetilde{h}$ denotes the $2\pi$-periodic function $\theta \mapsto h(\cos(\theta))$.
Let $f \in C([-1,1])$. We assume that the sequence $(d(f, V_n))_{n \in \mathbb{N}}$ has rapid decay.
The purpose of this question is to show that the function $f$ is of class $C^\infty$.
a) Let $k \in \mathbb{N}^*$. Show that, for $P \in E_{k-1}$, $a_k(\widetilde{f}) = a_k(\widetilde{f-P})$.
b) Deduce that the sequence $(a_n(\widetilde{f}))_{n \in \mathbb{N}}$ of Fourier coefficients of the function $\widetilde{f}$ has rapid decay.
c) Conclude.