Let $\left(f_n\right)_{n \geqslant 1}$ be a sequence of functions from $\mathbb{N}^*$ to $\mathbb{R}$ such that, for all $x \in \mathbb{N}^*$, the sequence $\left(f_n(x)\right)_{n \geqslant 1}$ converges to a real number $f(x)$ as $n$ tends to $+\infty$. We assume that there exists a function $h : \mathbb{N}^* \rightarrow [0, +\infty[$ such that $h(X)$ has finite expectation and such that $\left|f_n(m)\right| \leqslant h(m)$ for all $m$ and $n$ in $\mathbb{N}^*$. Justify that $E(f(X))$ has finite expectation and show that $$\lim_{n \rightarrow +\infty} E\left(f_n(X)\right) = E(f(X)).$$

Let $n$ be a non-zero natural number. Let $A \in \mathbb{C}_{2n}[X]$, split with simple roots, and $(\alpha_1, \ldots, \alpha_{2n})$ its roots. Show that $$\forall B \in \mathbb{C}_{2n-1}[X], \quad B(X) = \sum_{k=1}^{2n} B(\alpha_k) \frac{A(X)}{(X - \alpha_k) A'(\alpha_k)}$$

Let $n$ be a non-zero natural number. Let $A \in \mathbb{C}_{2n}[X]$, split with simple roots, and $(\alpha_1, \ldots, \alpha_{2n})$ its roots. Show that $$\forall B \in \mathbb{C}_{2n-1}[X], \quad B(X) = \sum_{k=1}^{2n} B(\alpha_k) \frac{A(X)}{(X - \alpha_k) A'(\alpha_k)}$$

We define, for all $x \in \mathbb{R}^{+*}$, $$\Gamma(x) = \int_0^{+\infty} t^{x-1} \mathrm{e}^{-t} \, \mathrm{d}t$$ Show that the function $\Gamma$ is continuous and strictly positive on $\mathbb{R}^{+*}$.

Let $n \in \mathbb{N}$. We consider $n+1$ distinct points in $I$, denoted $x_0 < x_1 < \cdots < x_n$, and the polynomials $L_0, \ldots, L_n$ defined in Q5.
Show that $(L_0, \ldots, L_n)$ is a basis of $\mathbb{R}_n[X]$.

Consider a quadrature formula $I_n(f) = \sum_{j=0}^n \lambda_j f(x_j)$ where $n \in \mathbb{N}$, $\lambda_0, \ldots, \lambda_n \in \mathbb{R}$ and $x_0 < x_1 < \cdots < x_n$ are $n+1$ distinct points in $I$. We assume that the coefficients $(\lambda_j)_{0 \leqslant j \leqslant n}$ are chosen as $$\forall j \in \llbracket 0, n \rrbracket, \quad \lambda_j = \int_I L_j(x) w(x)\,\mathrm{d}x,$$ where $(L_0, \ldots, L_n)$ is the Lagrange basis associated with the points $(x_0, \ldots, x_n)$. Thus, the formula $I_n(f)$ is of order $m \geqslant n$.
By reasoning with the polynomial $\prod_{i=0}^n (X - x_i)$, show that $m \leqslant 2n+1$.

Consider a quadrature formula $I_n(f) = \sum_{j=0}^n \lambda_j f(x_j)$ where the coefficients $(\lambda_j)_{0 \leqslant j \leqslant n}$ are chosen as $$\forall j \in \llbracket 0, n \rrbracket, \quad \lambda_j = \int_I L_j(x) w(x)\,\mathrm{d}x,$$ where $(L_0, \ldots, L_n)$ is the Lagrange basis associated with the points $(x_0, \ldots, x_n)$. Let $(p_n)_{n \in \mathbb{N}}$ be the sequence of orthogonal polynomials associated with the weight $w$.
Show that $m = 2n+1$ if and only if the $x_i$ are the roots of $p_{n+1}$.

In $\mathbb { R } [ X ]$ equipped with an inner product $( \cdot \mid \cdot )$, let $\left( V _ { n } \right) _ { n \in \mathbb { N } }$ be an orthogonal system (a sequence of polynomials that is orthogonal and where each $V_n$ is monic of degree $n$). Show that, for all $n \in \mathbb { N }$, the family $\left( V _ { 0 } , V _ { 1 } , \ldots , V _ { n } \right)$ is an orthogonal basis of the vector space $\mathbb { R } _ { n } [ X ]$ of polynomials with real coefficients of degree at most $n$.

We consider the case where $I = [-1,1]$ and $w(x) = 1$. Let $(p_n)_{n \in \mathbb{N}}$ be the sequence of orthogonal polynomials associated with the weight $w$ (monic, $\deg(p_n) = n$, orthogonal for $\langle f, g \rangle = \int_{-1}^1 f(x)g(x)\,\mathrm{d}x$).
Determine the first four orthogonal polynomials $(p_0, p_1, p_2, p_3)$ associated with the weight $w$.

Let $g : [ 0,1 ] \rightarrow [ 0,1 ]$ be an increasing function (but not necessarily continuous). Show that $g$ has at least one fixed point. Hint: one may consider the set $$E = \{ x \in [ 0,1 ] ; x \leqslant g ( x ) \} .$$

Let $[ a , b ]$ be a closed bounded interval of $\mathbb { R }$. If $\phi : [ a , b ] \rightarrow [ a , b ]$ is continuous, show that $\phi$ has at least one fixed point.

If $\phi : \mathbb { R } \rightarrow \mathbb { R }$ is of class $\mathcal { C } ^ { 1 }$ and satisfies $$\sup \left\{ \left| \phi ^ { \prime } ( x ) \right| ; x \in \mathbb { R } \right\} < 1$$ show that $\phi$ has at least one fixed point (one may study the sign of $x - \phi ( x )$ for $| x |$ sufficiently large). Show that this fixed point is unique.

By means of the function $\psi ( x ) = \sqrt { 1 + x ^ { 2 } }$, show that in the previous question hypothesis (1) cannot be replaced by $$\forall x \in \mathbb { R } , \left| \phi ^ { \prime } ( x ) \right| < 1$$

Let $\ell$ be a strictly positive integer. Let $F$ be a closed subset of $\mathbb { R } ^ { \ell }$ and let $\phi : F \rightarrow F$ be a map. We assume that there exists $k \in [ 0,1 [$ such that $$\forall x \in F , \forall y \in F , \quad \| \phi ( y ) - \phi ( x ) \| \leqslant k \| y - x \| .$$
(a) We choose a point $x _ { 0 } \in F$. Show that the formula $x _ { n + 1 } = \phi \left( x _ { n } \right)$ defines a sequence $\left( x _ { n } \right) _ { n \geqslant 0 }$ of elements of $F$, and that this sequence is convergent in $F$.
(b) Deduce that $\phi$ has a unique fixed point in $F$.
(c) This fixed point being denoted $x ^ { * }$, bound $\left\| x _ { n } - x ^ { * } \right\|$ as a function of $\left\| x _ { 0 } - x ^ { * } \right\|$.
(d) In what precedes, we assume that $$\phi = \underbrace { \theta \circ \cdots \circ \theta } _ { m \text { times } } ,$$ where $\theta : F \rightarrow F$ is a map and $m \geqslant 2$ is an integer. Show that $\theta$ has a fixed point, and a unique one, in $F$.

Let $g : [ 0,1 ] \rightarrow [ 0,1 ]$ be an increasing function (but not necessarily continuous). Show that $g$ has at least one fixed point. Hint: one may consider the set $$E = \{ x \in [ 0,1 ] ; x \leqslant g ( x ) \} .$$

Let $C$ be a non-empty, convex and closed subset of $\mathbb{R}^d$ and $x \in \mathbb{R}^d$, consider: $$\inf_{y \in C} \|x - y\|^2. \tag{1}$$ Show that (1) has a unique solution (that is, there exists a unique $y \in C$ such that $\|x - y\|^2 \leqslant \|x - z\|^2$ for all $z \in C$) which we will call the projection of $x$ onto $C$ and denote $\operatorname{proj}_C(x)$. Show that $x = \operatorname{proj}_C(x)$ if and only if $x \in C$.

Let $C$ be a non-empty, convex and closed subset of $\mathbb{R}^d$ and $x \in \mathbb{R}^d$, consider: $$\inf_{y \in C} \|x - y\|^2. \tag{1}$$ Show that (1) has a unique solution (that is, there exists a unique $y \in C$ such that $\|x - y\|^2 \leq \|x - z\|^2$ for all $z \in C$) which we will call the projection of $x$ onto $C$ and denote $\operatorname{proj}_C(x)$. Show that $x = \operatorname{proj}_C(x)$ if and only if $x \in C$.

Show the interpolation inequality $$\forall f \in \mathcal{C}^{1}([0,1]), \quad \|f\|_{\infty} \leqslant \left\|f^{\prime}\right\|_{\infty} + C\left|f\left(x_{1}\right)\right|$$ with $C = 1$.

Show the interpolation inequality $$\forall f \in \mathcal{C}^1([0,1]), \quad \|f\|_\infty \leqslant \left\|f^\prime\right\|_\infty + C\left|f\left(x_1\right)\right|$$ with $C = 1$.

Show that $\langle \cdot , \cdot \rangle$ is an inner product on $\mathbb { R } _ { n - 1 } [ X ]$, where $$\langle P , Q \rangle = \sum _ { k = 1 } ^ { n } P \left( a _ { k } \right) Q \left( a _ { k } \right).$$

Show that, for all $\alpha \in \mathbb { R } _ { + } ^ { * } , p _ { \alpha }$ belongs to $E$, where $E$ is the set of continuous functions $f$ from $\mathbb { R } _ { + } ^ { * }$ to $\mathbb { R }$ such that the integral $\int _ { 0 } ^ { + \infty } f ^ { 2 } ( t ) \frac { \mathrm { e } ^ { - t } } { t } \mathrm {~d} t$ converges, and $p_\alpha$ is the function $t \mapsto t^\alpha$.

Let the functions $f$, $g$ and $D$ be defined on $\mathbb{R} \backslash \mathbb{Z}$ by: $$f(x) = \pi \operatorname{cotan}(\pi x) = \pi \frac{\cos(\pi x)}{\sin(\pi x)}, \quad g(x) = \frac{1}{x} + \sum_{n=1}^{+\infty} \left(\frac{1}{x+n} + \frac{1}{x-n}\right).$$ We set $D = f - g$.
Show that the functions $g$ and $D$ are odd.

Let the functions $f$, $g$ and $D$ be defined on $\mathbb{R} \backslash \mathbb{Z}$ by: $$f(x) = \pi \operatorname{cotan}(\pi x) = \pi \frac{\cos(\pi x)}{\sin(\pi x)}, \quad g(x) = \frac{1}{x} + \sum_{n=1}^{+\infty} \left(\frac{1}{x+n} + \frac{1}{x-n}\right).$$ We set $D = f - g$.
Show that the functions $g$ and $D$ are periodic with period 1.

Let the functions $f$, $g$ and $D$ be defined on $\mathbb{R} \backslash \mathbb{Z}$ by: $$f(x) = \pi \operatorname{cotan}(\pi x) = \pi \frac{\cos(\pi x)}{\sin(\pi x)}, \quad g(x) = \frac{1}{x} + \sum_{n=1}^{+\infty} \left(\frac{1}{x+n} + \frac{1}{x-n}\right).$$ We set $D = f - g$.
Show that the functions $g$ and $D$ are continuous on $\mathbb{R} \backslash \mathbb{Z}$.

Let $f(x) = \pi \operatorname{cotan}(\pi x)$, $g(x) = \frac{1}{x} + \sum_{n=1}^{+\infty}\left(\frac{1}{x+n} + \frac{1}{x-n}\right)$, and $D = f - g$. Show that the functions $g$ and $D$ are odd.