In this question, we study the case $\lambda(t) = t^2$ and $f(t) = \dfrac{1}{1+t^2}$ for all $t \in \mathbb{R}^+$.
II.C.1) Determine $E$. What is the value of $Lf(0)$?
II.C.2) Prove that $Lf$ is differentiable.
II.C.3) Show the existence of a constant $A > 0$ such that for all $x > 0$, we have $$Lf(x) - (Lf)^{\prime}(x) = \frac{A}{\sqrt{x}}.$$
II.C.4) We denote $g(x) = e^{-x}Lf(x)$ for $x \geqslant 0$.
Show that for all $x \geqslant 0$, we have $$g(x) = \frac{\pi}{2} - A\int_0^x \frac{e^{-t}}{\sqrt{t}}\,dt.$$
II.C.5) Deduce from this the value of the integral $\displaystyle\int_0^{+\infty} e^{-t^2}\,dt$.

In this part, $\lambda(t) = t$ for all $t \in \mathbb{R}^+$ and $f(t) = \dfrac{t}{e^t - 1} - 1 + \dfrac{t}{2}$ for all $t \in \mathbb{R}^{+*}$.
Show that $f$ extends by continuity at 0.

$\mathcal{P}$ denotes the vector space of polynomial functions with complex coefficients. We denote for every pair $(P,Q) \in \mathcal{P}^2$, $$\langle P, Q \rangle = \int_0^{+\infty} e^{-t}\bar{P}(t)Q(t)\,dt.$$
Verify that $\langle \cdot, \cdot \rangle$ defines an inner product on $\mathcal{P}$.

$\mathcal{P}$ denotes the vector space of polynomial functions with complex coefficients, with inner product $\langle P, Q \rangle = \displaystyle\int_0^{+\infty} e^{-t}\bar{P}(t)Q(t)\,dt$.
We denote by $D$ the differentiation endomorphism and $U$ the endomorphism of $\mathcal{P}$ defined by $$U(P)(t) = e^t D\left(te^{-t}P^{\prime}(t)\right).$$
Verify that $U$ is an endomorphism of $\mathcal{P}$.

$\mathcal{P}$ denotes the vector space of polynomial functions with complex coefficients, with inner product $\langle P, Q \rangle = \displaystyle\int_0^{+\infty} e^{-t}\bar{P}(t)Q(t)\,dt$, and $U$ is the endomorphism of $\mathcal{P}$ defined by $U(P)(t) = e^t D\left(te^{-t}P^{\prime}(t)\right)$.
Show that for all $P$ and $Q$ in $\mathcal{P}$, we have $$\langle U(P), Q \rangle = \langle P, U(Q) \rangle.$$

$\mathcal{P}$ denotes the vector space of polynomial functions with complex coefficients, with inner product $\langle P, Q \rangle = \displaystyle\int_0^{+\infty} e^{-t}\bar{P}(t)Q(t)\,dt$, and $U$ is the endomorphism of $\mathcal{P}$ defined by $U(P)(t) = e^t D\left(te^{-t}P^{\prime}(t)\right)$.
Show that $U$ admits eigenvalues in $\mathbb{C}$, that they are real and that two eigenvectors associated with distinct eigenvalues are orthogonal.

$\mathcal{P}$ denotes the vector space of polynomial functions with complex coefficients, with inner product $\langle P, Q \rangle = \displaystyle\int_0^{+\infty} e^{-t}\bar{P}(t)Q(t)\,dt$, and $U$ is the endomorphism of $\mathcal{P}$ defined by $U(P)(t) = e^t D\left(te^{-t}P^{\prime}(t)\right)$.
Let $\lambda$ be an eigenvalue of $U$ and $P$ an eigenvector associated with it.
VIII.F.1) Show that $P$ is a solution of a simple linear differential equation that we will specify.
VIII.F.2) What is the relationship between $\lambda$ and the degree of $P$?

$\mathcal{P}$ denotes the vector space of polynomial functions with complex coefficients, with inner product $\langle P, Q \rangle = \displaystyle\int_0^{+\infty} e^{-t}\bar{P}(t)Q(t)\,dt$, and $U$ is the endomorphism of $\mathcal{P}$ defined by $U(P)(t) = e^t D\left(te^{-t}P^{\prime}(t)\right)$.
We consider on $[0, +\infty[$ the differential equation $$(E_n): \quad tP^{\prime\prime} + (1-t)P^{\prime} + nP = 0$$ with $n \in \mathbb{N}$ and unknown $P \in \mathcal{P}$.
VIII.G.1) By applying the transformation $L$ with $\lambda(t) = t$ to $(E_n)$, show that if $P$ is a solution of $(E_n)$ on $[0, +\infty[$, then its image $Q$ by $L$ is a solution of a differential equation $(E_n^{\prime})$ of order 1 on $]1, +\infty[$.
VIII.G.2) Solve the equation $(E_n^{\prime})$ on $]1, +\infty[$ and deduce from this the eigenvalues and eigenvectors of the endomorphism $U$.
VIII.G.3) What is the relationship between the above and the polynomial functions defined for $n \in \mathbb{N}$ by $P_n(t) = e^t D^n\left(e^{-t}t^n\right)$?

We denote by $\mathcal { C }$ the vector space of continuous functions from $[ 0,1 ]$ to $\mathbb { R }$, equipped with the supremum norm $\| f \| _ { \infty } = \sup _ { x \in [ 0,1 ] } | f ( x ) |$. For $f \in \mathcal { C }$ and $n \in \mathbb { N } ^ { * }$, the $n$-th Bernstein polynomial of $f$ is defined by $$B _ { n } ( f ) ( x ) = \sum _ { k = 0 } ^ { n } f \left( \frac { k } { n } \right) \binom { n } { k } x ^ { k } ( 1 - x ) ^ { n - k }$$ for all $x \in [ 0,1 ]$.
Let $f \in \mathcal { C }$. Show, for all $x \in [ 0,1 ]$, the relation $$B _ { n } ( f ) ( x ) - f ( x ) = \sum _ { k = 0 } ^ { n } \left( f \left( \frac { k } { n } \right) - f ( x ) \right) \binom { n } { k } x ^ { k } ( 1 - x ) ^ { n - k }$$

We denote by $\mathcal { C }$ the vector space of continuous functions from $[ 0,1 ]$ to $\mathbb { R }$, equipped with the supremum norm $\| f \| _ { \infty } = \sup _ { x \in [ 0,1 ] } | f ( x ) |$. For $f \in \mathcal { C }$ and $n \in \mathbb { N } ^ { * }$, the $n$-th Bernstein polynomial of $f$ is defined by $$B _ { n } ( f ) ( x ) = \sum _ { k = 0 } ^ { n } f \left( \frac { k } { n } \right) \binom { n } { k } x ^ { k } ( 1 - x ) ^ { n - k }$$ for all $x \in [ 0,1 ]$.
a) Show that if $f$ is $\delta$-Lipschitz, then $\left\| B _ { n } ( f ) - f \right\| _ { \infty } \leqslant \frac { \delta } { 2 \sqrt { n } }$ for all integer $n \geqslant 1$. b) Deduce that if $f$ is of class $C ^ { 1 }$, then there exists a real $c$ such that, for all $n \in \mathbb { N } ^ { * } , \left\| B _ { n } ( f ) - f \right\| _ { \infty } \leqslant \frac { c } { \sqrt { n } }$. c) Extend the previous result to the case where $f$ is a continuous function, piecewise $C ^ { 1 }$.

We denote by $\mathcal { C }$ the vector space of continuous functions from $[ 0,1 ]$ to $\mathbb { R }$, equipped with the supremum norm $\| f \| _ { \infty } = \sup _ { x \in [ 0,1 ] } | f ( x ) |$. For $f \in \mathcal { C }$ and $n \in \mathbb { N } ^ { * }$, the $n$-th Bernstein polynomial of $f$ is defined by $$B _ { n } ( f ) ( x ) = \sum _ { k = 0 } ^ { n } f \left( \frac { k } { n } \right) \binom { n } { k } x ^ { k } ( 1 - x ) ^ { n - k }$$ for all $x \in [ 0,1 ]$.
Let $f : [ 0,1 ] \rightarrow \mathbb { R }$ be a continuous function, piecewise $C ^ { 1 }$. Deduce from the above that, for all real $r > 0$, there exists a polynomial $P$ with real coefficients such that $\forall x \in [ 0,1 ] , f ( x ) - r \leqslant P ( x ) \leqslant f ( x ) + r$.

Let $\left( a _ { n } \right) _ { n \geqslant 0 }$ be a real sequence with $\sum a_n x^n$ having radius of convergence 1, with sum $f(x) = \sum_{n=0}^{+\infty} a_n x^n$ satisfying $f(x) \sim \frac{1}{1-x}$ as $x \to 1^-$, and with $a_n \geqslant 0$ for all $n \in \mathbb{N}$. We denote $A _ { n } = \sum _ { k = 0 } ^ { n } a _ { k }$.
For all $x \in \left[ 0,1 \left[ \right. \right.$ and all $n \in \mathbb { N }$, show that $f ( x ) \geqslant A _ { n } x ^ { n }$.

Let $g : [ 0,1 ] \rightarrow \mathbb { R }$ be the function such that $g ( x ) = 1 / x$ if $x \geqslant \mathrm { e } ^ { - 1 }$ and $g ( x ) = 0$ otherwise. We fix a real $\varepsilon \in ] 0 , \mathrm { e } ^ { - 1 } [$. We define two continuous applications $g ^ { + } , g ^ { - } : [ 0,1 ] \rightarrow \mathbb { R }$ as follows:

• $g ^ { + }$ is affine on $\left[ \mathrm { e } ^ { - 1 } - \varepsilon , \mathrm { e } ^ { - 1 } \right]$ and coincides with $g$ on $\left[ 0 , \mathrm { e } ^ { - 1 } - \varepsilon \right] \cup \left[ \mathrm { e } ^ { - 1 } , 1 \right]$;
• $g ^ { - }$ is affine on $\left[ \mathrm { e } ^ { - 1 } , \mathrm { e } ^ { - 1 } + \varepsilon \right]$ and coincides with $g$ on $\left[ 0 , \mathrm { e } ^ { - 1 } \left[ \cup \left[ \mathrm { e } ^ { - 1 } + \varepsilon , 1 \right] \right. \right.$.

Establish the existence of two polynomials $P , Q$ with real coefficients such that: $$\forall x \in [ 0,1 ] , \quad g ^ { - } ( x ) - \varepsilon \leqslant P ( x ) \leqslant g ( x ) \leqslant Q ( x ) \leqslant g ^ { + } ( x ) + \varepsilon$$

Justify the equality
$$\forall t \in \mathbb { R } \quad G _ { x } ( t ) = e ^ { i x \sin t } = \sum _ { n = - \infty } ^ { + \infty } \varphi _ { n } ( x ) e ^ { i n t }$$
What can be said about the convergence of the Fourier series of $G _ { x }$ ?

Show that for all $k$ in $\mathbb { N } ^ { * } , \left| \varphi _ { n } ( x ) \right| = o \left( \frac { 1 } { n ^ { k } } \right)$ as $n$ tends to $+ \infty$.
Use Fourier series of successive derivatives of $G _ { x }$.

By expressing $G _ { x } ( - t )$ in terms of $G _ { x } ( t )$, show that for $n$ in $\mathbb { Z } , \varphi _ { n } ( x ) \in \mathbb { R }$.

Express $G _ { x } ( t + \pi )$ and deduce the following equalities for $n$ in $\mathbb { Z }$ :
$$\varphi _ { n } ( - x ) = ( - 1 ) ^ { n } \varphi _ { n } ( x ) = \varphi _ { - n } ( x )$$
What can be said about the parity of $\varphi _ { n }$ for $n \in \mathbb { Z }$ ?

Calculate $\sum _ { n = - \infty } ^ { + \infty } \left| \varphi _ { n } ( x ) \right| ^ { 2 }$.

Justify that for real $x$, $\left| \varphi _ { n } ( x ) \right| \leqslant 1$.

We approximate $\varphi _ { n } ( x )$ using partial sums
$$S _ { m } = \sum _ { p = 0 } ^ { m } ( - 1 ) ^ { p } a _ { p } \quad \text { with } \quad m \in \mathbb { N } \quad \text { and } \quad a _ { p } = \frac { 1 } { p ! ( n + p ) ! } \left( \frac { x } { 2 } \right) ^ { n + 2 p }$$
Write a Maple or Mathematica function, CalculPhi, with arguments $(n, x, \varepsilon)$ returning an approximate value of $\varphi _ { n } ( x )$ to within $\varepsilon$. The coefficients $a _ { p }$ will be calculated by recursion.

We introduce the differential equation
$$z _ { 1 } ^ { \prime \prime } ( x ) + c ^ { 2 } z _ { 1 } ( x ) = 0 \quad \text { with } \quad c > 0 \tag{IV.1}$$
In this question, we fix $n \in \mathbb { N }$ and $c \in ] 0,1 [$. We set $z ( x ) = \sqrt { x } \varphi _ { n } ( x )$, for $x > 0$.
Justify that there exists a real $A > 0$ such that for $x > A , q ( x ) > c ^ { 2 }$ ($q$ defined in III.A.2).

Let $n \in \mathbb { N }$.
Show that we can order the zeros of $\varphi _ { n }$, that is, there exists a strictly increasing sequence $\left( \alpha _ { k } ^ { ( n ) } \right) _ { k \in \mathbb { N } }$ of zeros of $\varphi _ { n }$ such that $\varphi _ { n }$ does not vanish on $] 0 , \alpha _ { 0 } ^ { ( n ) } [$ and on every interval $] \alpha _ { k } ^ { ( n ) } , \alpha _ { k + 1 } ^ { ( n ) } [$ with $k$ in $\mathbb { N }$ and that $\lim _ { k \rightarrow \infty } \alpha _ { k } ^ { ( n ) } = + \infty$.
Construct the sequence $\left( \alpha _ { k } ^ { ( n ) } \right) _ { k \in \mathbb { N } }$ by induction on $k$ by showing that the set $\mathcal { Z } _ { k }$ of zeros of $\varphi _ { n }$ in the interval $] \alpha _ { k } ^ { ( n ) } , + \infty [$ has a smallest element.

Let $n \in \mathbb { N }$. Assuming the sequence $\left( \alpha _ { k } ^ { ( n ) } \right) _ { k \in \mathbb { N } }$ of zeros of $\varphi_n$ constructed in IV.D.1, deduce that the sequence $\left( \alpha _ { k } ^ { ( n ) } \right) _ { k \in \mathbb { N } }$ satisfies the asymptotic distribution property:
$$\forall c \in ] 0,1 \left[ , \quad \exists j \in \mathbb { N } \quad \text { such that } \quad \forall k \in \mathbb { N } , 0 < \alpha _ { j + k + 1 } ^ { ( n ) } - \alpha _ { j + k } ^ { ( n ) } < \frac { \pi } { c } \right.$$

Throughout the problem, $\mathbb { R } ^ { 2 }$ is equipped with the canonical Euclidean inner product denoted $\langle$,$\rangle$ and the associated norm $\| \|$. Let $f$ and $g$ be in $\mathcal { C } ^ { 1 } \left( \mathbb { R } ^ { 2 } , \mathbb { R } \right)$ satisfying the Cauchy-Riemann equations: $$\frac { \partial f } { \partial x } = \frac { \partial g } { \partial y } \quad \text { and } \quad \frac { \partial f } { \partial y } = - \frac { \partial g } { \partial x }$$ We define two functions on $\mathbb { R } _ { + } ^ { * } \times \mathbb { R }$ by $$\forall ( r , \theta ) \in \mathbb { R } _ { + } ^ { * } \times \mathbb { R } , \quad \widetilde { f } ( r , \theta ) = f ( r \cos \theta , r \sin \theta ) \quad \text { and } \quad \widetilde { g } ( r , \theta ) = g ( r \cos \theta , r \sin \theta )$$
Express $\frac { \partial \widetilde { f } } { \partial r } ( r , \theta )$ and $\frac { \partial \widetilde { f } } { \partial \theta } ( r , \theta )$ in terms of $r , \theta , \frac { \partial f } { \partial x } ( r \cos \theta , r \sin \theta )$ and $\frac { \partial f } { \partial y } ( r \cos \theta , r \sin \theta )$.

Throughout the problem, $\mathbb { R } ^ { 2 }$ is equipped with the canonical Euclidean inner product denoted $\langle$,$\rangle$ and the associated norm $\| \|$. Let $f$ and $g$ be in $\mathcal { C } ^ { 1 } \left( \mathbb { R } ^ { 2 } , \mathbb { R } \right)$ satisfying the Cauchy-Riemann equations: $$\frac { \partial f } { \partial x } = \frac { \partial g } { \partial y } \quad \text { and } \quad \frac { \partial f } { \partial y } = - \frac { \partial g } { \partial x }$$ We define two functions on $\mathbb { R } _ { + } ^ { * } \times \mathbb { R }$ by $$\forall ( r , \theta ) \in \mathbb { R } _ { + } ^ { * } \times \mathbb { R } , \quad \widetilde { f } ( r , \theta ) = f ( r \cos \theta , r \sin \theta ) \quad \text { and } \quad \widetilde { g } ( r , \theta ) = g ( r \cos \theta , r \sin \theta )$$
For all $( r , \theta ) \in \mathbb { R } _ { + } ^ { * } \times \mathbb { R }$, show that $\frac { \partial \widetilde { f } } { \partial r } ( r , \theta ) = \frac { 1 } { r } \times \frac { \partial \widetilde { g } } { \partial \theta } ( r , \theta )$ and $\frac { \partial \widetilde { g } } { \partial r } ( r , \theta ) = - \frac { 1 } { r } \times \frac { \partial \widetilde { f } } { \partial \theta } ( r , \theta )$.