Explain what the Python function \texttt{maxSp} defined by: \begin{verbatim} def maxSp(Q:np.ndarray, k:int, t:float) -> float: n = Q.shape[1] E = np.exp(t * np.array(range(n))) A = Q * E B = puiss2k(A, k) C = np.dot(A, B) return C.trace() / B.trace() \end{verbatim} does.

We are given a real $a > 0$. Let $E_4$ be the space of functions continuous on $[0,a]$, taking values in $\mathbb{R}$, of class $\mathcal{C}^1$ piecewise and furthermore satisfying $f(a) = 0$. Let $\varphi:[0,a] \rightarrow \mathbb{R}$ be of class $\mathcal{C}^1$ satisfying $\varphi(a) = 0$ and, for all $x \in [0,a]$, $\varphi'(x) < 0$. Determine an inner product on $E_4$ such that the function $(x,y) \mapsto \min(\varphi(x), \varphi(y))$ is a reproducing kernel on the pre-Hilbert space $E_4$.

Let $(E, \langle \cdot, \cdot \rangle)$ be a reproducing kernel Hilbert space on an interval $I$, with reproducing kernel $K$. For all $(x,y) \in I^2$, we set $k_x(y) = K(x,y)$. Let $x \in I$ and $V_x$ defined on $E$ by $V_x(f) = f(x)$. We set $$N(V_x) = \sup_{\|f\|=1} |f(x)|$$ Prove that $$N(V_x) = \sqrt{\langle k_x, k_x \rangle}.$$

Let $(E, \langle \cdot, \cdot \rangle)$ be a reproducing kernel Hilbert space on an interval $I$, with reproducing kernel $K$. For all $(x,y) \in I^2$, we set $k_x(y) = K(x,y)$. Suppose that $K$ is continuous on $I \times I$. Prove that all functions in $E$ are continuous.

We denote by $E$ the vector space of continuous functions defined on $[0,1]$ and taking values in $\mathbb{R}$ equipped with the inner product defined by $$\langle f, g \rangle = \int_0^1 f(t) g(t)\,\mathrm{d}t$$ We consider a function $A:[0,1] \times [0,1] \rightarrow \mathbb{R}$ continuous. We are interested in the application $T: E \rightarrow E$ defined by $$T(f)(x) = \int_0^1 A(x,t) f(t)\,\mathrm{d}t$$ We suppose that $\ker T$ is finite-dimensional. Justify that $T$ induces an isomorphism from $(\ker T)^\perp$ onto $\operatorname{Im} T$.

We denote by $E$ the vector space of continuous functions defined on $[0,1]$ and taking values in $\mathbb{R}$ equipped with the inner product defined by $$\langle f, g \rangle = \int_0^1 f(t) g(t)\,\mathrm{d}t$$ We consider a function $A:[0,1] \times [0,1] \rightarrow \mathbb{R}$ continuous. We are interested in the application $T: E \rightarrow E$ defined by $$T(f)(x) = \int_0^1 A(x,t) f(t)\,\mathrm{d}t$$ We suppose that $\ker T$ is finite-dimensional. We denote by $S$ the inverse bijection of the isomorphism induced by $T$ from $(\ker T)^\perp$ onto $\operatorname{Im} T$. We define the inner product $\varphi$ on $\operatorname{Im} T$ by setting, for all $(f,g) \in (\operatorname{Im} T)^2$, $$\varphi(f,g) = \langle S(f), S(g) \rangle$$ We consider the application $K$ defined on $[0,1]^2$ by $$K(x,y) = \int_0^1 A(x,t) A(y,t)\,\mathrm{d}t$$ Show that $(\operatorname{Im} T, \varphi)$ is a reproducing kernel Hilbert space, with kernel $K$.

In the case $I = [0,1]$ and $\forall x \in I, w(x) = 1$, we seek to approximate $\int_0^1 f(x)\,\mathrm{d}x$ when $f$ is a continuous function from $[0,1]$ to $\mathbb{R}$.
Determine the order of the quadrature formula $I_0(f) = f(0)$ and represent graphically the associated error $e(f)$.

Let $K$ be a closed, bounded and infinite subset of $\mathbb{C}$. Show that $\|P\|_K$ belongs to $\mathbb{R}$ for all $P \in \mathbb{C}[X]$.

Let $K$ be a closed, bounded and infinite subset of $\mathbb{C}$. Verify that $\|\cdot\|_K$ is a norm on $\mathbb{C}[X]$.

Let $K$ be a closed, bounded and infinite subset of $\mathbb{C}$. Let $Q$ and $R$ be two non-zero polynomials in $\mathbb{C}[X]$. Show that: $$\|Q\|_K \|R\|_K \geq \|QR\|_K.$$

Let $K$ be a closed, bounded and infinite subset of $\mathbb{C}$. Show that, if the inequality $\|Q\|_K \|R\|_K \geq \|QR\|_K$ is an equality, then there exists $z_0 \in K$ such that: $$\left|Q\left(z_0\right)\right| = \|Q\|_K, \quad \left|R\left(z_0\right)\right| = \|R\|_K \quad \text{and} \quad \left|Q\left(z_0\right)R\left(z_0\right)\right| = \|QR\|_K.$$

Let $K$ be a closed, bounded and infinite subset of $\mathbb{C}$. We fix two non-zero natural integers $n$ and $m$, and we set: $$C_{n,m}^K = \sup\left\{\left.\frac{\|Q\|_K \|R\|_K}{\|QR\|_K}\right\rvert\, Q \in \mathbb{C}_n[X]\backslash\{0\}, R \in \mathbb{C}_m[X]\backslash\{0\}\right\} \in \mathbb{R} \cup \{+\infty\}$$ Show that $C_{n,m}^K > 1$.
To do this, one may choose two distinct elements $a$ and $b$ in $K$ and verify that, for $\rho \in \mathbb{R}$ sufficiently large, we have $\left\|Q_\rho R_\rho\right\|_K < \left\|Q_\rho\right\|_K \left\|R_\rho\right\|_K$ with $Q_\rho(X) = X - (\rho(b-a)+a)$ and $R_\rho(X) = X - (\rho(a-b)+b)$.

Let $K$ be a closed, bounded and infinite subset of $\mathbb{C}$. We fix two non-zero natural integers $n$ and $m$. We introduce the $\mathbb{C}$-vector space $V = \mathbb{C}_n[X] \times \mathbb{C}_m[X]$ as well as the set: $$E = \left\{(Q,R) \in V \mid \|Q\|_K = \|R\|_K = 1\right\}.$$ Show that there exists a pair $\left(Q_0, R_0\right) \in E$ such that: $$\left\|Q_0 R_0\right\|_K = \inf\left\{\|QR\|_K \mid (Q,R) \in E\right\}.$$ To do this, one may equip $V$ with the norm defined by $$\|(Q,R)\| = \|Q\|_K + \|R\|_K$$ for $(Q,R) \in V$, then study the application $$\begin{aligned} f : E &\rightarrow \mathbb{R} \\ (Q,R) &\mapsto \|QR\|_K. \end{aligned}$$

Let $K$ be a closed, bounded and infinite subset of $\mathbb{C}$. We fix two non-zero natural integers $n$ and $m$, and we set: $$C_{n,m}^K = \sup\left\{\left.\frac{\|Q\|_K \|R\|_K}{\|QR\|_K}\right\rvert\, Q \in \mathbb{C}_n[X]\backslash\{0\}, R \in \mathbb{C}_m[X]\backslash\{0\}\right\} \in \mathbb{R} \cup \{+\infty\}$$ Deduce that there exist two monic polynomials $Q_1 \in \mathbb{C}_n[X]$ and $R_1 \in \mathbb{C}_m[X]$ such that: $$\frac{\left\|Q_1\right\|_K \left\|R_1\right\|_K}{\left\|Q_1 R_1\right\|_K} = C_{n,m}^K.$$

Let $Q \in \mathbb{C}[X]$ be a non-zero polynomial. Verify that the integral: $$\int_0^{2\pi} \ln\left|Q\left(e^{i\theta}\right)\right| d\theta$$ converges absolutely in the sense of Definition 1.
One may use the d'Alembert-Gauss theorem.

Let $Q \in \mathbb{C}[X]$ be a non-zero polynomial. We set for $p > 0$: $$M_p(Q) = \frac{1}{2\pi} \int_0^{2\pi} \left|Q\left(e^{i\theta}\right)\right|^p d\theta$$ Explain why $M_p(Q)$ is strictly positive for all $p > 0$.

Let $Q \in \mathbb{C}[X]$ be a non-zero polynomial. We define the function: $$\begin{aligned} \varphi : [0,+\infty[ &\rightarrow \mathbb{R} \\ p &\mapsto \begin{cases} \ln\left(M_p(Q)\right) & \text{if } p > 0 \\ 0 & \text{if } p = 0 \end{cases} \end{aligned}$$ where $M_p(Q) = \frac{1}{2\pi}\int_0^{2\pi}\left|Q(e^{i\theta})\right|^p d\theta$. Show that $\varphi$ is continuous on $[0,+\infty[$.

Let $I = [a,b]$ be a segment of $\mathbb{R}$, and let $n$ and $m$ be two non-zero natural integers. We are given two distinct real numbers $c$ and $d$ and we set: $$J = \begin{cases} [c,d] & \text{if } c < d \\ [d,c] & \text{if } d < c. \end{cases}$$ Let $A \in \mathbb{C}_n[X]$ and $B \in \mathbb{C}_m[X]$ be two non-zero polynomials. Show that there exist polynomials $C \in \mathbb{C}_n[X]$ and $D \in \mathbb{C}_m[X]$ satisfying the following properties: $$\begin{gathered} \|A\|_I = \|C\|_J, \quad \|B\|_I = \|D\|_J, \quad \|AB\|_I = \|CD\|_J \\ A(a) = C(c), \quad B(b) = D(d). \end{gathered}$$

We choose $I = [-1,1]$ and fix any very good extremal pair $(Q, R)$. We set $P = QR$ and denote by $x_1 \leq \ldots \leq x_{n+m}$ the roots of $P$ counted with multiplicity. We are given an integer $k \in \{m+1, m+2, \ldots, m+n-1\}$ and we set: $$S(X) = \left(X - x_k\right)\left(X - x_{k+1}\right).$$ Show that for all $\epsilon > 0$, there exists a polynomial $T \in \mathbb{R}[X]$ such that $S - T$ has degree 1 and: $$\begin{gathered} \|S - T\|_I \leq \epsilon \\ |T(-1)| = |S(-1)| \\ \forall x \in ]-1,1] \backslash ]x_k - \epsilon, x_{k+1} + \epsilon[, |T(x)| < |S(x)|. \end{gathered}$$ Deduce that there exists $y \in ]x_k, x_{k+1}[$ such that $|P(y)| = \|P\|_I$.
To handle this last point, one may proceed by contradiction, write $Q$ in the form $SU$ for a certain polynomial $U$, then verify that if $\epsilon$ is chosen appropriately, the pair $(TU, R)$ forms a very good extremal pair.

We choose $I = [-1,1]$ and fix any very good extremal pair $(Q, R)$. We set $P = QR$ and denote by $x_1 \leq \ldots \leq x_{n+m}$ the roots of $P$ counted with multiplicity. Following the method used in question 4.39, show that there exists an element $y \in ]x_m, x_{m+1}[$ such that $|P(y)| = \|P\|_I$.

We choose $I = [-1,1]$ and fix any very good extremal pair $(Q, R)$. We set $P = QR$. Show that $P$ satisfies the differential equation: $$\|P\|_I^2 - P^2 = \frac{1}{(n+m)^2}\left(1 - X^2\right)P'^2$$

We choose $I = [-1,1]$ and fix any very good extremal pair $(Q, R)$. We set $P = QR$. Deduce from question 4.41 that: $$\left(1 - X^2\right)P'' - XP' + (n+m)^2 P = 0$$

We define, for all $x \in \mathbb{R}^{+*}$, $$\Gamma(x) = \int_0^{+\infty} t^{x-1} \mathrm{e}^{-t} \, \mathrm{d}t$$ Justify that we thus define a function on $\mathbb{R}^{+*}$.

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 $\mathbb{R}^{+*}$.

We define, for all $x \in \mathbb{R}^{+*}$, $$\Gamma(x) = \int_0^{+\infty} t^{x-1} \mathrm{e}^{-t} \, \mathrm{d}t$$ Show that, for all $x \in \mathbb{R}^{+*}$, $$\Gamma(x+1) = x\Gamma(x)$$