grandes-ecoles 2021 Q26

grandes-ecoles · France · centrale-maths2__pc Sequences and Series Inner Product Spaces, Orthogonality, and Hilbert Space Structure on Sequence/Function Spaces

In this subsection, $I = ]{-1,1}[$ and $w(x) = \frac{1}{\sqrt{1-x^2}}$. For every integer $n \in \mathbb{N}$, let $Q_n$ denote the polynomial of $\mathbb{R}[X]$ that coincides with $x \mapsto \cos(n \arccos(x))$ on $[-1,1]$. Let $(p_n)_{n \in \mathbb{N}}$ be the sequence of orthogonal polynomials associated with the weight $w$.
Show that $$\begin{cases} p_0 = Q_0 \\ \forall n \in \mathbb{N}^*, \quad p_n = \dfrac{1}{2^{n-1}} Q_n \end{cases}$$

In this subsection, $I = ]{-1,1}[$ and $w(x) = \frac{1}{\sqrt{1-x^2}}$. For every integer $n \in \mathbb{N}$, let $Q_n$ denote the polynomial of $\mathbb{R}[X]$ that coincides with $x \mapsto \cos(n \arccos(x))$ on $[-1,1]$. Let $(p_n)_{n \in \mathbb{N}}$ be the sequence of orthogonal polynomials associated with the weight $w$.

Show that
$$\begin{cases} p_0 = Q_0 \\ \forall n \in \mathbb{N}^*, \quad p_n = \dfrac{1}{2^{n-1}} Q_n \end{cases}$$

Paper Questions

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