grandes-ecoles 2021 Q27

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 $w(x) = \frac{1}{\sqrt{1-x^2}}$. The orthogonal polynomials associated with $w$ satisfy $p_n = \frac{1}{2^{n-1}} Q_n$ for $n \geqslant 1$, where $Q_n(x) = \cos(n \arccos(x))$.
For $n \in \mathbb{N}$, explicitly determine the points $(x_j)_{0 \leqslant j \leqslant n}$ of $I$ such that the quadrature formula $I_n(f) = \sum_{j=0}^n \lambda_j f(x_j)$ has maximal order.

In this subsection, $I = ]{-1,1}[$ and $w(x) = \frac{1}{\sqrt{1-x^2}}$. The orthogonal polynomials associated with $w$ satisfy $p_n = \frac{1}{2^{n-1}} Q_n$ for $n \geqslant 1$, where $Q_n(x) = \cos(n \arccos(x))$.

For $n \in \mathbb{N}$, explicitly determine the points $(x_j)_{0 \leqslant j \leqslant n}$ of $I$ such that the quadrature formula $I_n(f) = \sum_{j=0}^n \lambda_j f(x_j)$ has maximal order.

Paper Questions

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