Quadratures de Txebixov a l’interval i Teorema de Bernstein

Abstract

[en] In this work we will prove a theorem that Bernstein proved in 1937. This theorem states that there are no quadrature formulas with equal weights (of Chebyshev) in the interval $[-1,1]$ $$ \int_{-1}^{1} f(x) d x \approx \frac{2}{n} \sum_{k=1}^{n} f\left(x_{k}\right) $$ that are true for polynomials $f$ of degree $\leq n$, with nodes $x_{k} \in[-1,1]$, if $n \geq 10$. We will also see some results related to the distribution of these nodes when $n$ is large.