Lobatto-type quadrature formula in rational spaces
Let \({\mathcal P}_m\) be the set of all real algebraic polynomials of degree at most \(m\) and \[ {\mathcal P}_m(a_1, a_2,\dots, a_n)= \left\{{P(x) \over \prod^n_{k=1} | x-a_k |^2}, P\in{\mathcal P}_m \right\}, \] where the nonreal elements in \(\{a_k\}^n_{k=1} \subset\mathbb{C} \setminus[-1,1]\) are paired by complex conjugation. The author constructs the Lobatto-type quadrature formula \[ \int^1_{-1} {f(x)\over \sqrt{1-x^2}} dx\approx Q_n(f): =\lambda_0 f(1)+ \sum^{n-1}_{k=1} \lambda_k f(x_k)+ \lambda_n f(-1), \] based on the \(n+1\) Chebyshev points \(-1=x_n< x_{n-1}< \cdots<x_1 <x_0=1\) such that \(T_n(x_k) =(-1)^k\), \(k=0,1, \dots,n\). He proves that it is a positive quadrature formula and it is exact for any element in \({\mathcal P}_{2n-1} (a_1,a_2, \dots,a_n)\). Moreover, a convergence result is given for \(f\in C[-1,1]\).
scientific article; zbMATH DE number 1326393
scientific article