On the universality of the Gaussian quadrature formula (Q2785846)

Let \(Q_n[f]=\sum_{\nu=1}^n a_\nu f(x_\nu)\) be a quadrature formula (q.f.) for \(I[f]=\int_{-1}^1 f(x)dx\) with error \(R_n[f]=I[f]-Q_n[f]\). Set \(\rho(Q_n,{\mathcal K})=\sup_{f\in{\mathcal K}}|R_n[f]|\), then \(\rho_n({\mathcal K})=\inf_{Q_n}\rho(Q_n,{\mathcal K})\) is the error of an optimal q.f.\ for the class \({\mathcal K}\). Here optimal is among all \(n\) point q.f. If a q.f.\ \(Q_n\) is used instead of the optimal q.f., one looses a factor \(\text{loss}(Q_n,{\mathcal K})=\rho(Q_n,{\mathcal K})/\rho_n({\mathcal K})\) in accuracy. A Gaussian q.f.\ \(Q_n^G\) is said to be universal (for \({\mathcal K}_s\)) if \(M(n)=\sup_{1\leq s\leq 2n}\text{loss}(Q_n^G,{\mathcal K}_s)\) is ``small''. The index \(s\) refers to the smoothness of the functions \(f\). In this paper NEWLINE\[NEWLINE{\mathcal K}_s=\{f\in C^s[-1,1]: \int_{-1}^1[f^{(s)}(x)]^2(1-x^2)^{s-1/2}dx<1\}NEWLINE\]NEWLINE. This notion of universal q.f.\ is an adaptation of a more general approach given by \textit{H. Brass}, ISNM 85, 16-24 (1988; Zbl 0648.41015)]. In this paper it is shown that \(M(n)\) as defined above is bounded by \((22/3)n\sqrt{n+1}\).