The minimal number of nodes in Chebyshev type quadrature formulas
The authors abstract is accurate: We study Chebyshev type quadrature formulas of degree \(n\) with respect to a weight function on \(\langle - 1,+1 \rangle\), i.e. formulas \[ {1 \over {\int_{-1}^{+1} w(t)dt}}\cdot \int_{-1}^{+1} f(t) w(t) dt={1\over N} \sum_{i=1}^ N f(x_ i)+ R(f) \] with nodes \(x_ i\in \langle -1,+1\rangle\) such that \(R(f)=0\) for every polynomial of degree \(\leq n\). It is known that for a Jacobi weight function \(w(t)= (1-t)^ \alpha (1+t)^ \beta\) the number of nodes has to satisfy the inequality \(N\geq K_ 1 n^{2+2\max(\alpha, \beta)}\) for some absolute constant \(K_ 1>0\). In this paper it is shown that for an ultraspherical weight function \(w(t)= (1-t^ 2)^ \alpha\) with \(\alpha\geq 0\), this lower bound is of the right order i.e. there exists a Chebyshev type quadrature formula of degree \(n\) with \(N\leq K_ 2 n^{2+2\alpha}\) nodes. Our method of proof is based on a method of S. N. Bernstein who obtained the result in case \(\alpha=0\). In general this method gives a large number of multiple nodes. It is also shown that the nodes can be chosen to be distinct.
scientific article; zbMATH DE number 510031