Chebyshev-type quadrature for analytic weights on the circle and the interval
Let \(\gamma\) be a probability measure on a compact subset \(K\) of the complex plane \(\mathbb{C}\). A Chebyshev-type quadrature formula for \(\gamma\) is a numerical integration of the form \[ \int \Phi (\zeta) d \gamma (\zeta) \approx {1 \over n} \sum^n_{j = 1} \Phi (\zeta_j), \] where the nodes \(\zeta_1, \zeta_2, \dots, \zeta_n\) are points in \(\mathbb{C}\). The algebraic degree of (1) is the maximal number \(p\) such that equality holds for all polynomials \(\Phi\) (in one complex variable) of degree \(\leq p\). We denote by \(N_k (p; \gamma)\) the minimal number \(n\) such that a Chebyshev-type quadrature formula of degree \(\geq p\) exists with all its \(n\) nodes in \(K\). We are interested in the behavior of \(N_K (p; \gamma)\) as \(p \to \infty \). If \(\nu\) is a probability measure on \([- 1,1] = I\) of the form \(d \nu (t) = {w(t) \over \pi \sqrt {1 - t^2}}\), where \(w\) is a positive analytic function on \([- 1,1]\), then the author proves: \[ \max_{t \in I} {1 \over 2 w(t)} \leq \lim_{p \to \infty} {N_I (p; \gamma) \over p} \leq \max_{t \in I} {1 \over w (t)}. \] In addition, some results on Chebyshev quadrature on certain algebraic curves are presented.
scientific article; zbMATH DE number 854421