On Birkhoff quadrature formulas. II (Q5906439)

[For part I see Proc. Am. Math. Soc. 97, 38-40 (1986; Zbl 0599.41049).] In 1974 P. Turán raised 89 problems on approximation theory. Some of them are on Birkhoff interpolation. Here we are concerned with the problem related to Birkhoff quadrature theory. Problem XXXII [\textit{P. Turán}, J. Approximation Theory 29, 23-85 (1980; Zbl 0454.41001)]. Determine the matrices \(A\), if any, for which \[ \int^ 1_{- 1}f(x)dx=\sum^ n_{k=1}f(x_{kn})\lambda^{(0)}_{kn}+\sum^ n_{k=1} f''(x_{kn})\lambda^{(1)}_{kn} \] is valid for all polynomials of degree \(\leq 2n\). We may also ask the following analogous problem: Determine the matrices \(A\), if any, for which \[ \int^{2\pi}_ 0f(\theta)d\theta=\sum^{n-1}_{k=0}f(\theta_{kn}) \lambda^{(0)}_{kn}+\sum^{n- 1}_{k=0}f^{(M)}(\theta_{kn})\lambda^{(1)}_{kn} \] is valid for all trigonometric polynomials of highest possible degree. The object of this paper is to provide the solution of this latter problem.