On the weights of Sard's quadrature formulas (Q1263766)

The weights of quadrature formulas are studied which are optimal in the sense of Sard. The nodes of such quadrature formulas have to fulfill certain restrictions. This conditions are not so restrictive (e.g. the nodes of the Gauss-Legendre formulas are possible). The main results obtained are an estimation of the size of the weights and a generalization of the first and second conjecture of \textit{L. F. Meyers} and \textit{A. Sard} [J. Math. Phys. Massachusetts 29, 118-123 (1950; Zbl 0039.342)] for the case of weighted integrals and for non-equidistant nodes. From the generalized first conjecture follows that the Sard's quadrature formulas satisfy the trapezoid and circle theorem [see \textit{P. J. Davis} and \textit{P. Rabinowitz}, J. Math. Anal. Appl. 2, 428-437 (1961; Zbl 0168.147)].