On a new type of quadrature formulas

Summary

In the paper quadrature formulas of the form$\int\limits_a^b f (x)dx = \sum\nolimits_{k = 1}^n {w_k I(a_k ,b_k ;f) + } E(f)$ are considered. HereI(a _k ,b _k ;f) is the average of the functionf over the interval [a _k ,b _k ],a _k ≦b _k . It is proved that for a set of anyn non-overlapping intervals there exists a quadrature formula of the above type exact at least for polynomials of degreen−1. The proof of the theorem uses a generalization of the notion of interpolation polynomial.