On the Chebyshev problem for quadrature formulas with derivatives in the integral (Q1100374)

The problem to construct to a given weight function W(x)a quadrature formula of interpolation type with equal coefficients (a so-called Chebyshev quadrature formula) is generalized to such quadrature formulae in which derivatives of the function at the knots occur: Under symmetry assumptions one is looking for knots \(x_ j\) and for coefficients \(A_ h\) such that the quadrature formula for the integral \(\int^{+1}_{- 1}W(x)f(x)dx\) looks like \(\sum^{r}_{h=0}A_ h\sum^{n}_{j=1}f^{(h)}(x_ j),\) the error being zero for all polynomials of degree at most \(r+n+1\). The authors give a necessary and sufficient condition of algebraic type for the existence of such formulae. They investigate the construction, the behavior as well as other properties of corresponding Peano kernels in order to get error estimations of the remainder. A number of special cases is discussed in some detail. At the end of the paper the authors give for the weight \(W\equiv 1\) theoretical results for \(r=1\) and \(r=2\). Furthermore tables of the corresponding values of the knots and the coefficients are presented.