Quadrature formulae connected to σ-orthogonal polynomials

Let dλ(t) be a given nonnegative measure on the real line R, with compact or infinite support, for which all moments μk=∫Rtk dλ(t), k=0,1,…, exist and are finite, and μ0>0. Quadrature formulas of Chakalov–Popoviciu type with multiple nodes∫Rf(t) dλ(t)=∑ν=1n ∑i=02sν Ai,νf(i)(τν)+R(f),where σ=σn=(s1,s2,…,sn) is a given sequence of nonnegative integers, are considered. A such quadrature formula has maximum degree of exactness dmax=2∑ν=1nsν+2n−1 if and only if∫R ∏ν=1n (t−τν)2sν+1tk dλ(t)=0, k=0,1,…,n−1.The proof of the uniqueness of the extremal nodes τ1,τ2,…,τn was given first by Ghizzetti and Ossicini (Rend. Mat. 6(8) (1975) 1–15). Here, an alternative simple proof of the existence and the uniqueness of such quadrature formulas is presented. In a study of the error term R(f), an influence function is introduced, its relevant properties are investigated, and in certain classes of functions the error estimate is given. A numerically stable iterative procedure, with quadratic convergence, for determining the nodes τν, ν=1,2,…,n, which are the zeros of the corresponding σ-orthogonal polynomial, is presented. Finally, in order to show a numerical efficiency of the proposed procedure, a few numerical examples are included.