Quadratures with multiple nodes, power orthogonality, and moment-preserving spline approximation

Quadrature formulas with multiple nodes, power orthogonality, and some applications of such quadratures to moment-preserving approximation by defective splines are considered. An account on power orthogonality (s- and σ-orthogonal polynomials) and generalized Gaussian quadratures with multiple nodes, including stable algorithms for numerical construction of the corresponding polynomials and Cotes numbers, are given. In particular, the important case of Chebyshev weight is analyzed. Finally, some applications in moment-preserving approximation of functions by defective splines are discussed.