Weighted nonstandard quadrature formulas based on values of linear differential operators

Beside an account on Gaussian quadrature formulas and several interesting generalizations and modifications, this paper is mainly devoted to weighted nonstandard quadrature formulas of Gaussian type based on values of certain linear differential operators at some nodes. In addition to their theoretical significance, such kinds of quadrature formulas can be interesting in applications when the operator values are available, instead of the values of the original integrand function. Several numerical examples, including the construction of such non-standard quadratures, are also presented.