Algebraic Polynomial Integration

The integration formulas are obtained by replacing the integral I F = ∫ a b F ( x ) d x , F ∈ C ( 0 ) [ a , b ] , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315195674/823ed023-dd6c-41bb-a456-22174b64a780/content/math1_1.tif"/> 1.1 by an expression ∫ a b P n F ( x ) d x , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315195674/823ed023-dd6c-41bb-a456-22174b64a780/content/math1_2.tif"/> 1.2 where P^F _n is the uniquely determined interpolating polynomial of degree n at the nodal system { x k } k = 0 , … , n ⊂ [ a , b ] https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315195674/823ed023-dd6c-41bb-a456-22174b64a780/content/imath1_1.tif"/> . Well-known from classical textbooks (see, e.g., P.J. Davis, P. Rabinowitz [1967], V.I. Krylov [1962], J. Stoer [1989], J. Werner [1991]) is the following theorem.