On generalized averaged Gaussian formulas. II

Source abstract

Recently, by following the results on characterization of positive quadrature formulae by Peherstorfer, we proposed a new ( 2 ℓ + 1 ) (2\ell +1) -point quadrature rule G ^ 2 ℓ + 1 \widehat G_{2\ell +1} , referred to as a generalized averaged Gaussian quadrature rule. This rule has 2 ℓ + 1 2\ell +1 nodes and the nodes of the corresponding Gauss rule G ℓ G_\ell with ℓ \ell nodes form a subset. This is similar to the situation for the ( 2 ℓ + 1 ) (2\ell +1) -point Gauss-Kronrod rule H 2 ℓ + 1 H_{2\ell +1} associated with G ℓ G_\ell . An attractive feature of G ^ 2 ℓ + 1 \widehat G_{2\ell +1} is that it exists also when H 2 ℓ + 1 H_{2\ell +1} does not. The numerical construction, on the basis of recently proposed effective numerical procedures, of G ^ 2 ℓ + 1 \widehat G_{2\ell +1} is simpler than the construction of H 2 ℓ + 1 H_{2\ell +1} . A disadvantage might be that the algebraic degree of precision of G ^ 2 ℓ + 1 \widehat G_{2\ell +1} is 2 ℓ + 2 2\ell +2 , while the one of H 2 ℓ + 1 H_{2\ell +1} is 3 ℓ + 1 3\ell +1 . Consider a (nonnegative) measure d σ d\sigma with support in the bounded interval [ a , b ] [a,b] such that the respective orthogonal polynomials, above a specific index r r , satisfy a three-term recurrence relation with constant coefficients. For ℓ ≥ 2 r − 1 \ell \ge 2r-1 , we show that G ^ 2 ℓ + 1 \widehat G_{2\ell +1} has algebraic degree of precision at least 3 ℓ + 1 3\ell +1 , and therefore it is in fact H 2 ℓ + 1 H_{2\ell +1} associated with G ℓ G_\ell . We derive some interesting equalities for the corresponding orthogonal polynomials.