On generalized averaged gaussian formulas. Ii

Abstract

Recently, by following the results on characterization of positive quadrature formulae by Peherstorfer, we proposed a new (2l + 1)-point quadrature rule (G) over cap (2l + 1), referred to as a generalized averaged Gaussian quadrature rule. This rule has 2l + 1 nodes and the nodes of the corresponding Gauss rule G(l) with l nodes form a subset. This is similar to the situation for the (2l + 1)-point Gauss-Kronrod rule H2l + 1 associated with G(l). An attractive feature of (G) over cap (2l + 1) is that it exists also when H2l + 1 does not. The numerical construction, on the basis of recently proposed effective numerical procedures, of (G) over cap (2l + 1) is simpler than the construction of H2l + 1. A disadvantage might be that the algebraic degree of precision of (G) over cap (2l + 1) is 2l + 2, while the one of H2l + 1 is 3l + 1. Consider a (nonnegative) measure ds with support in the bounded interval [a, b] such that the respective orthogonal polynomials, above a specific index r, satisfy a three-term recurrence relation with constant coefficients. For l >= 2r -1, we show that (G) over cap (2l + 1) has algebraic degree of precision at least 3l + 1, and therefore it is in fact H2l + 1 associated with G(l). We derive some interesting equalities for the corresponding orthogonal polynomials.