Positive interpolatory quadrature rules generated by some biorthogonal polynomials

Abstract

Interpolatory quadrature rules whose abscissas are zeros of a biorthogonal polynomial have proved to be useful, especially in numerical integration of singular integrands. However, the positivity of their weights has remained an open question, in some cases, since 1980. We present a general criterion for this positivity. As a consequence, we establish positivity of the weights in a quadrature rule introduced by the second author in 1980, generated by a polynomial that is biorthogonal to ( log ⁡ x ) j \left ( \log x\right ) ^{j} , 0 ≤ j ≤ n − 1 0\leq j\leq n-1 .