Inversely symmetric interpolatory quadrature rules (Q1591292)

Let \(d\phi (x)\) a symmetric distribution defined on \([-d,d]\), with \(d>0\). Further, let \(d\psi (t)\) be a strong weight distribution defined on \((\beta^2/b,b)\), where \(0< \beta< b\leq \infty\), and such that \[ d\psi (t)/\sqrt t=-d\psi (\beta ^2/t)/\sqrt {\beta ^2/t}. \] The authors consider the transformation \(x(t)=1/(2\sqrt \alpha)(\sqrt t -\beta /\sqrt t)\), with \(\alpha >0\), \(\beta >0\) and investigate the relations between the classical symmetric quadrature rule \[ \int _{-d}^d f(x)\,d\phi (x)=\sum _{j=1}^n \omega _{n,j}f(x_{n,j})+E_n(f), \] and the inverse symmetric quadrature rule \[ \int _{\beta ^2/b}^b F(t)\,d\psi (t)=\sum _{j=1}^n \nu _{n,j}F(t_{n,j})+\tilde E_n(F), \] where the distribution \(d\phi (t)\) and \(d\psi (t)\) are such that \[ d\psi (t)=C\frac t {t+\beta }d \phi(x(t)), \quad C>0. \] Interesting properties of the nodes and weights are obtained. Specific examples of rules with fixed nodes are also given.