On the remainder of Gaussian quadrature formulas for Bernstein-Szegő weight functions

Abstract:

We give an explicit expression for the kernel of the error functional for Gaussian quadrature formulas with respect to weight functions of Bernstein-Szegö type, i.e., weight functions of the form ${(1 - x)^\alpha }{(1 + x)^\beta }/\rho (x),\quad x \in ( - 1,1)$, where $\alpha ,\beta \in \{ - \tfrac {1}{2},\tfrac {1}{2}\}$ and $\rho$ is a polynomial of arbitrary degree which is positive on $[ - 1,1]$. With the help of this result the norm of the error functional can easily be calculated explicitly for a wide subclass of these weight functions.