The error norm of certain Gaussian quadrature formulae

Abstract:

We consider Gauss quadrature formulae ${Q_n}$, $n \in {\mathbf {N}}$, approximating the integral $I(f): = \smallint _{ - 1}^1w(x)f(x)\;dx$, $w = W/{p_i}$, $i = 1,2$, with $W(x) = {(1 - x)^\alpha }{(1 + x)^\beta }$, $\alpha ,\beta = \pm 1/2$ and ${p_1}(x) = 1 + {a^2} + 2ax$, ${p_2}(x) = (2b + 1){x^2} + {b^2}$, $b > 0$. In certain spaces of analytic functions the error functional ${R_n}: = I - {Q_n}$ is continuous. In [1] and [2] estimates for $\left \| {{R_n}} \right \|$ are given for a wide class of weight functions. Here, for a restricted class of weight functions, we calculate the norm of ${R_n}$ explicitly.