Maximum of the modulus of kernels of Gaussian quadrature formulae for one class of Bernstein–Szegő weight functions

Abstract

We continue with the study of the kernels K n ( z) in the remainder terms R n ( f) of the Gaussian quadrature formulae for analytic functions f inside elliptical contours with foci at ∓1 and a sum of semi-axes ρ > 1. The weight function w of Bernstein–Szegő type here is w ( t ) ≡ w γ ( - 1 / 2 ) ( t ) = 1 1 - t 2 · 1 - 4 γ ( 1 + γ ) 2 t 2 - 1 , t ∈ ( - 1 , 1 ) , γ ∈ ( - 1 , 0 ) . Sufficient conditions are found ensuring that the kernel attains its maximum absolute value at the intersection point of the contour with either the real or the imaginary axis. This leads to effective error bounds of the corresponding Gauss quadratures. The quality of the derived bounds is demonstrated by a comparison with other error bounds intended for the same class of integrands.