Error bounds of Gaussian quadrature formulae for one class of Bernstein-Szegő weights

Spalević, Miodrag

doi:10.1090/S0025-5718-2012-02667-6

Error bounds of Gaussian quadrature formulae for one class of Bernstein-Szegő weights
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by Miodrag M. Spalević PDF

Math. Comp. 82 (2013), 1037-1056 Request permission

Abstract:

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 $\mp 1$ and a sum of semi-axes $\rho >1$, when the weight function $w$ is of Bernstein-Szegő type \[ w(t)\equiv w_\gamma ^{(-1/2,1/2)}(t)=\displaystyle \sqrt {\frac {1+t}{1-t}}\cdot \frac {1}{1-\displaystyle \frac {4\gamma }{(1+\gamma )^2} t^2}, \quad t\in (-1,1),\quad \gamma \in (-1,0), \] are studied. Sufficient conditions are found ensuring that the kernel attains its maximal absolute value at the intersection point of the contour with the positive real semi-axis. This leads to effective error bounds of the corresponding Gauss quadratures. The quality of the derived bounds is analyzed by a comparison with other error bounds intended for the same class of integrands. In part our analysis is based on the well-known Cardano formulas, which are not very popular among mathematicians.

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