Fehlerabschätzungen für Gauß-Quadraturformeln

Summary

We consider Gauss quadrature formulaeQ _n ,n∈ℤ, approximating the integral$I(f): = \int\limits_{ - 1}^1 {w(x)f(x)dx}$,w an even weight function. Let$f(z) = \sum\limits_{k = 0}^\infty {\alpha _k^f z^k }$ be analytic inK _r :={z∈ℂ:|z|<r},r>1, and$|f|_r : = \mathop {sup}\limits_{k \geqq n} \{ |\alpha _{2k}^f |r^{2k} \}< \infty$. The error functionalR _n :=I-Q _n is continuous with respect to |·|_r and the relation$\parallel R_n \parallel = \sum\limits_{k = 0}^\infty {[R_n (q_{2k} )/r^{2k} ]}$, q_2k (x):=x ^2k holds.

In this paper estimates for ∥R _n ∥ are given. To this end we first derive two new representations of ∥R _n ∥ which are essential for our further investigations. The ∥R _n ∥=r ² R _n (Φ), with Φ(x):=1/(r ²-x ²), is estimated in various ways by using the best uniform approximation of Φ in P_2n-1, and also the expansion of Φ with respect to Chebyshe polynomials of the first and second kind. Forw(x)=(1-x ²)^α, α=±1/2, ∥R _n ∥ is calculated. The asymptotic behaviour, forr→1+, of ∥R _n ∥ and of the derived error bounds is also discussed. Finally, we compare different error bounds and give numerical examples.