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\(\parallel R_n \parallel = \sum\limits_{k = 0}^\infty {[R_n (q_{2k} )/r^{2k} ]} \), q2k (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 P2n-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.
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Akrivis, G. Fehlerabschätzungen für Gauß-Quadraturformeln. Numer. Math. 44, 261–278 (1984). https://doi.org/10.1007/BF01410110
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DOI: https://doi.org/10.1007/BF01410110