Error estimates of Gaussian-type quadrature formulae for analytic functions on ellipses-a survey of recent results

Abstract

This paper presents a survey of recent results on error estimates of Gaussian-type quadrature formulas for analytic functions on confocal ellipses.

A note on “Error bounds of Gaussian quadrature formulae with Legendre weight function for analytic integrands” by M. M. Spalević et al.

“…where l(Γ) denotes the length of the contour Γ (see [2]). A common choice for the contour Γ is one of the confocal ellipses with foci at the points ∓1, also known as Bernstein ellipses, and the sum of semi-axes ρ > 1,…”

“…A recent survey of error bounds (2.4) as well as of related bound, when Γ = E ρ , for Gaussian type q.f. can be found in [2].…”

On the Gauss-Kronrod quadrature formula for a modified weight function of Chebyshev type

“…Although statements of this kind are clearly false in general, in our cases they are justified by a simple result which was shown in the recent survey paper (cf. [5,Theorem 4.1]). To make the paper self-contained, the statement of this result is included.…”

“…Indeed, the expression above for f (ρ, θ 0 ) − f (ρ, θ) = P (ρ,θ) R(ρ,θ)R(ρ,θ 0 ) shows that it suffices to prove that P (ρ, θ) is positive for all θ = θ 0 , whenever ρ is large enough. For the complete proof see [5].…”

“…To end this introduction, let us say that the problem of estimating the quadrature error for Gauss-type rules has been thoroughly studied in the literature; see the references [12,18,19,22,23], and [26][27][28][29][30], to only cite a few. See also [5] for a very recent survey of the error estimates of Gaussian type quadrature formulae for analytic functions on ellipses.…”