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

Abstract

In paper D. Lj. Đjukić, R. M. Mutavdžić Đjukić, A. V. Pejčev, and M. M. Spalević, Error estimates of Gaussian-type quadrature formulae for analytic functions on ellipses – a survey of recent results, Electron. Trans. Numer. Anal., 53 (2020), pp. 352–382, Lemma 4.1 can be applied to show the asymptotic behaviour of the modulus of the complex kernel in the remainder term of all the quadrature formulas in the recent papers that are concerned with error estimates of Gaussian-type quadrature formulae for analytic functions on ellipses. However, in the paper D. R. Jandrlić, Dj. M. Krtinić, Lj. V. Mihić, A. V. Pejčev, M. M. Spalević, Error bounds of Gaussian quadrature formulae with Legendre weight function for analytic integrands, Electron. Trans. Anal. 55 (2022), pp. 424–437, which this note is concerned with, there is a kernel whose numerator contains an infinite series, and in this case the mentioned lemma cannot be applied. This note shows that the modulus of the latter kernel attains its maximum as conjectured in the latter paper.