Derivative-free error bounds in numerical integration of analytic functions.
(Ableitungsfreie Fehlerabschätzungen bei numerischer Integration holomorpher Funktionen.)

*(German)*Zbl 0809.41029
Karlsruhe: Univ. Karlsruhe, Fak. f. Mathematik, 123 S. (1994).

The thesis deals with remainder terms of quadratures rules with respect to nonnegative weight functions \(w\) over the interval \([-1,1]\). For integrands \(f\) that can be extended analytically in a domain \(G(\supset [-1,1])\) it is well-known that the remainder term can be expressed as a contour integral with a kernel \(K_ n\). This leads to derivative-free error bounds. The knowledge of the maximum modulus of the kernel \(K_ n\) on the contour is essential to obtain sharp error bounds. These maxima were first studied for Gauss quadrature rules by W. Gautschi and R. S. Varga [SIAM J. Numer. Anal. 20, 1170-1186 (1983; Zbl 0545.41040)]. They determined the maxima on circular contours \({\mathcal C}_ r\) for a large class of weight functions and considered elliptical contours \({\mathcal E}_ \rho\) (with foci \(\pm 1\) and semiaxis \({1 \over 2} (\rho \pm \rho^{-1}), \rho > 1)\) for the four Chebyshev weights. Some of these results were later enlarged to Gauss-Lobatto and Gauss-Radau quadrature rules (see e.g. W. Gautschi’s survey [NATO Adv. Res. Workshop, Bergen/Norw. 1991, NATO ASI Ser., C 357, 133-145 (1992; Zbl 0754.41026)]).

In this thesis the discussion of the kernel \(K_ n\) is extended for quadrature rules with respect to more general weight functions that have not yet been considered in the literature. For the Gauss-Lobatto quadrature rule with respect to \(w{(1/2)} (x) : = (1 - x^ 2)^{1/2}\) a conjecture of W. Gautschi [Rocky Mt. J. Math. 21, 209-226 (1991; Zbl 0749.41026)] concerning the maxima of \(K_ n\) on \({\mathcal E}_ \rho\) for \(\rho \geq \rho_{n,L}\) is proved. For Gauss quadrature rules with respect to symmetric weight functions \(w\) a new series representation of the kernel \(K_ n\) is developed. This is essential for the determination of the maximum of \(K_ n\) on ellipses \({\mathcal E}_ \rho\) with \(\rho \geq \rho_{n,G}\) for a large class of symmetric weight functions, including the Gegenbauer weights \(w^{(\alpha)} (x) : = (1 - x^ 2)^ \alpha\) for \(| \alpha | \geq {1 \over 2}\). The remaining cases \(| \alpha | < {1 \over 2}\) cannot be handled in this way, but using a well-known expansion of the kernel \(K_ n\) similar results are obtained. For the quadrature rules of Pólya-type (i.e. Clenshaw-Curtis, Pólya and Filippi rule) special expansions of the corresponding kernel \(K_ n\) are used in order to determine the maximum of the kernel \(K_ n\) on ellipses \({\mathcal E}_ \rho\), \(\rho \geq \rho_{n,X}\). All the parameters \(\rho_{n,L}\), \(\rho_{n,G}\) and \(\rho_{n,X}\) are explicitly given. The knowledge of the maxima of \(K_ n\) leads to practical derivative- free error bounds for the mentioned quadrature rules. Additionally, another type of useful derivative-free error bounds is presented, based on the estimation of the kernel \(K_ n\) with the BOGS-method [see e.g. W. Haußmann, E. Luik, and K. Zeller [Multivariate approximation theory II, Proc. Conf., Oberwolfach 1982 ISNM 61, 191-200 (1982; Zbl 0508.41027)]. Several numerical examples show that both methods lead to similar error bounds. Finally, the bivariate case is treated. Considering several assumptions on the region of analyticity of the bivariate integrand \(f\) derivative-free error bounds for tensor product and blending cubature formulae are derived.

In this thesis the discussion of the kernel \(K_ n\) is extended for quadrature rules with respect to more general weight functions that have not yet been considered in the literature. For the Gauss-Lobatto quadrature rule with respect to \(w{(1/2)} (x) : = (1 - x^ 2)^{1/2}\) a conjecture of W. Gautschi [Rocky Mt. J. Math. 21, 209-226 (1991; Zbl 0749.41026)] concerning the maxima of \(K_ n\) on \({\mathcal E}_ \rho\) for \(\rho \geq \rho_{n,L}\) is proved. For Gauss quadrature rules with respect to symmetric weight functions \(w\) a new series representation of the kernel \(K_ n\) is developed. This is essential for the determination of the maximum of \(K_ n\) on ellipses \({\mathcal E}_ \rho\) with \(\rho \geq \rho_{n,G}\) for a large class of symmetric weight functions, including the Gegenbauer weights \(w^{(\alpha)} (x) : = (1 - x^ 2)^ \alpha\) for \(| \alpha | \geq {1 \over 2}\). The remaining cases \(| \alpha | < {1 \over 2}\) cannot be handled in this way, but using a well-known expansion of the kernel \(K_ n\) similar results are obtained. For the quadrature rules of Pólya-type (i.e. Clenshaw-Curtis, Pólya and Filippi rule) special expansions of the corresponding kernel \(K_ n\) are used in order to determine the maximum of the kernel \(K_ n\) on ellipses \({\mathcal E}_ \rho\), \(\rho \geq \rho_{n,X}\). All the parameters \(\rho_{n,L}\), \(\rho_{n,G}\) and \(\rho_{n,X}\) are explicitly given. The knowledge of the maxima of \(K_ n\) leads to practical derivative- free error bounds for the mentioned quadrature rules. Additionally, another type of useful derivative-free error bounds is presented, based on the estimation of the kernel \(K_ n\) with the BOGS-method [see e.g. W. Haußmann, E. Luik, and K. Zeller [Multivariate approximation theory II, Proc. Conf., Oberwolfach 1982 ISNM 61, 191-200 (1982; Zbl 0508.41027)]. Several numerical examples show that both methods lead to similar error bounds. Finally, the bivariate case is treated. Considering several assumptions on the region of analyticity of the bivariate integrand \(f\) derivative-free error bounds for tensor product and blending cubature formulae are derived.

Reviewer: Thomas Schira